TITLE: For which $n$ can $S_n$ act transitively on $n+k$ elements?
QUESTION [7 upvotes]: It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$.
Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is fixed? For example, is it known whether there are only finitely many such $n$ for every $k$?

REPLY [9 votes]: Fix $k > 0$.
Suppose that $n > 6$ and $\frac{n(n-3)}{2} > k$. If $[S_n : H] \leq n+k$, then $H$ is one of the following: $S_n$, $A_n$, $S_{n-1}$, or $A_{n-1}$. So in particular if $[S_n : H] = n+k$, then $k = n$ and $H = A_{n-1}$.
See Theorem 5.2B in "Permutation Groups" by Dixon and Mortimer, and also this Math.SE question.