Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
242 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 103 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×C22⋊Q8 | C2 |
| prime radical | C23×D7 | C22 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | #543 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C22⋊Q8 | C2 | semidirect |
| 2. #6509 | C2 | semidirect |
| 4. #6487 | C2 | semidirect |
| 5. #6460 | C2 | semidirect |
| 6. #6459 | C2 | semidirect |
| 7. #6459 | C2 | semidirect |
| 9. #6441 | C2 | semidirect |
| 11. C2×C7×Q8 | C22 | semidirect |
| 16. C2×Dic14 | C22 | semidirect |
| 17. C2×Dic14 | C22 | semidirect |
| 18. C7×C4⋊C4 | C22 | semidirect |
| 19. C7×C4⋊C4 | C22 | semidirect |
| 26. C4⋊Dic7 | C22 | semidirect |
| 27. C4⋊Dic7 | C22 | semidirect |
| 28. Dic7⋊C4 | C22 | semidirect |
| 29. Dic7⋊C4 | C22 | semidirect |
| 30. Dic7⋊C4 | C22 | semidirect |
| 35. C22×D7 | Q8 | semidirect |
| 57. C22⋊Q8 | D7 | semidirect |
| 78. C2×Q8 | C2×D7 | semidirect |
| 88. C4⋊C4 | C2×D7 | semidirect |
| 89. C4⋊C4 | C2×D7 | semidirect |
| 120. C7 | #543 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C22⋊Q8 | 2 |
| #6509 | 2 |
| C2×(C2×D7)⋊C4 | 2 |
| #6487 | 2 |
| #6460 | 2 |
| #6459 | 2 |
| D7×C22⋊C4 | 2 |
| #6441 | 2 |
| #543 | 7 |