  Let $X$ be a quotient of a bounded domain in $\mathbb C^n$. Under suitable
assumptions, we prove that every subvariety of $X$ not included in the branch
locus of the quotient map is of log general type in some orbifold sense. This
generalizes a recent result by Boucksom and Diverio, which treated the case of
compact, \'etale quotients.
  Finally, in the case where $X$ is compact, we give a sufficient condition
under which there exists a proper analytic subset of $X$ containing all entire
curves and all subvarieties not of general type (meant this time in in the
usual sense as opposed to the orbifold sense).
