Let $A=\mathbb{Z}_{360}\oplus\mathbb{Z}_{150}\oplus\mathbb{Z}_{75}\oplus\mathbb{Z}_{3}$. I need to calculate the number of subgroups of $A$ which is isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$
I know that every such subgroup is generated by $a,b$ which satisfies $ord\left(a\right)=ord\left(b\right)=5,a\notin\left\langle b\right\rangle$, and I know how to count them. But the same subgroup can be generated by many pairs - my problem is to count the number of pairs which generated the same subgroup.