A note on groups with a large permodularly embedded subgroup

It is known that if G is a group such that the centre factor group G/ζ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G/\zeta (G)$$\end{document} is polycyclic, then also the commutator subgroup G′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'$$\end{document} is polycyclic. The aim of this paper is to describe this situation from a lattice point of view. It is proved that if G is a group admitting a permodularly embedded non-periodic subgroup P such that the interval [G/P] is a polycyclic lattice, then G contains a polycyclic normal subgroup N such that G/N is quasihamiltonian.