Does "higher symmetry" mean any symmetry group of higher order? What is the precise definition of a higher symmetry in group theory? I see this term scattered around, such as in Cotton's Chemical Applications of Group Theory:


  ...In the cyclic groups each of the operations, Cn, Cn2, Cn3, ..., , Cnn-1, constitutes a class by itself and we continue to use this notations. However, in all other groups of higher symmetry, the number of classes spanned by these operations will be reduced in the following way...


Does higher symmetry strictly mean a supergroup? Or is it a quantitative statement about something like the number of symmetry elements or number of irreducible representations (which are not always ordered the same way, as in the case of C3v and S4). 

I am failing to find intuition because, if it does mean strictly supergroups, we can't compare the "amount" of symmetry in C2 compared to C3 (or Td compared to Ci even!).
"Higher symmetry" per se does not mean anything, unless you define it to mean something.

Depending on the context, it usually means one of the following things:


Any symmetry that is a supergroup of the group in question. In this case $C_{4v}$ is higher than $C_4$, while $C_6$ is incomparable to both. Bear with it. Partially ordered sets are quite a thing in math.
(rarely used) Any symmetry group having more elements than the group in question, so that $C_{4v}$ is higher than $C_6$, which in turn is higher than $C_4$.
(apparently, the meaning used by Cotton) Any symmetry group which is not a cyclic rotation group. In this case $C_{4v}$ is higher, while $C_4$ and $C_6$ are lower.


See, there is an intuition, after all. Worse yet, there are quite a few different kinds of intuition.

So it goes.