Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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 | C4×C7×Q8 | C2 |
| prime radical | C22×D7 | C23 |
| hypercenter | C4×Q8 | D7 |
| hypocenter | C7 | #541 |
| 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. C4×C7×Q8 | C2 | semidirect |
| 3. #6507 | C2 | semidirect |
| 6. #6453 | C2 | semidirect |
| 7. #6452 | C2 | semidirect |
| 10. #6433 | C2 | semidirect |
| 11. C4×Dic14 | C2 | semidirect |
| 59. C4×Q8 | D7 | semidirect |
| 116. C7 | #541 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×Q8 | 2 |
| #6509 | 2 |
| #6507 | 2 |
| #6461 | 2 |
| #6460 | 2 |
| #6453 | 2 |
| #6452 | 2 |
| #6439 | 2 |
| #6435 | 2 |
| #6433 | 2 |
| C4×Dic14 | 2 |
| #541 | 7 |