The enumeration of coverings of closed orientable Euclidean manifolds G₃ and G₅

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of n-fold coverings over the orientable Euclidean manifolds G₃ and G₅, and calculate the numbers of non-equivalent coverings of each type. The manifolds G₃ and G₅ are uniquely determined among other forms by their homology groups H₁(G₃)=Z₃×Z and H₁(G₅)=Z. We classify subgroups in the fundamental groups π₁(G₃) and π₁(G₅) up to isomorphism. Given index n, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating functions for the above sequences.