Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 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 | C2×C22⋊D7 | C22 |
| hypercenter | C22⋊Q8 | D7 |
| hypocenter | C7 | #554 |
| 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 |
| 3. #6507 | C2 | semidirect |
| 5. C2×Dic7⋊C4 | C2 | semidirect |
| 6. #6459 | C2 | semidirect |
| 7. #6459 | C2 | semidirect |
| 8. D7×C4⋊C4 | C2 | semidirect |
| 9. #6452 | C2 | semidirect |
| 10. #6452 | C2 | semidirect |
| 11. #6450 | C2 | semidirect |
| 14. #6441 | C2 | semidirect |
| 15. #6441 | C2 | semidirect |
| 62. C22⋊D7 | Q8 | semidirect |
| 85. C22⋊Q8 | D7 | semidirect |
| 177. C7 | #554 | 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 |
| #6507 | 2 |
| C4×C22⋊D7 | 2 |
| C2×Dic7⋊C4 | 2 |
| #6459 | 2 |
| D7×C4⋊C4 | 2 |
| #6452 | 2 |
| #6450 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| #6441 | 2 |
| #554 | 7 |