On the Lattice of Hereditary Shemetkov Formations of Finite Groups in a Class X

A formation F of finite groups is called a Shemetkov formation in a class X if every X -group not belonging to F , all of whose proper subgroups belong to F , is either a Schmidt group or a group of prime order. In this paper, for a hereditary solvably saturated formation X , it is proved that the lattice of all hereditary Shemetkov formations of X -groups in the class 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.