I notice the following result in GTM163.
Consider $S_{n}$ act on $\lbrace 1,2,\cdots,n \rbrace$ in a natural way. Then the maximal subgroups $M$ of $S_{n}$ fall into three classes:
$(i)$ (intransitive) $M$ is the set stabilizer of some set of size $m$ with $1\leq m<\frac{n}{2}$ and so is isomorphic to $S_{m}\times S_{n-m}$;
$(ii)$(imprimitive) $M$ is the stabilizer of some partition of $\lbrace 1,2,\cdots, n\rbrace$into m equal parts of size $k$ with $1<m<n$, and so is isomorphic to the wreath product $S_{k}wr S_{m}$ in its imprimitive action;
$(iii)$ (primitive) $M=A_{n}$, or else is a proper primitive group.
And I have seen some examples in $(iii)$. My question is: have all primitive maximal subgroups of $S_{n}$ been found?