Flower arrangement of subgroups

Definition
A collection of subgroups $$H_i, i \in I$$ is said to form a flower arrangement of subgroups if, for any $$i \ne j \in I$$ and any $$k \ne l \in I$$, we have:

$$H_i \cap H_j = H_k \cap H_l$$.

($$i,j$$ may be equal to $$k,l$$).

The term is typically used when $$I$$ has at least two elements, in which case the intersection of any two of the subgroups (which is also, therefore, the intersection of all the subgroups) is termed the core of the flower.