  Motivated by the recent realization of a $^{52}$Cr Bose-Einstein condensate,
we consider the phase diagram of a general spin-three condensate as a function
of its scattering lengths. We classify each phase according to its ``reciprocal
spinor,'' using a method developed in a previous work. We show that such a
classification can be naturally extended to describe the vortices for a spinor
condensate by using the topological theory of defects. To illustrate, we
systematically describe the types of vortex excitations for each phase of the
spin-three condensate.
