Suppose $G$ is a group. Let’s call a subgroup $H$ of $G$ antinormal if $N_G(H) = H$, where $N$ stands for normaliser.
Any group is an antinormal subgroup of itself. For Dedekind groups this is the only possible antinormal subgroup.
Also a subgroup is antinormal iff the number of distinct subgroups conjugate with it is equal to its index.
A valid example of a proper antinormal subgroup is given by the "reflection subgroups" of odd dihedral groups $\langle b \rangle < \langle a, b| a^{2n + 1}, b^{-1}aba \rangle$.
However, my question is:
Is there some sort of classification of antinormal subgroups of $S_n$ for arbitrary $n$?
The ones that I see are the whole $S_n$ and its subgroups corresponding to the stabilisers of each point (there are $n$ of them, they are all isomorphic to $S_{n - 1}$ and pairwise conjugate; the index of each of them equals $n$). But maybe there is anything else…