Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D173 |
| commutator, socle, cosocle | C22×C173 | C25 |
| Fitting | #543×C173 | C2 |
| prime radical | C24×C173 | C23 |
| hypercenter | #543 | D173 |
| hypocenter | C173 | #3458 |
| LCS | G ▷ C22×C173 ▷ C173 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26012120 | C2 | semidirect |
| 3. #26012043 | C2 | semidirect |
| 7. #19149678 | C22 | semidirect |
| 8. #19149677 | C22 | semidirect |
| 9. #19149669 | C22 | semidirect |
| 12. Q8×Dic173 | C22 | semidirect |
| 13. #19149644 | C22 | semidirect |
| 22. C22×Dic173 | Q8 | semidirect |
| 61. C173 | #3458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26012120 | 2 |
| #26012117 | 2 |
| #26012043 | 2 |
| C2×#19149635 | 2 |
| #543×C173 | 2 |
| #3458 | 173 |