Abstract
A formation \(\mathfrak{F}\) of finite groups is called a Shemetkov formation in a class \(\mathfrak{X}\) if every \(\mathfrak{X}\) -group not belonging to \(\mathfrak{F}\) , all of whose proper subgroups belong to \(\mathfrak{F}\) , is either a Schmidt group or a group of prime order. In this paper, for a hereditary solvably saturated formation \(\mathfrak{X}\) , it is proved that the lattice of all hereditary Shemetkov formations of \(\mathfrak{X}\) -groups in the class \(\mathfrak{X}\) is lattice-isomorphic to the lattice of all subgraphs of some directed graph. As a corollary, a description of the lattice of all hereditary local Shemetkov formations of solvable groups in the class of all solvable groups is obtained, which was found by Ballester-Bolinches, Kamornikov, and Yi in 2024.
