Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
1348 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 213 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 | C4×C7×C22⋊C4 | C2 |
| hypercenter | C4×C22⋊C4 | D7 |
| hypocenter | C7 | #2433 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D28 | C2 | semidirect |
| 2. C2×#6493 | C2 | semidirect |
| 3. C2×#6445 | C2 | semidirect |
| 4. C2×#6437 | C2 | semidirect |
| 5. C4×C7×C22⋊C4 | C2 | semidirect |
| 6. #89711 | C2 | semidirect |
| 7. #89443 | C2 | semidirect |
| 15. C22×D28 | C22 | semidirect |
| 16. C22×D28 | C22 | semidirect |
| 18. C2×C42×C7 | C22 | semidirect |
| 20. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 21. C2×(C2×D7)⋊C4 | C22 | semidirect |
| 22. C2×C4⋊Dic7 | C22 | semidirect |
| 25. C7×#95 | C22 | semidirect |
| 35. C22×C4×C7 | D4 | semidirect |
| 52. C2×D28 | D4 | semidirect |
| 53. C7×C22⋊C4 | D4 | semidirect |
| 64. C4×C22⋊C4 | D7 | semidirect |
| 90. C2×C4×C7 | C2×D4 | semidirect |
| 114. C2×C42 | C2×D7 | semidirect |
| 115. #95 | C2×D7 | semidirect |
| 119. C22×C7 | #127 | semidirect |
| 136. C4×C7 | #120 | semidirect |
| 160. C22×C4 | D28 | semidirect |
| 167. C22⋊C4 | D28 | semidirect |
| 207. C2×C4 | C2×D28 | semidirect |
| 209. C7 | #2433 | semidirect |
| 223. C22 | #6437 | semidirect |
| 231. C4 | #6445 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D28 | 2 |
| C2×#6493 | 2 |
| C2×#6445 | 2 |
| C2×#6437 | 2 |
| C4×C7×C22⋊C4 | 2 |
| #89711 | 2 |
| #89443 | 2 |
| #2433 | 7 |