Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
248 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C22⋊D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C7×2−1+4 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | 2−1+4 | D7 |
| hypocenter | C7 | #578 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×2−1+4 | C2 | semidirect |
| 2. #6554 | C2 | semidirect |
| 3. #6514 | C2 | semidirect |
| 4. #6513 | C2 | semidirect |
| 5. #6511 | C2 | semidirect |
| 6. C2×#1144 | C2 | semidirect |
| 7. #6505 | C2 | semidirect |
| 43. 2−1+4 | D7 | semidirect |
| 93. C7 | #578 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×2−1+4 | 2 |
| #6554 | 2 |
| #6514 | 2 |
| #6513 | 2 |
| #6511 | 2 |
| C2×#1144 | 2 |
| #6505 | 2 |
| #578 | 7 |