  We introduce a natural way of associating oriented closed geodesics on the
modular curve to elements of $(\mathbb{Z}/q\mathbb{Z})^\times$ and prove that
the corresponding packets associated to sufficiently large subgroups
equidistribute in the unit tangent bundle as $q$ tends to infinity. This is a
$q$-orbit analogue of Duke's Theorem for real quadratic field as extended to
subgroups by Popa. We also show that the homology classes of the $q$-orbits of
oriented closed geodesics concentrate around the Eisenstein line and present
group theoretic applications thereof.
