LAUSR

Abstract 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{a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most a⁢n3/2{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 |G:H|-1{\lvert G:H\rvert-1}.