Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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 |
| prime radical | C2×C7×D4 | C22 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #552 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #6489 | C2 | semidirect |
| 5. C2×Dic7⋊C4 | C2 | semidirect |
| 6. #6453 | C2 | semidirect |
| 7. #6452 | C2 | semidirect |
| 8. #6452 | C2 | semidirect |
| 9. #6450 | C2 | semidirect |
| 10. #6442 | C2 | semidirect |
| 11. #6442 | C2 | semidirect |
| 12. #6441 | C2 | semidirect |
| 13. #6441 | C2 | semidirect |
| 14. #6440 | C2 | semidirect |
| 15. #6440 | C2 | semidirect |
| 169. C7 | #552 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#123 | 2 |
| #6498 | 2 |
| D4×Dic7 | 2 |
| #6489 | 2 |
| C2×Dic7⋊C4 | 2 |
| #6453 | 2 |
| #6452 | 2 |
| #6450 | 2 |
| #6442 | 2 |
| #6441 | 2 |
| #6440 | 2 |
| #552 | 7 |