The Number of Subgroups and Cyclic Subgroups of Finite Group and Its Application by GAP Program

In this paper, we present a novel approach for calculating the set of subgroups of a finite group, focusing on cyclic subgroups, and using it to establish the quantity of all subgroups in the direct product of two groups. Specifically, we consider the Dicyclic group of order $4n$ and the Cyclic group of order $p$. Let $\tau (n)$ denote the total number of divisors of $n$, and $\sigma (n)$ denote the summation of all divisors of $n$. Using these functions, we derive a formula for the number of subgroups in the group $T_{4n}\times C_p$. We then use the computer program GAP to find all $T_{4n}\times C_p$ with exactly $\mathrm{\mid }{T}_{4n}\times C_p\mathrm{\mid }-t$ cyclic subgroups for $t\ge 1$.