A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

We show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a vertical bar G : H vertical bar(3/2). In particular, a transitive permutation group of degree n has at most an(3/2) maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is, the number of maximal subgroups of G containing H is at most vertical bar G : H vertical bar - 1.