Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
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×D181 |
| commutator, socle, cosocle | C22×C181 | C25 |
| Fitting | H(3,𝔽4)×C181 | C2 |
| hypercenter | H(3,𝔽4) | D181 |
| hypocenter | C181 | #3529 |
| LCS | G ▷ C22×C181 ▷ C181 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26042029 | C2 | semidirect |
| 2. #26042023 | C2 | semidirect |
| 3. #26042018 | C2 | semidirect |
| 4. #26041828 | C2 | semidirect |
| 5. #26041808 | C2 | semidirect |
| 6. #26041785 | C2 | semidirect |
| 7. H(3,𝔽4)×C181 | C2 | semidirect |
| 8. C2×#15668450 | C22 | semidirect |
| 10. C23×Dic181 | C22 | semidirect |
| 11. #19153814 | C22 | semidirect |
| 13. #19153809 | C22 | semidirect |
| 14. #19153808 | C22 | semidirect |
| 15. #19153804 | C22 | semidirect |
| 16. #19153800 | C22 | semidirect |
| 17. #19153798 | C22 | semidirect |
| 19. #19153793 | C22 | semidirect |
| 20. #19153792 | C22 | semidirect |
| 21. #19153789 | C22 | semidirect |
| 24. #19153785 | C22 | semidirect |
| 25. Q8×Dic181 | C22 | semidirect |
| 27. #19153776 | C22 | semidirect |
| 28. #19153772 | C22 | semidirect |
| 29. C42×D181 | C22 | semidirect |
| 97. C181 | #3529 | semidirect |
| 101. H(3,𝔽4) | D181 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26042029 | 2 |
| #26042023 | 2 |
| #26042018 | 2 |
| #26041828 | 2 |
| #26041808 | 2 |
| #26041785 | 2 |
| H(3,𝔽4)×C181 | 2 |
| #3529 | 181 |