  A Lie-algebraic classification of the variable coefficient cubic-quintic
nonlinear Schr\"odinger equations involving 5 arbitrary functions of space and
time is performed under the action of equivalence transformations. It is shown
that their symmetry group can be at most four-dimensional in the genuine
cubic-quintic nonlinearity. It is only five-dimensional (isomorphic to the
Galilei similitude algebra gs(1)) when the equations are of cubic type, and
six-dimensional (isomorphic to the Schr\"odinger algebra sch(1)) when they are
of quintic type.
