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 . In particular, a transitive permutation group of degree n has at most 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 .
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