  The class of all quasigroups is covered by six classes: the class of all
asymmetric quasigroups and five varieties of quasigroups (commutative, left
symmetric, right symmetric, semi-symmetric and totally symmetric). Each of
these classes is characterized by symmetry groups of its quasigroups.
  In this article, criteria of belonging of group isotopes to each of these
classes are found, including the corollaries for linear, medial and central
quasigroups etc. It is established that an isotope of a noncommutative group is
either semi-symmetric or asymmetric, each non-medial T-quasigroup is asymmetric
etc. The obtained results are applied for the classification of linear group
isotopes of prime orders, taking into account their up to isomorphism
description.
