Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 250 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 | #538 |
| 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. #6501 | C2 | semidirect |
| 3. #6500 | C2 | semidirect |
| 4. #6493 | C2 | semidirect |
| 5. C4×C22⋊D7 | C2 | semidirect |
| 6. #6461 | C2 | semidirect |
| 7. #6460 | C2 | semidirect |
| 8. #6457 | C2 | semidirect |
| 9. #6456 | C2 | semidirect |
| 10. #6448 | C2 | semidirect |
| 11. #6448 | C2 | semidirect |
| 12. #6447 | C2 | semidirect |
| 13. #6445 | C2 | semidirect |
| 14. D7×C22⋊C4 | C2 | semidirect |
| 15. #6442 | C2 | semidirect |
| 18. C22×C4×C7 | C22 | semidirect |
| 22. C2×C22⋊D7 | C22 | semidirect |
| 27. C2×C4×D7 | C22 | semidirect |
| 28. C2×Dic14 | C22 | semidirect |
| 29. C7×C4⋊C4 | C22 | semidirect |
| 30. C7×C4⋊C4 | C22 | semidirect |
| 33. C7×C22⋊C4 | C22 | semidirect |
| 36. C22⋊Dic7 | C22 | semidirect |
| 43. (C2×D7)⋊C4 | C22 | semidirect |
| 44. (C2×D7)⋊C4 | C22 | semidirect |
| 45. C4⋊Dic7 | C22 | semidirect |
| 46. C4⋊Dic7 | C22 | semidirect |
| 47. Dic7⋊C4 | C22 | semidirect |
| 48. Dic7⋊C4 | C22 | semidirect |
| 49. C4×Dic7 | C22 | semidirect |
| 50. C4×Dic7 | C22 | semidirect |
| 87. #123 | D7 | semidirect |
| 136. C22×C4 | C2×D7 | semidirect |
| 141. C4⋊C4 | C2×D7 | semidirect |
| 142. C4⋊C4 | C2×D7 | semidirect |
| 156. C22⋊C4 | C2×D7 | semidirect |
| 197. C7 | #538 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#123 | 2 |
| #6501 | 2 |
| #6500 | 2 |
| #6493 | 2 |
| C4×C22⋊D7 | 2 |
| #6461 | 2 |
| #6460 | 2 |
| #6457 | 2 |
| #6456 | 2 |
| #6448 | 2 |
| #6447 | 2 |
| #6445 | 2 |
| D7×C22⋊C4 | 2 |
| #6442 | 2 |
| #538 | 7 |