Abstract. We characterize the groups which do not have non-trivial per-fect sections and such that any strictly descending chain of non-“nilpotent-by-finite ” subgroups is finite. 0. Let X be a property pertaining to subgroups. One of approaches to study the structure of groups is to investigate the groups in which the set of non-X-subgroups is small in some sense (or in other words, the groups which have many X-subgroups). There is increasing interest in groups with many nilpo-tent subgroups. Problems of this type have been considered in several papers. For instance, the examples of non-nilpotent groups with nilpotent and subnor-mal proper subgroups (the so-called groups of Heineken-Mohamed type) were constructed by Heineken and Mohamed , Hartley , Menegazzo  and others. In this way Bruno –, Bruno and Phillips  and Asar  have stud-ied the minimal non-“nilpotent-by-finite ” groups (i.e. non-“nilpotent-by-finite
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