Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
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×D7 |
| commutator, socle, cosocle | C22×C7 | C25 |
| Fitting | C7×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D7 |
| hypocenter | C7 | #3529 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×H(3,𝔽4) | C2 | semidirect |
| 2. #90375 | C2 | semidirect |
| 3. #90371 | C2 | semidirect |
| 4. #90363 | C2 | semidirect |
| 5. #90359 | C2 | semidirect |
| 6. #90286 | C2 | semidirect |
| 7. #90277 | C2 | semidirect |
| 8. C23×Dic7 | C22 | semidirect |
| 9. C2×#1161 | C22 | semidirect |
| 14. #6516 | C22 | semidirect |
| 15. #6509 | C22 | semidirect |
| 16. Q8×Dic7 | C22 | semidirect |
| 17. #6507 | C22 | semidirect |
| 23. #6449 | C22 | semidirect |
| 24. #6447 | C22 | semidirect |
| 25. #6446 | C22 | semidirect |
| 26. (C22⋊D7)⋊C4 | C22 | semidirect |
| 27. #6442 | C22 | semidirect |
| 28. #6441 | C22 | semidirect |
| 29. #6440 | C22 | semidirect |
| 30. #6438 | C22 | semidirect |
| 31. C42×D7 | C22 | semidirect |
| 32. #6433 | C22 | semidirect |
| 64. H(3,𝔽4) | D7 | semidirect |
| 179. C7 | #3529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(3,𝔽4) | 2 |
| #90375 | 2 |
| #90371 | 2 |
| #90363 | 2 |
| #90359 | 2 |
| #90286 | 2 |
| #90277 | 2 |
| #3529 | 7 |