  Magneto-transport measurements in a wide GaAs quantum well in which we can
tune the Fermi energy ($E_F$) to lie in different Landau levels of the two
occupied electric subbands reveal a remarkable pattern for the appearance and
disappearance of fractional quantum Hall states at $\nu$ = 10/3, 11/3, 13/3,
14/3, 16/3, and 17/3. The data provide direct evidence that the $q/3$ states
are stable and strong even at such high fillings as long as $E_F$ lies in a
ground-state (N=0) Landau level of either of the two electric subbands,
regardless of whether that level belongs to the symmetric or the anti-symmetric
subband. Evidently, the node in the out-of-plane direction of the
anti-symmetric subband does not destabilize the $q/3$ fractional states. On the
other hand, when $E_F$ lies in an excited ($N>0$) Landau level of either
subband, the wavefunction node(s) in the in-plane direction weaken or
completely destabilize the $q/3$ fractional quantum Hall states. Our data also
show that the $q/3$ states remain stable very near the crossing of two Landau
levels belonging to the two subbands, especially if the levels have parallel
spins.
