Frattini-type and fitting-type subgroups

Many generalizations of the Frattini and Fitting subgroups can be found in the literature. In this paper we study subgroups defined by choosing a certain type of chief factors in the line of Phi(p) (G) or F-p (G). This subgroups have properties analogues to the Frattini and the Fitting subgroups and they influence on the structure of the group. The main result of this paper is the following: Let X be a class of simple groups of characteristic pi and let H the class of the pi' groups which have not section in X; if G is a group such that every composition factor of G belongs to X boolean OR R, then Phi(X) (G) = Phi(X) (G mod G(H)).