We numerically study the fractional quantum Hall effect at filling factors nu = 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 the nu = 3/5 fractional quantum Hall effect relative to nu = 12/5. By contrast, we find both nu = 2/5 and (its particle-hole conjugate) nu = 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.