Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 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 |
| prime radical | C2×C7×D4 | C22 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #563 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #6487 | C2 | semidirect |
| 5. #6452 | C2 | semidirect |
| 6. #6451 | C2 | semidirect |
| 7. #6442 | C2 | semidirect |
| 8. #6442 | C2 | semidirect |
| 9. #6441 | C2 | semidirect |
| 106. C7 | #563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#123 | 2 |
| #6499 | 2 |
| #6498 | 2 |
| #6487 | 2 |
| #6452 | 2 |
| #6451 | 2 |
| #6442 | 2 |
| #6441 | 2 |
| #563 | 7 |