In work it is shown that tile of non-tile-transitive tiling of the Lobachevsky space can have an arbitrarily large number hyperfaces. In the review [4] among the unresolved questions the following is specified also: whether there is an upper (superior) bound for the number of faces of three-dimensional tile in monohedral tiling. Thus, a generalized construction of Boroczky’s tiling us to prove the following theorem: Theorem. In monohedral non-tile-transitive (both face-to-face and non-face-to-face) tiling of n-dimensional Lobachevsky space Λn there is no global (i.e. some constant c(n) depending only on n) upper bound of number of hyperfaces on n-dimensional hyperbolic tile, i.e. whatever number L > 0 you can specify such tiling from considered classes, the tile which will be by number of hyperfaces, greater than L.