  We show that quantum circuits restricted by a symmetry inherit the properties
of the whole special unitary group $SU(2^n)$, in particular composition,
algebraic and topological closedness and connectedness. It extends prior work
on symmetric states to the operators and shows that the operator space follows
the same structure as the state space. The well-behavedness is independent of
the symmetry requirement imposed on the subgroup. We provide an example of a
permutation invariance across all qubits.
