On the coverings of closed orientable Euclidean manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$

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 $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups $π_1(\mathcal{G}_{2})$ and $π_1(\mathcal{G}_{4})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{2})=\mathbb{Z}_2\times \mathbb{Z}_2 \times \mathbb{Z}$ and $H_1(\mathcal{G}_{4})=\mathbb{Z}_2 \times \mathbb{Z}$.

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