Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D7 |
| commutator, socle, cosocle | C22×C7 | C24 |
| Fitting | C7×#123 | C2 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #540 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#123 | C2 | semidirect |
| 2. #6503 | C2 | semidirect |
| 3. #6502 | C2 | semidirect |
| 4. #6493 | C2 | semidirect |
| 5. C4×C22⋊D7 | C2 | semidirect |
| 6. #6458 | C2 | semidirect |
| 7. #6457 | C2 | semidirect |
| 8. #6455 | C2 | semidirect |
| 9. #6452 | C2 | semidirect |
| 10. #6449 | C2 | semidirect |
| 11. #6447 | C2 | semidirect |
| 12. #6447 | C2 | semidirect |
| 13. #6447 | C2 | semidirect |
| 14. #6446 | C2 | semidirect |
| 15. (C22⋊D7)⋊C4 | C2 | semidirect |
| 87. #123 | D7 | semidirect |
| 196. C7 | #540 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#123 | 2 |
| #6503 | 2 |
| #6502 | 2 |
| #6493 | 2 |
| C4×C22⋊D7 | 2 |
| #6458 | 2 |
| #6457 | 2 |
| #6455 | 2 |
| #6452 | 2 |
| #6449 | 2 |
| #6447 | 2 |
| #6446 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| #540 | 7 |