Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D4×D7 |
| commutator, cosocle | C4×C7 | C24 |
| Frattini | C4 | C23×D7 |
| Fitting | C2×C7×Q16 | C2 |
| socle | C2×C7 | C22×D4 |
| hypercenter | C2×Q16 | D7 |
| hypocenter | C7 | #578 |
| LCS | G ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×Q16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6551 | C2 | semidirect |
| 2. C2×C7×Q16 | C2 | semidirect |
| 3. #6505 | C2 | semidirect |
| 4. #6482 | C2 | semidirect |
| 5. #6481 | C2 | semidirect |
| 6. D7×Q16 | C2 | semidirect |
| 7. #6467 | C2 | semidirect |
| 8. #6464 | C2 | semidirect |
| 54. C2×Q16 | D7 | semidirect |
| 106. C7 | #578 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6551 | 2 |
| C2×C7×Q16 | 2 |
| #6505 | 2 |
| #6482 | 2 |
| #6481 | 2 |
| D7×Q16 | 2 |
| #6467 | 2 |
| #6464 | 2 |
| #578 | 7 |