Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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 | #537 |
| 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. C2×#1159 | C2 | semidirect |
| 2. C2×#1157 | C2 | semidirect |
| 3. C7×#121 | C2 | semidirect |
| 4. #6502 | C2 | semidirect |
| 5. #6501 | C2 | semidirect |
| 6. #6501 | C2 | semidirect |
| 7. #6487 | C2 | semidirect |
| 8. #6460 | C2 | semidirect |
| 9. D7×C4⋊C4 | C2 | semidirect |
| 10. #6447 | C2 | semidirect |
| 11. #6441 | C2 | semidirect |
| 60. C4×D7 | D4 | semidirect |
| 72. #121 | D7 | semidirect |
| 151. C7 | #537 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1159 | 2 |
| C2×#1157 | 2 |
| C7×#121 | 2 |
| #6502 | 2 |
| #6501 | 2 |
| #6487 | 2 |
| #6460 | 2 |
| D7×C4⋊C4 | 2 |
| #6447 | 2 |
| #6441 | 2 |
| #537 | 7 |