Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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 | C22×C7×D4 | C2 |
| hypercenter | C22×D4 | D7 |
| hypocenter | C7 | #535 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C7×D4 | C2 | semidirect |
| 2. C2×#1157 | C2 | semidirect |
| 3. #6516 | C2 | semidirect |
| 4. #6503 | C2 | semidirect |
| 5. #6501 | C2 | semidirect |
| 6. #6499 | C2 | semidirect |
| 7. #6489 | C2 | semidirect |
| 14. C2×C22⋊D7 | C22 | semidirect |
| 16. C2×D28 | C22 | semidirect |
| 17. C2×C4×D7 | C22 | semidirect |
| 18. C2×Dic14 | C22 | semidirect |
| 20. C22⋊Dic7 | C22 | semidirect |
| 21. C4⋊Dic7 | C22 | semidirect |
| 22. C4×Dic7 | C22 | semidirect |
| 31. C2×C4×C7 | D4 | semidirect |
| 42. C22×D4 | D7 | semidirect |
| 94. C2×C4 | C22⋊D7 | semidirect |
| 97. C7 | #535 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×D4 | 2 |
| C2×#1157 | 2 |
| #6516 | 2 |
| #6503 | 2 |
| #6501 | 2 |
| #6499 | 2 |
| #6489 | 2 |
| #535 | 7 |