The number of subgroups of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times   \Bbb{Z}_r \times \Bbb{Z}_s$
  We deduce direct formulas for the total number of subgroups and the number of subgroups of a given order of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times \Bbb{Z}_r \times \Bbb{Z}_s$, where $m,n,r,s\in \Bbb{N}$. The proofs are by some simple group theoretical and number theoretical arguments based on Goursat's lemma for groups. Two conjectures are also formulated. 
L\'aszl\'o T\'oth
math.GR math.CO math.NT