The Number of Subgroups of a Given Type in Certain Finite Groups

The number of subgroups, normal subgroups and characteristic subgroups of a finite group G are denoted by Sub(G), NSub(G) and CSub(G), respectively. The main goal of this paper is to present a matrix model for computing these positive integers for dicyclic groups, semi-dihedral groups, and three sequences U6n, V8n and H(n) of groups that can be presented as follows: U6n = ⟨a, b | a 2n = b 3 = e, bab = a⟩, V8n = ⟨a, b | a 2n = b 4 = e, aba = b −1 , ab−1a = b⟩, H(n) = ⟨a, b, c | a 2 n−2 = b 2 = c 2 = e, [a, b] = [b, c] = e, ac = ab⟩. For each group, a matrix model containing all information is given.