Bounding the order of complex linear groups and permutation groups with selected composition factors

Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of $\mathrm{G}\mathrm{L}(n,\mathbb{C})$ with restricted composition factors (as usual, modulo Abelian normal subgroups). We place the question in a wider context, and find other applications.