Generalized Frattini Subgroups as Coradicals of Groups

The paper deals with finite solvable groups only. It is established that the class of all regular subgroup $m$-functors coincides with the class of all $X$-abnormal $m$-functors, where $X$ ranges over all subclasses of the class of all primitive groups. The properties of the lattice of all regular subgroup $m$-functors are studied and the atoms and coatoms of this lattice are described. It is proved that the generalized Frattini subgroup of $G$ corresponding to a regular $m$-functor coincides with the $X$-coradical of $G$ for some $R_0$-closed class $X$.