Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
140 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D7 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#102 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | #102 | D7 |
| hypocenter | C7 | #459 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #102 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6503 | C2 | semidirect |
| 2. C2×#1141 | C2 | semidirect |
| 3. C2×D56 | C2 | semidirect |
| 4. #6456 | C2 | semidirect |
| 5. C7×#102 | C2 | semidirect |
| 6. Dic7⋊C8 | C2 | semidirect |
| 7. #6383 | C2 | semidirect |
| 36. D28 | D4 | semidirect |
| 46. #102 | D7 | semidirect |
| 61. Dic7 | D8 | semidirect |
| 99. C7 | #459 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6503 | 2 |
| C2×#1141 | 2 |
| C2×D56 | 2 |
| #6456 | 2 |
| C7×#102 | 2 |
| Dic7⋊C8 | 2 |
| #6383 | 2 |
| #459 | 7 |