In this paper we obtain a classification of those subgroups of the finite general linear group $\mbox{GL}_d (q)$ with orders divisible by a primitive prime divisor of $q^e - 1$ for some $e > \frac{1}{2}d$. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups.