Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 97 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×#121 | C2 |
| hypercenter | #121 | D7 |
| hypocenter | C7 | #553 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#121 | C2 | semidirect |
| 2. #6501 | C2 | semidirect |
| 3. #6501 | C2 | semidirect |
| 4. #6498 | C2 | semidirect |
| 5. D4×Dic7 | C2 | semidirect |
| 6. C4×C22⋊D7 | C2 | semidirect |
| 7. #6489 | C2 | semidirect |
| 8. D7×C4⋊C4 | C2 | semidirect |
| 9. #6453 | C2 | semidirect |
| 10. (C22⋊D7)⋊C4 | C2 | semidirect |
| 11. #6442 | C2 | semidirect |
| 60. #121 | 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 |
|---|---|
| C7×#121 | 2 |
| #6501 | 2 |
| #6498 | 2 |
| D4×Dic7 | 2 |
| C4×C22⋊D7 | 2 |
| #6489 | 2 |
| D7×C4⋊C4 | 2 |
| #6453 | 2 |
| (C22⋊D7)⋊C4 | 2 |
| #6442 | 2 |
| #553 | 7 |