This paper is a new one in a series of papers dealing with the study of the structure of a finite group in terms of the normal index of maximal subgroups in the sense of W. E. Deskins [Proc. Symp. Pure Math. 1, 100-104 (1959; Zbl 0096.248)]. This series started with a paper of the first author [Ill. J. Math. 19, 173-178 (1975; Zbl 0303.20014)]. Other references may be found in the present authors’ paper [Pac. J. Math. 132, 143-149 (1988; Zbl 0649.20019)].
In the paper under review, the influence of the normal indices of certain classes of maximal subgroups of a given group on the structure of the group is studied. As a consequence, various characterizations of nilpotent, supersolvable and other classes of finite groups are given in these terms.