Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
140 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 140 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 | #459 |
| 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. D4×Dic7 | C2 | semidirect |
| 4. C2×#1141 | C2 | semidirect |
| 5. #6406 | C2 | semidirect |
| 6. #6395 | C2 | semidirect |
| 7. Dic7⋊C8 | C2 | semidirect |
| 28. C7×D4 | D4 | semidirect |
| 41. C2×D8 | D7 | semidirect |
| 61. Dic7 | D8 | semidirect |
| 91. D4 | C22⋊D7 | semidirect |
| 99. C7 | #459 | 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 |
| D4×Dic7 | 2 |
| C2×#1141 | 2 |
| #6406 | 2 |
| #6395 | 2 |
| Dic7⋊C8 | 2 |
| #459 | 7 |