Arrangement of normal subgroups in ordered groups

We look into the problem of determining the arrangement of normal (not necessarily relatively complex) subgroups $A$ of an l.o. group $G$ with respect to a system $\mathfrak L(G)$ of convex subgroups and describing the structure of quotient groups $G/A$ with the aid of notions of the theory of l.o. groups. A characterization of quotient groups is obtained for groups possessing an infrainvariant system of subgroups. For the class of l.o. groups, as well as for certain of the classes close to it, answers to some known particular questions have been found.