  We study the magneto-conductance of a $1.4~\mathrm{\mu m}$-wide quantum dot
in the fractional quantum Hall regime. For a filling factor $\approx 2/3$ and
$\gtrsim 1/3$ in the quantum dot the observed Coulomb resonances show a
periodic modulation in magnetic field. This indicates a non-trivial
reconstruction of the 2/3 fractional quantum Hall state in the quantum dot. We
present a model for the charge stability diagram of the system assuming two
compressible regions separated by an incompressible stripe of filling factor
$2/3$ and $1/3$, respectively. From the dependence of the magnetic field period
on total magnetic field we construct the zero-field charge density distribution
in the quantum dot. The tunneling between the two compressible regions exhibits
fractional Coulomb blockade. For both filling factor regions, we extract a
fractional charge $e^*/e = 0.32 \pm 0.03$ by comparing to measurements at
filling factor 2. With their close relation to quantum Hall Fabry-P\'{e}rot
interferometers, our investigations on quantum dots in the fractional quantum
Hall regime extend and complement interference experiments investigating the
nature of anyonic fractional quantum Hall quasiparticles.
