A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.

It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, 1964.) that the primary parallelohedra are the cube, hexagonal prism, elongated dodecahedron, rhombic dodecahedron, and truncated octahedron.

Is there is a classification for any higher dimensions? What are the primary d-parallelotopes?

The following is a conjecture of mine regarding the case of $d=4$.

Conjecture: There are exactly 7 primary 4-parallelotopes:

(1) Hypercube

(2) 16-cell

(3) 24-cell

(4) Hexagonal Square Duoprism

(5) Prismatic Elongated Dodecahedron

(6) Prismatic Rhombic Dodecahedron

(7) Prismatic Truncated Octahedron