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A transitive <permutation> <group> is primitive when it preserves no nontrivial partition into blocks. A <point stabilizer> is maximal precisely when the action is primitive. The orbits of any <normal subgroup> form a <block system>, so every nontrivial <normal subgroup> of a faithful primitive <group> is transitive. This turns the classification of primitive <groups> into a study of their <minimal normal subgroups> and their <group socle>.
The <socle of a finite group> is generated by its minimal nontrivial <normal subgroups>. A <minimal normal subgroup> is characteristically simple, hence a direct power of a <simple group>. Distinct <minimal normal subgroups> commute. If a transitive <normal subgroup> is abelian it is regular: its <point stabilizer>, being the same at all points, lies in the <group homomorphism kernel>. In that case it is elementary abelian, and the ambient <group> is affine. If two <minimal normal subgroups> occur, each centralizes the other; the <centralizer> of a transitive <permutation> <group> is semiregular, so both are regular and the two regular <groups> are mutual left/right partners. These observations are central to the <O'Nan–Scott theorem>.
The theorem organizes finite primitive <groups> into the following socle/action types. In the affine type HA, the <group socle> is $C_p^d$, degree $p^d$, and $G=C_p^d\rtimes H$ with $H\le GL_d(p)$ irreducible. In almost-simple type AS the <group socle> is one nonabelian <simple group> $T$ and $T\le G\le\operatorname{Aut}(T)$; a <maximal subgroup> not containing $T$ specifies the primitive coset action.
In simple-diagonal type SD the <group socle> is $T^r$, $r\ge2$, and its intersection with a <point stabilizer> is a full diagonal copy of $T$. The degree is $|T|^{r-1}$, and the action includes the appropriate <permutation> of the simple factors. Compound-diagonal type CD replaces this one diagonal by a product of diagonals in disjoint collections of factors; it is a product version of diagonal type. In product-action type PA one embeds in a <wreath product> with a primitive nonregular almost-simple base, acting on a Cartesian power $\Delta^r$, so the degree is $|\Delta|^r$ and the <group socle> is a corresponding power of $T$.
Types HS and HC have two regular <minimal normal subgroups>, each isomorphic respectively to $T$ or to $T^r$ with $r>1$. Their socle consists of the left and right regular <groups>; the degrees are $|T|$ and $|T|^r$. Finally TW, twisted-wreath type, has a unique regular nonabelian <minimal normal subgroup> $T^r$ and a <point stabilizer> acting through a twisted <permutation> of its factors. This has degree $|T|^r$; primitivity imposes restrictive conditions, in particular $r\ge6$. These eight modern labels refine the older affine/almost-simple/diagonal/product description. The theorem reduces the problem to <simple groups>, their <maximal subgroups>, irreducible linear actions and the specified product constructions; it does not make every primitive <group> maximal in a <symmetric group>.
For <maximal subgroups> of $S_n$ there are first two elementary families. An intransitive <maximal subgroup> is the stabilizer $S_a\times S_{n-a}$ of an unordered distinguished subset, with $1\le a<n/2$; replacing the subset by its complement gives the same class. When $a=n/2$, this <subgroup> sits properly inside the stabilizer of the two equal blocks. A transitive imprimitive <maximal subgroup> is a full block-system stabilizer $S_a\wr S_b$, with $ab=n$, $a,b>1$, in its imprimitive action.
The <alternating group> $A_n$ is itself maximal. A remaining primitive maximum must contain odd <permutations>. The primitive part of O'Nan–Scott places it among full affine, diagonal, product-action, or almost-simple overgroups. Full affine <groups> have degree $p^d$; full diagonal <groups> have degree $|T|^{r-1}$ and socle $T^r$; full product-action wreath <groups> have degree $d^r$. Almost-simple cases require the actual simple-group <permutation> degree and its full <permutation> <normalizer>. Regular nonabelian types are contained in the corresponding holomorph/diagonal or product overgroups, so do not supply extra maximal families. Parity and exceptional containments must be checked before calling any of these candidates maximal.
In degree sixteen the complete list, up to $S_{16}$-conjugacy, is
$$
\boxed{A_{16};\quad S_a\times S_{16-a}\ (1\le a\le7);\quad S_2\wr S_8,\ S_4\wr S_4,\ S_8\wr S_2.}
$$
The three <wreath products> here act on blocks, not Cartesian coordinates. The other proper primitive <groups> of this degree are affine and lie in $AGL_4(2)$. A nonzero translation of its sixteen points is eight <transpositions>, and a generating linear <transvection> is four <transpositions>, so the full affine <group> lies in $A_{16}$. Even the tempting product-action $S_4\wr S_2$ on $4^2$ points is contained in this affine <group>: identify each four-point coordinate with $\mathbb F_2^2$. Alternatively, each base <transposition> repeats four times and coordinate exchange makes six <transpositions>, both even. Thus none is an additional maximum of $S_{16}$. There are \b[eleven conjugacy classes] in the displayed list.
In degree sixty the complete list is
$$
\boxed{\begin{gathered}
A_{60};\qquad S_a\times S_{60-a}\quad(1\le a\le29);\\
S_a\wr S_{60/a}\quad(a=2,3,4,5,6,10,12,15,20,30);\\
PGL_2(59)\text{ on }\mathbb P^1(\mathbb F_{59});\qquad D(A_5,2)\text{ on }A_5.
\end{gathered}}
$$
The notation $D(A_5,2)$ denotes the full diagonal <normalizer>, not just $A_5\times A_5$. It is generated on the set $A_5$ by $x\mapsto axb^{-1}$, an <outer automorphism> induced by odd <conjugation> in $S_5$, and $x\mapsto x^{-1}$. Its order is
$$
|D(A_5,2)|=60^2\cdot2\cdot2=14400.
$$
Its two alternating factors form the <group socle>. The <point stabilizer> intersects them in the diagonal $A_5$, of index sixty. Odd <conjugation> by a <transposition> of $S_5$ fixes its six commuting elements in $A_5$, hence makes $(60-6)/2=27$ <transpositions> and is odd. Inversion fixes the identity and the fifteen <involutions>, so makes twenty-two <transpositions> and is even. This verifies that the full diagonal <group> is not lost inside $A_{60}$.
The almost-simple maximum $PGL_2(59)$ has order $59(59^2-1)=205320$ and its usual two-transitive projective-line action. Multiplication by a primitive element of $\mathbb F_{59}^*$ makes one 58-cycle and fixes zero and infinity, so it too contains odd <permutations>. The simple <subgroup> $PSL_2(59)$ alone is not maximal. Degree sixty is neither a prime power nor a proper power, excluding affine and product/compound types; its diagonal possibility is $T=A_5,r=2$. The almost-simple primitive possibilities, besides the natural alternating/<symmetric groups>, have socle $PSL_2(59)$. Their full <normalizer> is the displayed $PGL_2(59)$. These degree-specific classification facts supply the exhaustive primitive part of the list. Altogether there are \b[forty-two conjugacy classes of <maximal subgroups> of $S_{60}$]: twenty-nine intransitive, ten imprimitive, the <alternating group>, and two further primitive classes.
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