Subgroups of $S_{n}$ of index $n$

Question

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups lying inside of $S_{n}$. Are there any more subgroups of index $n$, or do these characterize all subgroups of $S_{n}$ of index $n$?