  We study geometric properties of the action of the Picard modular group
$\Gamma=PU(2,1,\mathcal{O}_7)$ on the complex hyperbolic plane
$H^2_\mathbb{C}$, where $\mathcal{O}_7$ denotes the ring of algebraic integers
in $\mathbb{Q}(i\sqrt{7})$. We list conjugacy classes of maximal finite
subgroups in $\Gamma$ and give an explicit description of the Fuchsian
subgroups that occur as stabilizers of mirrors of complex reflections in
$\Gamma$. As an application, we describe an explicit torsion-free subgroup of
index $336$ in $\Gamma$.
