  Suppose that $M$ is a closed, connected, and oriented Riemannian
$n$-manifold, $f \colon \mathbb{R}^n \to M$ is a quasiregular map automorphic
under a discrete group $\Gamma$ of Euclidean isometries, and $f$ has finite
multiplicity in a fundamental cell of $\Gamma$. We show that if $\Gamma$ has a
sufficiently large translation subgroup $\Gamma_T$, then $\dim \Gamma \in \{0,
n-1, n\}$. If $f$ is strongly automorphic and induces a non-injective
Latt\`es-type uniformly quasiregular map, then the same holds without the
assumption on the size of $\Gamma_T$. Moreover, an even stronger restriction
holds in the Latt\`es case if $M$ is not a rational cohomology sphere.
