There exist subgroups of any given subnormal depth for any given tuple of nontrivial groups as quotient groups in a subnormal series
such that each is normal in and and such that the subnormal depth of in is exactly , i.e., has no shorter subnormal series in .
- Normality is not transitive for any pair of nontrivial quotient groups: This is the case.
- There exist subgroups of arbitrarily large subnormal depth
For the proof, see the linked Math Overflow question page.