Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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 | H(3,𝔽4)×C173 | C2 |
| hypercenter | H(3,𝔽4) | D173 |
| hypocenter | C173 | #3529 |
| LCS | G ▷ C22×C173 ▷ C173 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26012241 | C2 | semidirect |
| 2. #26012235 | C2 | semidirect |
| 3. #26012230 | C2 | semidirect |
| 4. #26012040 | C2 | semidirect |
| 5. #26012020 | C2 | semidirect |
| 6. #26011997 | C2 | semidirect |
| 7. H(3,𝔽4)×C173 | C2 | semidirect |
| 8. C2×#14583827 | C22 | semidirect |
| 10. C23×Dic173 | C22 | semidirect |
| 11. #19149677 | C22 | semidirect |
| 13. #19149672 | C22 | semidirect |
| 14. #19149671 | C22 | semidirect |
| 15. #19149667 | C22 | semidirect |
| 16. #19149663 | C22 | semidirect |
| 17. #19149661 | C22 | semidirect |
| 19. #19149656 | C22 | semidirect |
| 20. #19149655 | C22 | semidirect |
| 21. #19149652 | C22 | semidirect |
| 24. #19149648 | C22 | semidirect |
| 25. Q8×Dic173 | C22 | semidirect |
| 27. #19149639 | C22 | semidirect |
| 28. #19149635 | C22 | semidirect |
| 29. C42×D173 | C22 | semidirect |
| 97. C173 | #3529 | semidirect |
| 101. H(3,𝔽4) | D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26012241 | 2 |
| #26012235 | 2 |
| #26012230 | 2 |
| #26012040 | 2 |
| #26012020 | 2 |
| #26011997 | 2 |
| H(3,𝔽4)×C173 | 2 |
| #3529 | 173 |