  From a procedure to calculate the $C_5$-cone of a reduced complex analytic
curve $X \subset \mathbb{C}^n$ at a singular point $0 \in X$, we extract a
collection of integers that we call {\it auxiliary multiplicities} and we prove
they characterize the Lipschitz type of complex curve singularities. We then
use them to improve the known bounds for the number of irreducible components
of the $C_5$-cone. We finish by giving an example showing that in a Lipschitz
equisingular family of curves the number of planes in the $C_5$-cone may not be
constant.
