For any group G, denote by φf(G respectively L(G)) the intersection of all maximal subgroups of finite index (respectively finite nonprime index) in G, with the usual provision that the subgroup concerned equals Gif no such maximals exist. The subgroup φf(G) was discussed in [1] in connection with a property v possessed by certain groups: a group G has v if and only if every nonnilpotent, normal subgroup of G has a finite, nonnilpotent G-image. It was shown there, for instance, that G/φf(G) has v for all groups G. The subgroup L(G), in the case where G is finite, was investigated at some length in [3], one of the main results being that L(G) is supersoluble. (A published proof of this result appears as Theorem 3 of [4]). The present paper is concerned with the role of L(G) in groups G having property v or a related property a, the definition of which is obtained by replacing “nonnilpotent” by “nonsupersoluble” in the definition of v. We also present a result (namely Theorem 4) which displays a close relationship between the subgroups L(G) and φf(G) in an arbitrary group G. Some of the results for finite groups in [3] are found to hold with rather weaker hypotheses and, in fact, remain true for groups with v or a. We recall that if a group has a it also has v ([2]Theorem 2) but not conversely. For example, G = (x, y: y-1xy = x2)has v but not a. It is a well-known result of Gaschütz ([8], 5.2.15) that, in a finite group G, if His a normal subgroup containing φ(G) such that H/φ(G) is nilpotent than His nilpotent. This remains true in the case where G is any group with v [1, Proposition 1]. Our first result is in a similar vein and is a generalization of Theorem 9 of [7] and Theorem 1.2.9 of [3], the latter of which states that, for a finite group G, if G/L(G) is supersoluble, then so is G.