# Is there an established name for the system of all orbit of all subgroups $H\le G$?

Is there an established name for the system of all orbit of all subgroups of transitive permutation group $G$? Or, at least, was this or equivalent concept examined in a some book/article?

For instance, the cyclic group $G=C_{6}$ has an unique orbit $\{0,1,2,3,4,5\}$, but $G$ contain also subgroups with orbits $\{0\},\{1\},\{2\},\{3\},\{4\},\{5\}$ and $\{0,2,4\},\{1,3,5\}$

In general, this does not the same as a set of blocks $G$

• This doesn't answer your question but the orbits of any normal subgroup form a block system. – Derek Holt Aug 9 '18 at 9:04
• Yes. But for the efficient design of specific biomedical experiments, it will be important to know the orbits of all subgroups. Maybe. – Slepecky Mamut Aug 9 '18 at 9:22