All abelian symmetries of Landau-Ginzburg potentials

We present an algorithm for determining all inequivalent abelian symmetries of non-degenerate quasi-homogeneous polynomials and apply it to the recently constructed complete set of Landau-Ginzburg potentials for N=2 superconformal field theories with c=9. A complete calculation of the resulting orbifolds without torsion increases the number of known spectra by about one third. The mirror symmetry of these spectra, however, remains at the same low level as for untwisted Landau-Ginzburg models. This happens in spite of the fact that the subclass of potentials for which the Berglund-Huebsch construction works features perfect mirror symmetry. We also make first steps into the space of orbifolds with Z2 torsions by including extra trivial fields. (orig.)