  This paper gives a complete parametrization of the commensurability classes
of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b}
= (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all
Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for
any $n\geq 1$, examples of manifolds/Shimura varieties with precisely $n$
commensurability classes of totally geodesic submanifolds/Shimura subvarieties.
This is in stark contrast with the previously studied cases of arithmetic
hyperbolic $3$-manifolds and quaternionic Shimura surfaces, where the presence
of one commensurability class of geodesic submanifolds implies the existence of
infinitely many classes.
