Direct product of cyclic group of prime-square order and cyclic group of prime order

Definition
Let $$p$$ be a prime number. This group of order $$p^3$$ is defined as the defining ingredient::external direct product of the defining ingredient::cyclic group of prime-square order and the defining ingredient::cyclic group of prime order.

Arithmetic functions of a counting nature
Note that since the group is abelian, the number of subgroups equals the number of conjugacy classes of subgroups as well as the number of normal subgroups.