Normal subgroups and quotients of the symmetric group on three letters

The conjugacy classes of the symmetric group on three letters have sizes $1,3,2$: the identity, the three transpositions and the two 3-cycles. A normal subgroup is a union of whole conjugacy classes and contains the identity. Its candidate orders are therefore $1,4,3,6$. Lagrange's theorem excludes four, leaving precisely $\{e\}$, $A_3$ and $S_3$. The order-three subgroup is the cyclic group of the two 3-cycles and the identity; it is also the kernel of a group homomorphism of the sign map.

The first isomorphism theorem gives the corresponding quotients $S_3$, $C_2$ and the trivial group. Thus these are exactly the possible images of surjective group homomorphisms from $S_3$. Each occurs, by the identity map, the sign map and the constant identity map, respectively.