  The fractional quantum Hall effect (FQHE) is studied in the semiclassical
limit in the framework of the Hofstadter model with a short-range interaction
between fermions. In the mean-field approximation, the repulsion between
fermions leads to a periodic potential. Numerical calculations show that in the
case of the periodic potential with a period that is a multiple on
$\frac{1}{\nu}$ of the magnetic cell ($\nu$ is filling of a separated band)
composite Hofstadter bands (HBs) are formed. The composite HBs are split into
$\frac{1}{\nu}$ subbands, which are separated by the Dirac points. The Chern
number of $\gamma$-full filled composite HBs is equal to the Chern number
$C_{\gamma}$ of the corresponding HB. The Chern number, equal to $\nu
C_{\gamma}$, corresponds to $\nu$-filling of $\gamma$-composite HB. Thus, FQHE
is realized by fractional filling of composite HBs.
