Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
144 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 43 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 | C2×C7×D8 | C2 |
| socle | C22×C7 | C2×D4 |
| hypercenter | C2×D8 | D7 |
| hypocenter | C7 | #496 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C7×D8 | C2 | semidirect |
| 2. #6503 | C2 | semidirect |
| 3. #6499 | C2 | semidirect |
| 4. C2×#1142 | C2 | semidirect |
| 5. C2×#1141 | C2 | semidirect |
| 6. C2×#1132 | C2 | semidirect |
| 7. #6389 | C2 | semidirect |
| 37. C7×C8 | D4 | semidirect |
| 38. C7⋊C8 | D4 | semidirect |
| 42. C2×D8 | D7 | semidirect |
| 91. C8 | C22⋊D7 | semidirect |
| 92. C7 | #496 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×D8 | 2 |
| #6503 | 2 |
| #6499 | 2 |
| C2×#1142 | 2 |
| C2×#1141 | 2 |
| C2×#1132 | 2 |
| #6389 | 2 |
| #496 | 7 |