Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#127 | C2 |
| hypercenter | #127 | D7 |
| hypocenter | C7 | #550 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1159 | C2 | semidirect |
| 2. C7×#127 | C2 | semidirect |
| 3. #6503 | C2 | semidirect |
| 4. #6502 | C2 | semidirect |
| 5. #6501 | C2 | semidirect |
| 6. D4×Dic7 | C2 | semidirect |
| 7. C4×D28 | C2 | semidirect |
| 8. C4×Dic14 | C2 | semidirect |
| 43. Dic14 | D4 | semidirect |
| 46. #127 | D7 | semidirect |
| 61. C4×C7 | #41 | semidirect |
| 104. C7 | #550 | semidirect |
| 117. C4 | #1159 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1159 | 2 |
| C7×#127 | 2 |
| #6503 | 2 |
| #6502 | 2 |
| #6501 | 2 |
| D4×Dic7 | 2 |
| C4×D28 | 2 |
| C4×Dic14 | 2 |
| #550 | 7 |