Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 248 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 | #6500 | C2 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #551 |
| 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 |
| 3. D4×Dic7 | C2 | semidirect |
| 4. C2×(C2×D7)⋊C4 | C2 | semidirect |
| 5. #6489 | C2 | semidirect |
| 6. #6461 | C2 | semidirect |
| 7. #6461 | C2 | semidirect |
| 8. #6460 | C2 | semidirect |
| 9. #6455 | C2 | semidirect |
| 10. #6449 | C2 | semidirect |
| 11. #6448 | C2 | semidirect |
| 12. #6446 | C2 | semidirect |
| 13. (C22⋊D7)⋊C4 | C2 | semidirect |
| 14. D7×C22⋊C4 | C2 | semidirect |
| 15. #6441 | C2 | semidirect |
| 22. C22×Dic7 | C22 | semidirect |
| 25. C2×Dic14 | C22 | semidirect |
| 26. C7×C4⋊C4 | C22 | semidirect |
| 27. C7×C4⋊C4 | C22 | semidirect |
| 29. C7×C22⋊C4 | C22 | semidirect |
| 30. C7×C22⋊C4 | C22 | semidirect |
| 33. C22⋊Dic7 | C22 | semidirect |
| 42. C4⋊Dic7 | C22 | semidirect |
| 43. C4⋊Dic7 | C22 | semidirect |
| 44. C4⋊Dic7 | C22 | semidirect |
| 45. C4⋊Dic7 | C22 | semidirect |
| 46. Dic7⋊C4 | C22 | semidirect |
| 47. Dic7⋊C4 | C22 | semidirect |
| 48. C4×Dic7 | C22 | semidirect |
| 49. C4×Dic7 | C22 | semidirect |
| 50. C4×Dic7 | C22 | semidirect |
| 84. #123 | D7 | semidirect |
| 135. C4⋊C4 | C2×D7 | semidirect |
| 136. C4⋊C4 | C2×D7 | semidirect |
| 149. C22⋊C4 | C2×D7 | semidirect |
| 150. C22⋊C4 | C2×D7 | semidirect |
| 188. C7 | #551 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#123 | 2 |
| #6500 | 2 |
| D4×Dic7 | 2 |
| C2×(C2×D7)⋊C4 | 2 |
| #6489 | 2 |
| #6461 | 2 |
| #6460 | 2 |
| #6455 | 2 |
| #6449 | 2 |
| #6448 | 2 |
| #6446 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| D7×C22⋊C4 | 2 |
| #6441 | 2 |
| #551 | 7 |