Coupled-microdisk photonic molecules with high-symmetry geometries that support nondegenerate whispering gallery modes with high quality factors are numerically designed, based on a two-dimensional model. Dramatic enhancement of calculated Q factors of nondegenerate modes as compared with single-microdisk whispering gallery modes of the same radial order is achieved by tuning the intercavity coupling distances. Potential applications of this work to the design of single-mode coupled-microdisk structures with large spontaneous emission factors and low threshold lasing operation and to random laser simulations are discussed.