Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
1536 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 549 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 | C22×#41×C173 | C2 |
| hypercenter | C22×#41 | D173 |
| hypocenter | C173 | #3487 |
| LCS | G ▷ C22×C173 ▷ C173 | |
| UCS | C1 ◁ C22 ◁ C22×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19149727 | C2 | semidirect |
| 2. #26012160 | C2 | semidirect |
| 3. #26012148 | C2 | semidirect |
| 4. #26012058 | C2 | semidirect |
| 5. #26012002 | C2 | semidirect |
| 6. #26011969 | C2 | semidirect |
| 7. C22×#41×C173 | C2 | semidirect |
| 9. C2×#14583827 | C22 | semidirect |
| 11. C2×Q8×D173 | C22 | semidirect |
| 12. #19149675 | C22 | semidirect |
| 13. #19149666 | C22 | semidirect |
| 15. #19149658 | C22 | semidirect |
| 16. #19149653 | C22 | semidirect |
| 17. #19149650 | C22 | semidirect |
| 19. Q8×Dic173 | C22 | semidirect |
| 20. C4×C22⋊D173 | C22 | semidirect |
| 43. C2×Q8×C173 | D4 | semidirect |
| 87. C173 | #3487 | semidirect |
| 90. C22×#41 | D173 | semidirect |
| 121. C2×Q8 | C22⋊D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19149727 | 2 |
| #26012160 | 2 |
| #26012148 | 2 |
| #26012058 | 2 |
| #26012002 | 2 |
| #26011969 | 2 |
| C22×#41×C173 | 2 |
| #3487 | 173 |