  Transport of the Brownian particles driven by L\'evy flights coexisting with
subdiffusion in asymmetric periodic potentials is investigated in the absence
of any external driving forces. Using the Langevin-type dynamics with
subordination techniques, we obtain the group velocity which can measure the
transport. It is found that the group velocity increases monotonically with the
subdiffusive index and there exists an optimal value of the L\'evy index at
which the group velocity takes its maximal value. There is a threshold value of
the subdiffusive index below which the ratchet effects will disappear. The
nonthermal character of the L\'evy flights and the asymmetry of the potential
are necessary to obtain the directed transport. Some peculiar phenomena induced
by the competition between L\'evy flights and subdiffusion are also observed.
The pseudonormal diffusion will appear on the level of the median.
