Maximal

We study embeddings of P mathvariant="normal">S mathvariant="normal">L 2 ( p a stretchy="false">) \mathrm {PSL}_2(p^a) into exceptional groups alttext="upper G b G b encoding="application/x-tex">G(p^b) for equals F 4 comma E 6 squared 7"> = F 4 , E 6 7 encoding="application/x-tex">G=F_4,E_6,{}^2\!E_6,E_7 , and alttext="p"> encoding="application/x-tex">p prime with alttext="a b"> encoding="application/x-tex">a,b positive integers. With few possible exceptions, we prove that any almost simple group socle is maximal inside an Lie type 4"> encoding="application/x-tex">F_4 6"> encoding="application/x-tex">E_6 alttext="squared encoding="application/x-tex">{}^2\!E_6 encoding="application/x-tex">E_7 the fixed points under Frobenius map corresponding closed subgroup A 1"> A 1 encoding="application/x-tex">A_1 algebraic group. Together recent result Burness Testerman Coxeter number plus one, this proves all subgroups these finite are known, three exceptions ( 8 25 encoding="application/x-tex">p^a=7,8,25 ). In remaining cases provide considerable information about potential subgroup.