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The fractional quantum Hall (FQH) effect arises from strong Coulomb interactions and correlations among electrons, resulting in quasiparticles with fractional elementary charge. While recent research has focused heavily on the FQH state at a filling factor of 5/2 due to its potential non-Abelian properties, many other FQH states with odd denominators have also been observed. With improvements in sample quality, experimental signatures of FQH states have been detected in an increasing number of two-dimensional systems, many of which have subsequently been verified. Theoretical frameworks, such as composite fermion theory, have successfully explained the existence of these fractions, providing a paradigm for understanding this many-body system. However, the relative stability between any two arbitrary fractions often requires detailed calculations for clarification. In this work, we observe FQH states in ultrahigh-mobility GaAs/AlGaAs quantum well samples based on quantum transport measurements conducted in nuclear demagnetization and dilution refrigerators. Feature of longitudinal resistance dips for new FQH states have been found, including 17/33 and 15/31. We also report indications of previously unobserved FQH states under tilted magnetic fields. By combining these findings with known FQH states, we present a summary of the observed fractions, propose a pattern in their experimental appearance, and suggest trends in the stability of possible FQH states.