For every $n$-group $(Q,A)$, $n\geq3$, there is an algebra $(Q,\{\cdot,\varphi,b\})$ [of the type $\langle2,1,0\rangle]$ such that the following statements hold: $1^\circ$ $(Q,\cdot)$ is a group; $2^\circ$ $\varphi\in Aut(Q,\cdot)$, $3^\circ$ $\varphi(b)=b$; $4^\circ$ for every $x\in Q$ $\varphi(x)\cdot b=b\cdot x$; and $5^\circ$ for every $x^n_1\in Q$ $A(x^n_1)=x_1\cdot\varphi(x_2)\cdot\dots\cdot\varphi^{n-1}\cdot b$ [Hosszu-Gluskin Theorem [2, 3]]. We say that an algebra $(Q,\{\cdot,\varphi,b\})$ is a Hosszu-Gluskin algebra of order $n$ $(n\geq3)$ [briefly: $n$HG-algebra] iff the statements $l6\circ$-$4^\circ$ hold. In addition, we say that an $n$HG-algebra $(Q,\{\cdot,\varphi,b\})$ is associated to an $n$-group $(Q,A)$ iff $5^\circ$ holds, [in [10], all $n$HG-algebras associated to the given $n$-group are described]. One of the main results of the paper is the following proposition: Let $n\geq3$, and let $(Q,A)$ be an $n$-group. Further on, let $(Q,\{\cdot,\varphi,b\})$ be its arbitrary associated $n$HG-algebra. Then $Con(Q,A)=Con(Q,\cdot)\cap Can(Q,\varphi)$. In addition, in the present paper we prove that the congruence lattice of an $n$-group $(Q,A)$ is a sublattice of the congruence lattice of the group $(Q,\cdot)$ and that it is isomorphic with the lattice of normal subgroups $(H,\cdot)$ of the group $(Q,\cdot)$ for which $\varphi(H)=H$. [In [4], Monk and Sioson described the congruence lattice of the $n$-group $(n\geq3)$, up to an isomorphism, in the scope of the Post covering group. (Remark 5.3).] In this paper, we also prove the following proposition: Let $n\geq3$ and let $(Q,A)$ be an $n$-group. Further on, let $\theta$ be an arbitrary element of the set $Con(Q, A)$. Then, for every $C_t\in Q/\theta$ there is an $n$HG-algebra $(Q,\{\cdot,\varphi,b\})$ associated to the $n$-group $(Q,A)$ such that the following statements hold: (i) $(C_t,\cdot)\triangleleft(Q,\cdot)$; (ii) $(C_t,\varphi)$ is a 1-groupoid; and (iii) $(C_t,A)$ is an $n$-subgroup of the $n$-group $(Q,A)$ iff $b\in C_t$.