Maximal PSL2 Subgroups of Exceptional Groups of Lie Type

We study embeddings of PSL2(p(a)) into exceptional groups G(p(b)) for G = F-4, E-6, E-2(6), E-7, and p a prime with a, b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(p(a)), that is maximal inside an almost simple exceptional group of Lie type F-4, E-6, E-2(6) and E-7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A(1) inside the algebraic group. Together with a recent result of Burness and Testerman for p the Coxeter number plus one, this proves that all maximal subgroups with socle PSL2(p(a)) inside these finite almost simple groups are known, with three possible exceptions (p(a) = 7, 8, 25 for E-7). In the three remaining cases we provide considerable information about a po-tential maximal subgroup.