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×#117 | C2 |
| prime radical | #6445 | C2 |
| hypercenter | #117 | D7 |
| hypocenter | C7 | #551 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#117 | C2 | semidirect |
| 2. #6492 | C2 | semidirect |
| 3. C2×(C2×D7)⋊C4 | C2 | semidirect |
| 4. #6461 | C2 | semidirect |
| 5. #6459 | C2 | semidirect |
| 6. #6457 | C2 | semidirect |
| 7. #6457 | C2 | semidirect |
| 9. (C22⋊D7)⋊C4 | C2 | semidirect |
| 10. D7×C22⋊C4 | C2 | semidirect |
| 11. #6441 | C2 | semidirect |
| 12. #6439 | C2 | semidirect |
| 13. #6438 | C2 | semidirect |
| 14. C4×D28 | C2 | semidirect |
| 15. #6435 | C2 | semidirect |
| 24. C2×C4×D7 | C22 | semidirect |
| 25. C2×Dic14 | C22 | semidirect |
| 26. C7×C4⋊C4 | C22 | semidirect |
| 27. C7×C4⋊C4 | C22 | semidirect |
| 30. C42×C7 | C22 | semidirect |
| 31. C42×C7 | C22 | semidirect |
| 41. (C2×D7)⋊C4 | C22 | semidirect |
| 42. (C2×D7)⋊C4 | C22 | semidirect |
| 43. (C2×D7)⋊C4 | C22 | semidirect |
| 44. C4⋊Dic7 | C22 | semidirect |
| 45. Dic7⋊C4 | C22 | semidirect |
| 46. Dic7⋊C4 | C22 | semidirect |
| 47. Dic7⋊C4 | C22 | semidirect |
| 48. Dic7⋊C4 | C22 | semidirect |
| 49. Dic7⋊C4 | C22 | semidirect |
| 50. C4×Dic7 | C22 | semidirect |
| 91. #117 | D7 | semidirect |
| 135. C4⋊C4 | C2×D7 | semidirect |
| 136. C4⋊C4 | C2×D7 | semidirect |
| 152. C42 | C2×D7 | semidirect |
| 153. C42 | 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×#117 | 2 |
| #6492 | 2 |
| C2×(C2×D7)⋊C4 | 2 |
| #6461 | 2 |
| #6459 | 2 |
| #6457 | 2 |
| #6445 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| D7×C22⋊C4 | 2 |
| #6441 | 2 |
| #6439 | 2 |
| #6438 | 2 |
| C4×D28 | 2 |
| #6435 | 2 |
| #551 | 7 |