Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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 | #560 |
| 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. #6503 | C2 | semidirect |
| 3. #6500 | C2 | semidirect |
| 4. #6493 | C2 | semidirect |
| 5. #6461 | C2 | semidirect |
| 6. #6458 | C2 | semidirect |
| 7. #6447 | C2 | semidirect |
| 8. #6446 | C2 | semidirect |
| 9. #6445 | C2 | semidirect |
| 12. C22×C4×C7 | C22 | semidirect |
| 15. C2×C22⋊D7 | C22 | semidirect |
| 18. C2×D28 | C22 | semidirect |
| 19. C2×C4×D7 | C22 | semidirect |
| 20. C7×C4⋊C4 | C22 | semidirect |
| 22. C7×C22⋊C4 | C22 | semidirect |
| 27. (C2×D7)⋊C4 | C22 | semidirect |
| 28. C4⋊Dic7 | C22 | semidirect |
| 29. Dic7⋊C4 | C22 | semidirect |
| 30. C4×Dic7 | C22 | semidirect |
| 55. #123 | D7 | semidirect |
| 87. C22×C4 | C2×D7 | semidirect |
| 90. C4⋊C4 | C2×D7 | semidirect |
| 97. C22⋊C4 | C2×D7 | semidirect |
| 125. C7 | #560 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#123 | 2 |
| #6503 | 2 |
| #6500 | 2 |
| #6493 | 2 |
| #6461 | 2 |
| #6458 | 2 |
| #6447 | 2 |
| #6446 | 2 |
| #6445 | 2 |
| #560 | 7 |