|Abstract:||Some time ago we developed a five dimensional SO(5) superrotor model  which describes the energy term diagram of the lowest energy states of CH$_5^+$ surprisingly well . The basis for this model lies in the S$_5$ permutation symmetry of the five identical protons in this molecule. In this contribution we discuss possible sub-group structures,
in particular the embedded S$_4$$\otimes$S$_2$ $\subseteq$ S$_5$ $\subset$ SO(5), which potentially simplifies the current model and more importantly can be associated with the internal motions (vibration/rotation/internal-rotation) of the molecule. As a result the ad-hoc SO(5) model motivated by the simple super-rotor energies can be replaced by a model justified by the underlying physics.
 H. Schmiedt, P. Jensen, S. Schlemmer, Rotation-vibration motion of extremely flexible molecules – the molecular superrotor, Chem. Phys. Lett. 672 (2017), p. 34-46\newline
 O. Asvany, K. M. T. Yamada, S. Br\"unken, A. Potapov, S. Schlemmer,
Experimental ground-state combination differences of CH$_5^+$, Science, 347 (2015), pp. 1346-1349