Groups that are not generated by certain $\mathfrak X$-abnormal subgroups

In the theory of formations the concept of an $\mathfrak X$-normal maximal subgroup is widely used, being some formation [1]. In this note the concept of $\mathfrak X$ normality is introduced for arbitrary subgroups. Subgroups that are not $\mathfrak X$-normal are called $\mathfrak X$-abnormal. Finite groups that are not generated by $S$-abnormal $n$-th maximal subgroups ($n=1,2,3,4$) are studied, where $S$ is the formation of all supersolvable groups. The general nature of the results of [3–6] is brought out.