Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
188 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 50 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 |
| hypercenter | C23 | D3×D7 |
| hypocenter | C3×C7 | #121 |
| 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. C2×#4268 | C2 | semidirect |
| 2. C2×C7×#67 | C2 | semidirect |
| 3. C2×#4247 | C2 | semidirect |
| 4. C2×D21⋊C4 | C2 | semidirect |
| 5. C3×C22⋊Dic7 | C2 | semidirect |
| 6. Dic3⋊Dic7 | C2 | semidirect |
| 7. (C2×D3)⋊Dic7 | C2 | semidirect |
| 25. C22⋊Dic7 | D3 | semidirect |
| 34. C2×D21 | D4 | semidirect |
| 41. C7×Dic3 | D4 | semidirect |
| 54. C2×#67 | D7 | semidirect |
| 92. C3×C7 | #121 | semidirect |
| 119. Dic3 | C22⋊D7 | semidirect |
| 131. C7 | #884 | semidirect |
| 152. C3 | #6501 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4268 | 2 |
| C2×C7×#67 | 2 |
| C2×#4247 | 2 |
| C2×D21⋊C4 | 2 |
| C3×C22⋊Dic7 | 2 |
| Dic3⋊Dic7 | 2 |
| (C2×D3)⋊Dic7 | 2 |
| #6501 | 3 |
| #884 | 7 |