Quantum 10, 2123 (2026). https://doi.org/10.22331/q-2026-06-01-2123 We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra – the symmetric subalgebra under a finite Abelian group symmetry GG. For systems where each site carries a regular representation of GG, we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subal