The geometrical application of multiple antisymmmetry groups and Mackay groups for the derivation of multidimensional subperiodic groups is considered and illustrated by the direct derivation of 4-dimensional groups of the category $G_{4321}$ from the category $G_{21}$ by using Mackay 2-multiple antisymmetry groups. In general, symmetry groups of the category $G_{(r+2)(r+1)r\ldots}$ treated as a subcategory of the category $G_{(r+2)r\ldots}$ can be derived directly by using Mackay 2-multiple antisymmetry groups.