The Wielandt–Hartley theorem for submaximal X -subgroups

A nonempty class X of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We consider two definitions of submaximal X-subgroups suggested by H. Wielandt and discuss which one better suits the task of determining maximal X-subgroups. We prove that these definitions are not equivalent yet the Wielandt–Hartley theorem holds true for either definition of X-submaximality. We also give some applications of the strong version of the Wielandt–Hartley theorem.