The symmetric groups $S_n$ for $n \leq 4$ are solvable. But for $n >4,$ the only proper non-trivial normal subgroup of $S_n$ is the alternating group $A_n.$ Hence the only factor groups of symmetric groups, other than $S_n$ itself and the trivial group are: the cyclic group of order $2$ (a non-trivial factor group of every $S_n$ with $n >2)$, and the group $S_3$ (which occurs in a non-trivial way as a factor group of $S_4).$ These elementary facts can be found (at least implicitly) in almost any algebra text.