p-Nilpotent maximal subgroups in finite groups

Let G be a finite group, p a prime, and suppose that every maximal subgroup of G is p-nilpotent or has prime index in G. We prove, relying in the Classification of Finite Simple Groups, that if p is odd and p 5, then G is p-solvable, and the p-length of G is at most 2. For p=5, however, a group G satisfying the same conditions need not be 5-solvable, and in that case we show that G/S₅ (G) PSL₂ (11), where S₅ (G) is the 5-solvable radical of G. For p=2, groups satisfying our conditions need not be solvable either. We prove, among other properties, that a unique simple group of Lie type, which belongs to the family PSL₂ (r^2ᵃ) for certain values of the prime r, can be involved in the structure of such groups.