Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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×C4⋊Q8 | C2 |
| prime radical | C22×D7 | C23 |
| hypercenter | C4⋊Q8 | D7 |
| hypocenter | C7 | #533 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C4⋊Q8 | C2 | semidirect |
| 3. Q8×Dic7 | C2 | semidirect |
| 6. #6453 | C2 | semidirect |
| 7. #6450 | C2 | semidirect |
| 9. #6432 | C2 | semidirect |
| 40. C4×D7 | Q8 | semidirect |
| 43. C4⋊Q8 | D7 | semidirect |
| 98. C7 | #533 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C4⋊Q8 | 2 |
| #6509 | 2 |
| Q8×Dic7 | 2 |
| #6460 | 2 |
| #6455 | 2 |
| #6453 | 2 |
| #6450 | 2 |
| C42×D7 | 2 |
| #6432 | 2 |
| #533 | 7 |