The following proposition is well known in the group theory: if $(Q,A)$ is a group (2-group) and $\theta$ its congruence relation, then there is exactly one $C_a\in Q/\theta$ such that $(C_a,A)$ is a subgroup of the group $(Q,A)$. However, for $n\geq3$, for instance, there are $n$-groups $(Q,A)$ and their congruences $\theta$ such that for any $C_a\in Q/\theta$ the pair $(C_a,A)$ is not an $n$-group (4.1), (4.3). The main results of the paper are Theorems 3.1, 3.2 and 5.1.