Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D72 |
| commutator, cosocle | C2×C72 | C23×C3 |
| Frattini | C2 | C2×C3×D72 |
| Fitting | C22×C3×C72 | C22 |
| socle | C2×C3×C72 | C23 |
| hypercenter | C22×C3 | D72 |
| hypocenter | C72 | C3×#41 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C22⋊(C7⋊D7) | C2 | semidirect |
| 2. C3×C7×C22⋊D7 | C2 | semidirect |
| 3. C2×C3×C7×Dic7 | C2 | semidirect |
| 4. C3×#87642 | C2 | semidirect |
| 5. C3×#87641 | C2 | semidirect |
| 6. C3×#87639 | C2 | semidirect |
| 7. C3×D7×Dic7 | C2 | semidirect |
| 8. #11716108 | C3 | direct (N × H) |
| 16. C22⋊(C7⋊D7) | C2×C3 | semidirect |
| 17. C7×C22⋊D7 | C2×C3 | semidirect |
| 18. C2×C7×Dic7 | C2×C3 | semidirect |
| 19. #87642 | C2×C3 | semidirect |
| 20. #87641 | C2×C3 | semidirect |
| 21. #87639 | C2×C3 | semidirect |
| 22. D7×Dic7 | C2×C3 | semidirect |
| 42. C3×C22⋊D7 | D7 | semidirect |
| 44. C2×C3×Dic7 | D7 | semidirect |
| 51. C3×C72 | #41 | semidirect |
| 81. C22⋊D7 | C3×D7 | semidirect |
| 83. C2×Dic7 | C3×D7 | semidirect |
| 90. C72 | C3×#41 | semidirect |
| 128. C3×C7 | #1157 | semidirect |
| 129. C3×C7 | #1159 | semidirect |
| 157. C7 | C3×#1157 | semidirect |
| 158. C7 | C3×#1159 | semidirect |
| 170. C3 | #11716108 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C22⋊(C7⋊D7) | 2 |
| C3×C7×C22⋊D7 | 2 |
| C2×C3×C7×Dic7 | 2 |
| C3×#87642 | 2 |
| C3×#87641 | 2 |
| C3×#87639 | 2 |
| C3×D7×Dic7 | 2 |
| #11716108 | 3 |
| C3×#1159 | 7 |
| C3×#1157 | 7 |