Abstract :
Let G = G(q) be a finite almost simple exceptional group of Lie type over the field of q elements,
where q = pa and p is prime. The main result of the paper determines all maximal subgroups M of
G(q) such that M is an almost simple group which is also of Lie type in characteristic p, under the
condition that rank(M) > 1
2 rank(G). The conclusion is that either M is a subgroup of maximal
rank, or it is of the same type as G over a subfield of Fq, or (G,M) is one of (E
6(q), F4(q)),
(E
6 (q), C4(q)), (E7(q), 3D4(q)). This completes work of the first author with Saxl and
Testerman, in which the same conclusion was obtained under some extra assumptions.
