Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
108 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6447 |
| commutator, cosocle | C2×C4×C7 | C23 |
| Frattini | C2×C4 | C22×D7 |
| Fitting | C7×#100 | C2 |
| socle | C2×C7 | #121 |
| hypercenter | #100 | D7 |
| hypocenter | C7 | #472 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #100 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6495 | C2 | semidirect |
| 2. C2×#1141 | C2 | semidirect |
| 3. #6471 | C2 | semidirect |
| 4. #6470 | C2 | semidirect |
| 5. C7×#100 | C2 | semidirect |
| 6. #6397 | C2 | semidirect |
| 7. #6396 | C2 | semidirect |
| 46. #100 | D7 | semidirect |
| 92. C7 | #472 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6495 | 2 |
| C2×#1141 | 2 |
| #6471 | 2 |
| #6470 | 2 |
| C7×#100 | 2 |
| #6397 | 2 |
| #6396 | 2 |
| #472 | 7 |