Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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 | C7×#390 | C2 |
| hypercenter | #390 | D7 |
| hypocenter | C7 | #2431 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#1159 | C2 | semidirect |
| 2. C2×#6503 | C2 | semidirect |
| 3. C2×#6458 | C2 | semidirect |
| 4. C2×#6438 | C2 | semidirect |
| 5. C7×#390 | C2 | semidirect |
| 6. #89746 | C2 | semidirect |
| 7. #89732 | C2 | semidirect |
| 8. #89730 | C2 | semidirect |
| 9. C4×(C2×D7)⋊C4 | C2 | semidirect |
| 39. C2×C7×D4 | D4 | semidirect |
| 63. C2×C4×D7 | D4 | semidirect |
| 66. C2×Dic14 | D4 | semidirect |
| 80. #390 | D7 | semidirect |
| 119. C2×Dic7 | #41 | semidirect |
| 193. C2×D4 | D28 | semidirect |
| 262. C7 | #2431 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#1159 | 2 |
| C2×#6503 | 2 |
| C2×#6458 | 2 |
| C2×#6438 | 2 |
| C7×#390 | 2 |
| #89746 | 2 |
| #89732 | 2 |
| #89730 | 2 |
| C4×(C2×D7)⋊C4 | 2 |
| #2431 | 7 |