We numerically study the fractional quantum Hall effect at filling factors  = 12=5 and 13/5 (the particle-hole conjugate of 12/5) in high-quality two-dimensional GaAs heterostructures via exact diagonalization including finite well width and Landau level mixing. We find that Landau level mixing suppresses  = 13=5 fractional quantum Hall effect relative to  = 12=5. By contrast, we find both  = 2=5 and (its particle-hole conjugate)  = 3=5 fractional quantum Hall effects in the lowest Landau level to be robust under Landau level mixing and finite well-width corrections. Our results provide a possible explanation for the experimental absence of the 13/5 fractional quantum Hall state as caused by Landau level mixing effects.