Linear groups with orders having certain large prime divisors

In this paper we obtain a classification of those subgroups of the finite g eneral linear group GL(d)(q) with orders divisible by a primitive prime div isor of q(e) - 1 for some e>1/2d. In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In p articular, in the last section, we obtain substantial extensions of the res ults of Landazuri and Seitz on small cross-characteristic representations o f some of the finite classical groups.