Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D173 |
| commutator, socle, cosocle | C2×C173 | C26 |
| Fitting | #585×C173 | C2 |
| hypercenter | #585 | D173 |
| hypocenter | C173 | H(7,𝔽2) |
| LCS | G ▷ C2×C173 ▷ C173 | |
| UCS | C1 ◁ C2 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26012328 | C2 | semidirect |
| 2. #26012327 | C2 | semidirect |
| 3. C2×#19149725 | C2 | semidirect |
| 4. H(5,𝔽2)×D173 | C2 | semidirect |
| 5. #585×C173 | C2 | semidirect |
| 6. #19149727 | C22 | semidirect |
| 7. #19149726 | C22 | semidirect |
| 8. #19149725 | C22 | semidirect |
| 9. #19149725 | C22 | semidirect |
| 10. #19149724 | C22 | semidirect |
| 11. C2×#14583827 | C22 | semidirect |
| 12. C2×#14583826 | C22 | semidirect |
| 13. #41×D173 | C22 | semidirect |
| 14. C22×D692 | C22 | semidirect |
| 15. C2×D4×D173 | C22 | semidirect |
| 16. 2−1+4×C173 | C22 | semidirect |
| 17. H(5,𝔽2)×C173 | C22 | semidirect |
| 18. C2×#41×C173 | C22 | semidirect |
| 19. C23×D173 | C23 | semidirect |
| 20. #14583827 | C23 | semidirect |
| 21. #14583826 | C23 | semidirect |
| 22. #14583826 | C23 | semidirect |
| 23. #14583825 | C23 | semidirect |
| 24. C2×Dic346 | C23 | semidirect |
| 25. Q8×D173 | C23 | semidirect |
| 26. C2×D692 | C23 | semidirect |
| 27. C2×D692 | C23 | semidirect |
| 28. C2×C22⋊D173 | C23 | semidirect |
| 29. D4×D173 | C23 | semidirect |
| 30. C2×C4×D173 | C23 | semidirect |
| 31. #41×C173 | C23 | semidirect |
| 32. #41×C173 | C23 | semidirect |
| 33. C2×Q8×C173 | C23 | semidirect |
| 34. C2×D4×C173 | C23 | semidirect |
| 35. C22×C4×C173 | C23 | semidirect |
| 60. C173 | H(7,𝔽2) | semidirect |
| 64. #585 | D173 | semidirect |
| 69. 2−1+4 | C2×D173 | semidirect |
| 73. H(5,𝔽2) | C2×D173 | semidirect |
| 77. C2×#41 | C2×D173 | semidirect |
| 85. #41 | C22×D173 | semidirect |
| 86. #41 | C22×D173 | semidirect |
| 89. C2×Q8 | C22×D173 | semidirect |
| 94. C2×D4 | C22×D173 | semidirect |
| 96. C22×C4 | C22×D173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26012328 | 2 |
| #26012327 | 2 |
| C2×#19149725 | 2 |
| H(5,𝔽2)×D173 | 2 |
| #585×C173 | 2 |
| H(7,𝔽2) | 173 |