Groups having the same Hall-Senior genus

Definition
Two groups $$G$$ and $$H$$ are said to have the same Hall-Senior genus if there is a lattice isomorphism between their respective lattices of normal subgroups (i.e., a bijection that preserves the containment relationship) such that under this lattice isomorphism, every group is mapped to an isoclinic group.

The term was used in the classification, by Hall and Senior, of the groups of order $$2^k, 0 \le k \le 6$$, and in this context, $$G$$ and $$H$$ are taken to be groups of order $$2^k$$ for the same $$k$$.

Stronger equivalence relations

 * Isomorphic groups

Weaker equivalence relations

 * Isoclinic groups