Answer: 19.

Solution: Midpoints of 12 edges, centers of 6 faces, and the center of the cube.
7. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. We will call a point "equidistant" if there exist two vertices of the cube for which this point is the midpoint of the segment. How many "equidistant" points are there in the cube