This is an exercise in Dummit and Foote (4.6.3) I'm doing for revision: prove that for $n \geq 5$, $A_n$ is the only proper subgroup of $S_n$ such that $|S_n/G| < n$. ($A_n$ is the alternating group on $n$ elements and $S_n$ is the symmetric group on $n$ elements).
Not sure how to get started - given the condition that $n \geq 5$ it seems that the fact that $A_n$ is simple should be useful.