Just like the title says, this is probably quite elementary but I couldn't quite get started.
Not "the" but "one" maximal subgroup, since there are others, say the alternating subgroup $\displaystyle A_{n+1}$.
Now, $\displaystyle S_n$ as subset of $\displaystyle S_{n+1}$ can be identified as the set (in fact subgroup) of all permutations which leave one number
out of $\displaystyle \{1,2,...,n\}$ fixed, so if $\displaystyle S_n< N\leq S_{n+1}$ , then N must move all the numbers in $\displaystyle \{1,2,...,n\}$ . Try to use this to show that then $\displaystyle N=S_{n+1}$
Tonio