Finite Group DatabasebetaFinite Group DatabasebetaOrder 3402.
384 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C2×C3×C7 |
| Frattini | C32 | C3×C7×C3⋊D3 |
| Fitting | C7×C32⋊H(3,𝔽3) | C2 |
| socle | C32×C7 | C3×C3⋊D3 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C3×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C32⋊H(3,𝔽3) | C2 | semidirect |
| 3. C7×C3⋊#272 | C3 | semidirect |
| 5. C34×C7 | D3 | semidirect |
| 8. C3×C7×H(3,𝔽3) | D3 | semidirect |
| 9. C3×C7×H(3,𝔽3) | C2×C3 | semidirect |
| 10. #92300 | C7 | direct (N × H) |
| 18. C32⋊H(3,𝔽3) | C2×C7 | semidirect |
| 25. C33×C7 | C3×D3 | semidirect |
| 32. C3⋊#272 | C3×C7 | semidirect |
| 42. C34 | D3×C7 | semidirect |
| 45. C3×H(3,𝔽3) | D3×C7 | semidirect |
| 46. C3×H(3,𝔽3) | C2×C3×C7 | semidirect |
| 63. C32×C7 | (C3⋊D3)⋊C3 | semidirect |
| 64. C32×C7 | #272 | semidirect |
| 84. C33 | C3×D3×C7 | semidirect |
| 123. C32 | C7×(C3⋊D3)⋊C3 | semidirect |
| 124. C32 | C7×#272 | semidirect |
| 126. C7 | #92300 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C32⋊H(3,𝔽3) | 2 |
| C32×C7×C3⋊D3 | 3 |
| C7×C3⋊#272 | 3 |
| C3×C7×(C3⋊D3)⋊C3 | 3 |
| #92300 | 7 |