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For a finite group G, let τ( G) denote a set of primes such that a prime p belongs to τ( G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions: (1) G has a p-complement for each p∈τ( G); (2) ∣τ( G)∣=2; (3) the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ( G); then G either is solvable or G/Sol( G)≅(2, 7) where Sol( G) is the largest solvable normal subgroup of ( G). [ABSTRACT FROM AUTHOR]