  We completely characterize possible indices of quasi-cyclic subcodes in a
cyclic code for a very broad class of cyclic codes. We present enumeration
results for quasi-cyclic subcodes of a fixed index and show that the problem of
enumeration is equivalent to enumeration of certain vector subspaces in finite
fields. In particular, we present enumeration results for quasi-cyclic subcodes
of the simplex code and duals of certain BCH codes. Our results are based on
the trace representation of cyclic codes.
