If I have a many-body Hamiltonian that is invariant under the representation of a certain group $G$ i.e [H,U(g)] = 0, how many phases can exist through spontaneous symmetry breaking (SSB) of this symmetry by the ground state? Specifically, if a certain subgroup $H \in G$ can occur in $G$ multiple times, does SSB to each of these represent different phases?

To make my question clear with an example, consider a Hamiltonian invariant under some representation of $\mathbb{Z}_2 \times \mathbb{Z}_2 =\{e, a, x, ax \}$. There are three ways the ground state can break down to $\mathbb{Z}_2:$

$1) \mathbb{Z}_2^{A}: \{e,a\},\\2)~\mathbb{Z}_2^{B}: \{e,x\},\\3) ~\mathbb{Z}_2^{C}: \{e,ax\}$.

Does the residual symmetry being the above three $\mathbb{Z}_2$ representations represent different phases of matter? Or are they the same?