Lattices of subgroup and subsystem functors

Lattices of subgroup and subsystem functors are investigated. In particular, it is proved that for the case where ${\mathcal X}$ is a formation of finite groups and width of the lattice $F_0({\mathcal X})$ is at most $|\pi ({\mathcal X})|$, the formation ${\mathcal X}$ is metanilpotent and $|\pi({\mathcal X})| \leqslant 3$.