Finite Group DatabasebetaFinite Group DatabasebetaOrder 17792.
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×D139 |
| commutator, socle, cosocle | C22×C139 | C25 |
| Fitting | H(3,𝔽4)×C139 | C2 |
| hypercenter | H(3,𝔽4) | D139 |
| hypocenter | C139 | #3529 |
| LCS | G ▷ C22×C139 ▷ C139 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22581703 | C2 | semidirect |
| 2. #22581697 | C2 | semidirect |
| 3. #22581692 | C2 | semidirect |
| 4. #22581502 | C2 | semidirect |
| 5. #22581482 | C2 | semidirect |
| 6. #22581459 | C2 | semidirect |
| 7. H(3,𝔽4)×C139 | C2 | semidirect |
| 8. C2×#14555786 | C22 | semidirect |
| 10. C23×Dic139 | C22 | semidirect |
| 11. #17984840 | C22 | semidirect |
| 13. #17984835 | C22 | semidirect |
| 14. #17984834 | C22 | semidirect |
| 15. #17984830 | C22 | semidirect |
| 16. #17984826 | C22 | semidirect |
| 17. #17984824 | C22 | semidirect |
| 19. #17984819 | C22 | semidirect |
| 20. #17984818 | C22 | semidirect |
| 21. #17984815 | C22 | semidirect |
| 24. #17984811 | C22 | semidirect |
| 25. Q8×Dic139 | C22 | semidirect |
| 27. (C22⋊D139)⋊C4 | C22 | semidirect |
| 28. #17984798 | C22 | semidirect |
| 29. C42×D139 | C22 | semidirect |
| 97. C139 | #3529 | semidirect |
| 101. H(3,𝔽4) | D139 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22581703 | 2 |
| #22581697 | 2 |
| #22581692 | 2 |
| #22581502 | 2 |
| #22581482 | 2 |
| #22581459 | 2 |
| H(3,𝔽4)×C139 | 2 |
| #3529 | 139 |