Let G be a finite group and p an odd prime. Let ψ*p(G) be the set of proper subgroups M of G with |G: M| not a prime power and |G: M|p = 1. In this paper,we investigate the structure of G if every member of ψ*p(G) is nilpotent. In particular, a new characterization of PSL(2, 7) is obtained. The proof of the theorem depends on the classification of finite simple groups.