Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 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×#123 | C2 |
| prime radical | C2×#1159 | C2 |
| hypercenter | #123 | D7 |
| hypocenter | C7 | #536 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Dic14 | C2 | semidirect |
| 3. C7×#123 | C2 | semidirect |
| 4. #6498 | C2 | semidirect |
| 5. #6492 | C2 | semidirect |
| 6. #6459 | C2 | semidirect |
| 7. #6451 | C2 | semidirect |
| 8. #6449 | C2 | semidirect |
| 9. #6448 | C2 | semidirect |
| 10. #6441 | C2 | semidirect |
| 11. #6440 | C2 | semidirect |
| 67. #123 | D7 | semidirect |
| 138. C7 | #536 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1159 | 2 |
| C22×Dic14 | 2 |
| C7×#123 | 2 |
| #6498 | 2 |
| #6492 | 2 |
| #6459 | 2 |
| #6451 | 2 |
| #6449 | 2 |
| #6448 | 2 |
| #6441 | 2 |
| #6440 | 2 |
| #536 | 7 |