Maximal $\textrm {PSL}_2$ Subgroups of Exceptional Groups of Lie Type

With a few possible exceptions, the authots prove that any almost simple group with socle PSL2(pa), that is maximal inside an almost simple exceptional group of Lie type F4, E6, 2E6 and E7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1 inside the algebraic group.
We study embeddings of PSL2(pa) into exceptional groups G(pb)forG = F4,E6,2E6,E7,andp aprimewitha,b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(pa), that is maximal inside an almost simpleexceptional group of Lie type F4, E6, 2E6 and E7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1 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(pa) inside these finite almost simple groups are known, with three possible exceptions (pa = 7, 8,25 for E7). In the three remaining cases we provide considerable information about a potential maximal subgroup.

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