Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
292 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 113 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 | C2×C7×#41 | C2 |
| hypercenter | C2×#41 | D7 |
| hypocenter | C7 | #547 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×#41 | C2 | semidirect |
| 2. C2×#1159 | C2 | semidirect |
| 3. #6509 | C2 | semidirect |
| 4. #6507 | C2 | semidirect |
| 5. #6502 | C2 | semidirect |
| 6. #6498 | C2 | semidirect |
| 7. D4×Dic7 | C2 | semidirect |
| 8. #6492 | C2 | semidirect |
| 9. C4×C22⋊D7 | C2 | semidirect |
| 10. #6487 | C2 | semidirect |
| 11. C2×Dic7⋊C4 | C2 | semidirect |
| 43. C7×D4 | D4 | semidirect |
| 61. C2×#41 | D7 | semidirect |
| 88. Dic7 | #41 | semidirect |
| 128. D4 | C22⋊D7 | semidirect |
| 142. C7 | #547 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×#41 | 2 |
| C2×#1159 | 2 |
| #6509 | 2 |
| #6507 | 2 |
| #6502 | 2 |
| #6498 | 2 |
| D4×Dic7 | 2 |
| #6492 | 2 |
| C4×C22⋊D7 | 2 |
| #6487 | 2 |
| C2×Dic7⋊C4 | 2 |
| #547 | 7 |