For a finite group G and an arbitrary prime p, let S P ( G ) denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we set S P ( G ) = G. Some properties of G are considered involving S P ( G ) . In particular, we obtain a characterization of G when each M in the definition of S P ( G ) is nilpotent.