  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 orientable Euclidean
manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$, and calculate the numbers of
non-equivalent coverings of each type. We classify subgroups in the fundamental
groups $\pi_1(\mathcal{G}_{3})$ and $\pi_1(\mathcal{G}_{5})$ up to isomorphism
and calculate the numbers of conjugated classes of each type of subgroups for
index $n$. The manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$ are uniquely
determined among the others orientable forms by their homology groups
$H_1(\mathcal{G}_{3})=\ZZ_3\times \ZZ$ and $H_1(\mathcal{G}_{5})= \ZZ$.

Algebraic Topology;Group Theory