On the lattice of all solvable regular transitive subgroup functors

The properties of the lattice $\mathrm{R\,T}(\mathfrak{S})$ of all regular transitive subgroup functors are investigated. The notion of $\theta$-subnormal subgroup functor is introduced. It is proved that the set $\mathrm{SUM}(\mathfrak{S})$ of all $\theta$-subnormal subgroup functors is a sublattice and ideal of the lattice $\mathrm{R\,T}(\mathfrak{S})$. The connection of lattices $\mathrm{R\,T}(\mathfrak{S})$ and $\mathrm{SUM}(\mathfrak{S})$ is investigated. The existence of a congruence $\Psi$ defined on $\mathrm{R\,T}(\mathfrak{S})$ such that the lattices $\mathrm{R\,T}(\mathfrak{S})/\Psi$ and $\mathrm{SUM}(\mathfrak{S})$ are isomorphic, in particular, is proved.