Solution
= Solution
Let
$$
f=(X^2-2)(X^2-3),
\qquad
g=X^4-10X^2+1.
$$
Both have splitting field $\mathbb Q(\sqrt2,\sqrt3)$. On the roots of $f$, the Galois <group> has two orbits of size two and embeds as
$$
\{1,(12),(34),(12)(34)\}.
$$
The roots of $g$, namely $\pm\sqrt2\pm\sqrt3$, form one orbit and give the regular Klein-four <subgroup>
$$
\{1,(12)(34),(13)(24),(14)(23)\}.
$$
The <subgroups> are not conjugate because their orbit decompositions differ.