2-minimal subgroups in finite exceptional groups of Lie type

Suppose G is a finite group, S a Sylow 2-subgroup of G and B = N G ( S ) . Then a subgroup P of G is a 2-minimal subgroup of G with respect to B if and only if B is contained in a unique maximal subgroup of P . Here the 2-minimal subgroups of the finite simple groups of exceptional Lie type are classified. This classification yields detailed descriptions of the 2-minimal subgroups and, by way of illustration, we list explicitly the 2-minimal subgroups for G 2 ( 3 ) , F 4 ( 3 ) , E 6 ( 19 ) , E 7 ( 5 3 ) and E 8 ( 11 ) .