A note on theta pairs for maximal subgroups

Let M be a maximal subgroup of a finite group G. A pair of subgroups (C, D) of G is called a theta-pair of M if it satisfies the following conditions: (a) D less than or equal to C and D triangle G; (b) (M, C) = C and D less than or equal to M;(c) CID has no proper normal subgroup of G/D. A theta-pair (C,D) of M is said to be maximal if M has no theta-pair (C', D') such that a < C'. In this paper we obtain several results on maximal theta-pairs which imply G to be solvable or supersolvable.