I am dealing with a problem relating to monolithic group.
Let $L$ be a monolithic group with socle $N = S^r$, where S is a nonabelian simple group. Consider the projection $p:N \to S$. A maximal subgroup $H$ of $L$ is of product type if $HN=L$ and $ 1< p( H\cap N ) < S $.
I am considering four simple groups : $PSL_3(2), PSL_4(2), PSL_2(11)$ and $PSp_4(3)$. If $S$ is one of those, then do all $L$ with socle component $S$ have property that all maximal subgroups of $L$ are of product type?
Do we have any classification of L in general? And if they do exist answers, could you please tell me some source to find them.
Thanks a lot in advance.