Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
156 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C23 |
| Fitting | C23×C3×C7 | C22 |
| prime radical | C2×C7×#67 | C2 |
| hypercenter | C23 | D3×D7 |
| hypocenter | C3×C7 | #123 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Dic21 | C2 | semidirect |
| 3. C3×C22⋊Dic7 | C2 | semidirect |
| 4. Dic7⋊Dic3 | C2 | semidirect |
| 5. (C2×D3)⋊Dic7 | C2 | semidirect |
| 17. C22⋊Dic7 | D3 | semidirect |
| 60. C3×C7 | #123 | semidirect |
| 85. C7 | #886 | semidirect |
| 98. C3 | #6498 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Dic21 | 2 |
| C2×C7×#67 | 2 |
| C3×C22⋊Dic7 | 2 |
| Dic7⋊Dic3 | 2 |
| (C2×D3)⋊Dic7 | 2 |
| #6498 | 3 |
| #886 | 7 |