Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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, prime radical | C7×#120 | C2 |
| hypercenter | #120 | D7 |
| hypocenter | C7 | #551 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C22⋊Dic7 | C2 | semidirect |
| 3. #6499 | C2 | semidirect |
| 4. #6498 | C2 | semidirect |
| 5. #6498 | C2 | semidirect |
| 6. D4×Dic7 | C2 | semidirect |
| 7. #6442 | C2 | semidirect |
| 8. #6441 | C2 | semidirect |
| 9. #6440 | C2 | semidirect |
| 15. C22×Dic7 | C22 | semidirect |
| 16. C2×Dic14 | C22 | semidirect |
| 23. C22⋊Dic7 | C22 | semidirect |
| 24. C22⋊Dic7 | C22 | semidirect |
| 25. C4⋊Dic7 | C22 | semidirect |
| 26. Dic7⋊C4 | C22 | semidirect |
| 27. Dic7⋊C4 | C22 | semidirect |
| 28. Dic7⋊C4 | C22 | semidirect |
| 29. C4×Dic7 | C22 | semidirect |
| 30. C4×Dic7 | C22 | semidirect |
| 116. C7 | #551 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#120 | 2 |
| C2×C22⋊Dic7 | 2 |
| #6499 | 2 |
| #6498 | 2 |
| D4×Dic7 | 2 |
| #6442 | 2 |
| #6441 | 2 |
| #6440 | 2 |
| #551 | 7 |