Finite Group DatabasebetaFinite Group DatabasebetaOrder 2352.
136 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 36 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 | #87641 |
| commutator, cosocle | C4×C72 | C22×C3 |
| Frattini | C4 | C3×D72 |
| Fitting | C3×C4×C72 | C22 |
| socle | C2×C3×C72 | D4 |
| hypercenter | C3×C4 | D72 |
| hypocenter | C72 | C3×D8 |
| LCS | G ▷ C4×C72 ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C4⋊(C7⋊D7) | C2 | semidirect |
| 2. C3×C7×D28 | C2 | semidirect |
| 3. C3×C7×C7⋊C8 | C2 | semidirect |
| 4. #11716033 | C3 | direct (N × H) |
| 8. C4⋊(C7⋊D7) | C2×C3 | semidirect |
| 9. C7×D28 | C2×C3 | semidirect |
| 10. C7×C7⋊C8 | C2×C3 | semidirect |
| 23. C3×D28 | D7 | semidirect |
| 25. C3×C7⋊C8 | D7 | semidirect |
| 26. C3×C72 | D8 | semidirect |
| 44. D28 | C3×D7 | semidirect |
| 46. C7⋊C8 | C3×D7 | semidirect |
| 47. C72 | C3×D8 | semidirect |
| 69. C3×C7 | D56 | semidirect |
| 70. C3×C7 | #1141 | semidirect |
| 87. C7 | C3×D56 | semidirect |
| 88. C7 | C3×#1141 | semidirect |
| 95. C3 | #11716033 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C4⋊(C7⋊D7) | 2 |
| C3×C7×D28 | 2 |
| C3×C7×C7⋊C8 | 2 |
| #11716033 | 3 |
| C3×#1141 | 7 |
| C3×D56 | 7 |