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 | G: H | 3 / 2 alvert G:Hrvert3/2. In particular, a transitive permutation group of degree n has at most a ⁢ n 3 / 2 an3/2 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 | G: H | - 1 lvert G:Hrvert-1.