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In this paper, orbit colorings of a lattice Λ were considered, extending the idea of obtaining lattice colorings based on cosets of a sublattice. Given a subgroup  S  of finite index in the symmetry group  G  of the lattice Λ, if  T ( S ) is the subgroup of translations in  G  and  L  is the sublattice of Λ determined by  T ( S ), then a homomorphism  f  from  N  =  N   G   ( T ( S )) to the subgroup  S   C   of permutations of the cosets of  L  in Λ is defined. This homomorphism was used to obtain  S -orbit colorings of Λ and the corresponding color group  H  and color fixing group  K . Moreover,  S  ker  f  ≤  K  ≤  H  ≤  N .

Colorings of lattices based on subgroup orbits