Double coset-ordering subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed double coset-ordering if given any two double cosets $$HxH$$ and $$HyH$$ one cannot find $$g,h,g',h'$$ such that $$gHxHg' \subset HyH$$ and $$hHyHh' \subset HxH$$. In other words, the relation:

$$HxH \le HyH \iff \exists g,g' \qquad gHxHg' \subset HyH$$

is a partial order.

Stronger properties

 * Subgroup of double coset index two which is not normal