Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
238 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 99 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×D4 | C2 |
| hypercenter | C4×D4 | D7 |
| hypocenter | C7 | #553 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×C7×D4 | C2 | semidirect |
| 2. #6502 | C2 | semidirect |
| 3. #6493 | C2 | semidirect |
| 4. #6493 | C2 | semidirect |
| 5. #6489 | C2 | semidirect |
| 6. C2×C4⋊Dic7 | C2 | semidirect |
| 7. #6461 | C2 | semidirect |
| 8. #6446 | C2 | semidirect |
| 9. (C22⋊D7)⋊C4 | C2 | semidirect |
| 10. C4×D28 | C2 | semidirect |
| 11. #6433 | C2 | semidirect |
| 63. C4×D4 | D7 | semidirect |
| 132. C7 | #553 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C7×D4 | 2 |
| #6502 | 2 |
| #6493 | 2 |
| #6489 | 2 |
| C2×C4⋊Dic7 | 2 |
| #6461 | 2 |
| #6446 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| C4×D28 | 2 |
| #6433 | 2 |
| #553 | 7 |