For a finite group $G$, let $d(G)$ be the largest $k$ such that $G$ admits a subgroup isomorphic to a direct product of $k$ non-trivial groups. I am interested in families $G_n$ of finite simple groups such that $d(G_n)$ goes to infinity.
Examples are $S_n$ or groups of Lie type in dimension $n$. In both cases, any prime power eventually divides the order of these groups.
I was wondering: it possible to have a family of finite simple groups $G_n$ with $d(G_n)$ going to infinity while this does not happen? So there exists a prime power $q$ such that $q$ never divides the order of $G_n$?
Looking at the list it seems to me that this should not be possible. I don't have a specific requirement on how the $G_n$ should be related to one another but of course it would be nicer to have examples where there is some connection.