Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
1748 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 557 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 | C4×#41×C181 | C2 |
| hypercenter | C4×#41 | D181 |
| hypocenter | C181 | #3498 |
| LCS | G ▷ C22×C181 ▷ C181 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#19153862 | C2 | semidirect |
| 2. #26041951 | C2 | semidirect |
| 3. D4×D724 | C2 | semidirect |
| 4. #26041848 | C2 | semidirect |
| 5. #26041829 | C2 | semidirect |
| 6. C2×#19153813 | C2 | semidirect |
| 7. C4×#41×C181 | C2 | semidirect |
| 9. C2×#15668450 | C22 | semidirect |
| 10. C2×#15668449 | C22 | semidirect |
| 11. C22×D724 | C22 | semidirect |
| 12. C22×D724 | C22 | semidirect |
| 13. C2×D4×D181 | C22 | semidirect |
| 14. #19153816 | C22 | semidirect |
| 15. #19153813 | C22 | semidirect |
| 16. #19153813 | C22 | semidirect |
| 17. #19153813 | C22 | semidirect |
| 18. #19153804 | C22 | semidirect |
| 19. #19153791 | C22 | semidirect |
| 20. #19153790 | C22 | semidirect |
| 21. #19153786 | C22 | semidirect |
| 22. C4×D724 | C22 | semidirect |
| 24. C4×Q8×C181 | C22 | semidirect |
| 25. C4×D4×C181 | C22 | semidirect |
| 26. #117×C181 | C22 | semidirect |
| 27. C2×C42×C181 | C22 | semidirect |
| 43. #41×C181 | D4 | semidirect |
| 65. Q8×C181 | C2×D4 | semidirect |
| 66. D4×C181 | C2×D4 | semidirect |
| 68. C2×C4×C181 | C2×D4 | semidirect |
| 82. C4×C181 | H(5,𝔽2) | semidirect |
| 88. C181 | #3498 | semidirect |
| 96. C4×#41 | D181 | semidirect |
| 112. C4×Q8 | C2×D181 | semidirect |
| 114. C4×D4 | C2×D181 | semidirect |
| 115. #117 | C2×D181 | semidirect |
| 116. C2×C42 | C2×D181 | semidirect |
| 120. #41 | D724 | semidirect |
| 147. Q8 | C2×D724 | semidirect |
| 153. D4 | C2×D724 | semidirect |
| 157. C2×C4 | C2×D724 | semidirect |
| 172. C4 | #19153862 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#19153862 | 2 |
| #26041951 | 2 |
| D4×D724 | 2 |
| #26041848 | 2 |
| #26041829 | 2 |
| C2×#19153813 | 2 |
| C4×#41×C181 | 2 |
| #3498 | 181 |