Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
704 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 165 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×C4⋊C4 | C2 |
| prime radical | C2×#6437 | C2 |
| hypercenter | C4×C4⋊C4 | D7 |
| hypocenter | C7 | #2544 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#6438 | C2 | semidirect |
| 2. C2×#6438 | C2 | semidirect |
| 4. C2×C4×D28 | C2 | semidirect |
| 5. C4×C7×C4⋊C4 | C2 | semidirect |
| 6. #89763 | C2 | semidirect |
| 7. #89711 | C2 | semidirect |
| 8. #89701 | C2 | semidirect |
| 9. #89437 | C2 | semidirect |
| 58. C42×C7 | D4 | semidirect |
| 73. C4×C4⋊C4 | D7 | semidirect |
| 183. C42 | D28 | semidirect |
| 225. C7 | #2544 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#6438 | 2 |
| C2×#6437 | 2 |
| C2×C4×D28 | 2 |
| C4×C7×C4⋊C4 | 2 |
| #89763 | 2 |
| #89711 | 2 |
| #89701 | 2 |
| #89437 | 2 |
| #2544 | 7 |