  A permutation group is said to be quasiregular if every its transitive
constituent is regular, and a quasiregular coherent configuration can be
thought as a combinatorial analog of such a group: the transitive constituents
are replaced by the homogeneous components. In this paper, we are interested in
the question when the configuration is schurian, i.e., formed by the orbitals
of a permutation group, or/and separable, i.e., uniquely determined by the
intersection numbers. In these terms, an old result of Hanna Neumann is, in a
sense, dual to the statement that the quasiregular coherent configurations with
cyclic homogeneous components are schurian. In the present paper, we (a)
establish the duality in a precise form and (b) generalize the latter result by
proving that a quasiregular coherent configuration is schurian and separable if
the groups associated with homogeneous components have distributive lattices of
normal subgroups.
