Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×C22⋊D7 |
| commutator, cosocle | C4×C7 | C25 |
| Frattini | C4 | C24×D7 |
| Fitting | C7×#585 | C2 |
| socle | C2×C7 | C23×D4 |
| hypercenter | #585 | D7 |
| hypocenter | C7 | #3586 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#585 | C2 | semidirect |
| 2. C2×#6554 | C2 | semidirect |
| 3. #90529 | C2 | semidirect |
| 4. #90527 | C2 | semidirect |
| 5. C2×#6514 | C2 | semidirect |
| 6. #90513 | C2 | semidirect |
| 7. #90510 | C2 | semidirect |
| 9. C7×2−1+4 | C22 | semidirect |
| 11. C7×H(5,𝔽2) | C22 | semidirect |
| 16. #6554 | C22 | semidirect |
| 19. C2×#1157 | C22 | semidirect |
| 20. C22×Dic14 | C22 | semidirect |
| 21. #6514 | C22 | semidirect |
| 22. #6514 | C22 | semidirect |
| 23. #6513 | C22 | semidirect |
| 24. #6512 | C22 | semidirect |
| 25. #6511 | C22 | semidirect |
| 26. C2×#1144 | C22 | semidirect |
| 27. #6505 | C22 | semidirect |
| 28. C2×#1142 | C22 | semidirect |
| 29. #6495 | C22 | semidirect |
| 30. C2×#1136 | C22 | semidirect |
| 68. #585 | D7 | semidirect |
| 106. 2−1+4 | C2×D7 | semidirect |
| 108. H(5,𝔽2) | C2×D7 | semidirect |
| 207. C7 | #3586 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#585 | 2 |
| C2×#6554 | 2 |
| #90529 | 2 |
| #90527 | 2 |
| C2×#6514 | 2 |
| #90513 | 2 |
| #90510 | 2 |
| #3586 | 7 |