Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1152 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D7 |
| commutator, cosocle | C22×C7 | C25 |
| Frattini | C22 | C24×D7 |
| Fitting | C2×C7×#125 | C2 |
| socle | C23×C7 | C24 |
| hypercenter | C2×#125 | D7 |
| hypocenter | C7 | C2×#562 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#125 | C2 | semidirect |
| 2. #90391 | C2 | direct (N × H) |
| 3. C2×#6459 | C2 | semidirect |
| 4. C2×#6458 | C2 | semidirect |
| 5. C2×#6457 | C2 | semidirect |
| 6. C2×#6438 | C2 | semidirect |
| 11. C7×#125 | C22 | semidirect |
| 19. #6459 | C22 | semidirect |
| 20. #6458 | C22 | semidirect |
| 21. #6457 | C22 | semidirect |
| 22. #6438 | C22 | semidirect |
| 45. C2×#125 | D7 | semidirect |
| 71. #125 | C2×D7 | semidirect |
| 114. C2×C7 | #562 | semidirect |
| 135. C7 | C2×#562 | semidirect |
| 148. C2 | #90391 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#125 | 2 |
| #90391 | 2 |
| C2×#6459 | 2 |
| C2×#6458 | 2 |
| C2×#6457 | 2 |
| C2×#6438 | 2 |
| C2×#562 | 7 |