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 | C2×C7×C22⋊C4 | C2 |
| hypercenter | C2×C22⋊C4 | D7 |
| hypocenter | C7 | #546 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊D7 | C2 | semidirect |
| 2. C2×#1157 | C2 | semidirect |
| 3. C2×C7×C22⋊C4 | C2 | semidirect |
| 4. #6516 | C2 | semidirect |
| 5. #6493 | C2 | semidirect |
| 6. #6492 | C2 | semidirect |
| 7. C4×C22⋊D7 | C2 | semidirect |
| 8. #6448 | C2 | semidirect |
| 9. #6447 | C2 | semidirect |
| 10. #6447 | C2 | semidirect |
| 11. #6446 | C2 | semidirect |
| 12. #6445 | C2 | semidirect |
| 13. (C22⋊D7)⋊C4 | C2 | semidirect |
| 14. D7×C22⋊C4 | C2 | semidirect |
| 15. #6441 | C2 | semidirect |
| 18. C22×C4×C7 | C22 | semidirect |
| 19. C22×C4×C7 | C22 | semidirect |
| 26. C2×C22⋊D7 | C22 | semidirect |
| 27. C2×C22⋊D7 | C22 | semidirect |
| 30. C2×D28 | C22 | semidirect |
| 31. C2×D28 | C22 | semidirect |
| 32. C2×C4×D7 | C22 | semidirect |
| 33. C2×C4×D7 | C22 | semidirect |
| 34. C2×C4×D7 | C22 | semidirect |
| 35. C2×Dic14 | C22 | semidirect |
| 38. C7×C22⋊C4 | C22 | semidirect |
| 39. C7×C22⋊C4 | C22 | semidirect |
| 41. C22⋊Dic7 | C22 | semidirect |
| 42. C22⋊Dic7 | C22 | semidirect |
| 43. (C2×D7)⋊C4 | 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. Dic7⋊C4 | C22 | semidirect |
| 50. Dic7⋊C4 | C22 | semidirect |
| 51. Dic7⋊C4 | C22 | semidirect |
| 52. C4×Dic7 | C22 | semidirect |
| 78. C22⋊D7 | D4 | semidirect |
| 105. C2×C22⋊C4 | D7 | semidirect |
| 115. C22×C7 | #41 | semidirect |
| 154. C22×C4 | C2×D7 | semidirect |
| 155. C22×C4 | C2×D7 | semidirect |
| 170. C22⋊C4 | C2×D7 | semidirect |
| 171. C22⋊C4 | C2×D7 | semidirect |
| 219. C7 | #546 | semidirect |
| 238. C22 | #1157 | 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×#1157 | 2 |
| C2×C7×C22⋊C4 | 2 |
| #6516 | 2 |
| #6493 | 2 |
| #6492 | 2 |
| C4×C22⋊D7 | 2 |
| #6448 | 2 |
| #6447 | 2 |
| #6446 | 2 |
| #6445 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| D7×C22⋊C4 | 2 |
| #6441 | 2 |
| #546 | 7 |