++ Stems depressed-globose to ovate.

Radials 4 to 6, rigid.
uncinatus (9).

Radials 8 to 12.
wrightii* (15).

Radials 15 to 30.
grahami* (19), eschanzieri (21).

Radials 50 to 60.
barbatus (18).

**** Central spines more than one, and none of them hooked.

+ Slender or sometimes stout cylindrical plants, branching at base: Lower Californian.

brandegei (3), setispinus (24), halei (25).

++ Depressed-globose to ovate and stout cylindrical.

++ Radials few (3 to 12) and rigid: Mexican.