Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
924 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D7 |
| commutator, socle, cosocle | C23×C7 | C24 |
| Fitting | C2×C7×#121 | C2 |
| hypercenter | C2×#121 | D7 |
| hypocenter | C7 | #2856 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#121 | C2 | semidirect |
| 2. C2×#6503 | C2 | semidirect |
| 3. C2×#6502 | C2 | semidirect |
| 4. C2×#6493 | C2 | semidirect |
| 5. C22×C4×Dic7 | C2 | semidirect |
| 6. #89993 | C2 | semidirect |
| 7. #89760 | C2 | semidirect |
| 8. #89748 | C2 | semidirect |
| 9. #89729 | C2 | semidirect |
| 43. C22×C4×C7 | D4 | semidirect |
| 54. C22×Dic7 | D4 | semidirect |
| 70. C4×Dic7 | D4 | semidirect |
| 74. C2×#121 | D7 | semidirect |
| 150. C22×C7 | #127 | semidirect |
| 163. C4×C7 | #121 | semidirect |
| 166. Dic7 | #121 | semidirect |
| 191. C22×C4 | C22⋊D7 | semidirect |
| 252. C7 | #2856 | semidirect |
| 267. C22 | #6503 | semidirect |
| 278. C4 | #6502 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#121 | 2 |
| C2×#6503 | 2 |
| C2×#6502 | 2 |
| C2×#6493 | 2 |
| C22×C4×Dic7 | 2 |
| #89993 | 2 |
| #89760 | 2 |
| #89748 | 2 |
| #89729 | 2 |
| #2856 | 7 |