Classifying fuzzy (normal) subgroups of the group D2p x Zq and finite groups of order n ≤ 20

One of the problems in fuzzy group theory is concerned with classifying the fuzzy (normal) subgroups of a finite group. In this paper, we classify the fuzzy (normal) subgroups of D-2p x Z(q) (p, q primes) and all finite groups of order n <= 20, by using the natural equivalence relation. In this case, the corresponding equivalence classes of fuzzy (normal) subgroups of a group G are closely connected to the chains of (normal) subgroups in G. In fact, for determining the number of these classes, we calculate the number of all chains of (normal) subgroups of G that terminate in G.