Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#120 | C2 |
| hypercenter | #120 | D7 |
| hypocenter | C7 | #546 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊D7 | C2 | semidirect |
| 2. C2×#1159 | C2 | semidirect |
| 3. C7×#120 | C2 | semidirect |
| 4. C2×C22⋊Dic7 | C2 | semidirect |
| 5. #6502 | C2 | semidirect |
| 6. #6502 | C2 | semidirect |
| 7. #6501 | C2 | semidirect |
| 8. #6500 | C2 | semidirect |
| 9. D4×Dic7 | 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 |
| 19. C2×C7×D4 | C22 | semidirect |
| 20. C2×C7×D4 | C22 | semidirect |
| 26. C2×C22⋊D7 | C22 | semidirect |
| 27. C2×C22⋊D7 | C22 | semidirect |
| 29. C22×Dic7 | C22 | semidirect |
| 30. C22×Dic7 | C22 | semidirect |
| 31. C22×Dic7 | C22 | semidirect |
| 33. C2×C4×D7 | C22 | semidirect |
| 34. C2×C4×D7 | C22 | semidirect |
| 35. C2×Dic14 | C22 | semidirect |
| 37. C7×C22⋊C4 | C22 | semidirect |
| 38. C7×C22⋊C4 | C22 | semidirect |
| 41. C22⋊Dic7 | C22 | semidirect |
| 42. C22⋊Dic7 | C22 | semidirect |
| 43. C22⋊Dic7 | C22 | semidirect |
| 44. (C2×D7)⋊C4 | C22 | semidirect |
| 45. (C2×D7)⋊C4 | C22 | semidirect |
| 46. (C2×D7)⋊C4 | C22 | semidirect |
| 47. (C2×D7)⋊C4 | C22 | semidirect |
| 48. C4⋊Dic7 | C22 | semidirect |
| 49. C4⋊Dic7 | C22 | semidirect |
| 50. Dic7⋊C4 | C22 | semidirect |
| 51. Dic7⋊C4 | C22 | semidirect |
| 52. C4×Dic7 | C22 | semidirect |
| 77. C22⋊D7 | D4 | semidirect |
| 101. #120 | D7 | semidirect |
| 118. C22×C7 | #41 | semidirect |
| 148. C2×D4 | C2×D7 | semidirect |
| 149. C2×D4 | C2×D7 | semidirect |
| 170. C22⋊C4 | C2×D7 | semidirect |
| 171. C22⋊C4 | C2×D7 | semidirect |
| 219. C7 | #546 | semidirect |
| 239. C22 | #1159 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊D7 | 2 |
| C2×#1159 | 2 |
| C7×#120 | 2 |
| C2×C22⋊Dic7 | 2 |
| #6502 | 2 |
| #6501 | 2 |
| #6500 | 2 |
| D4×Dic7 | 2 |
| #6449 | 2 |
| #6448 | 2 |
| #6446 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| D7×C22⋊C4 | 2 |
| #6441 | 2 |
| #546 | 7 |