Finite Group DatabasebetaFinite Group DatabasebetaOrder 6720.
352 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#91886 | C2 | semidirect |
| 4. SL(2,𝔽3)×C7⋊AGL(1,𝔽5) | C2 | semidirect |
| 7. C5⋊(C7⋊GL(2,𝔽3)) | C4 | semidirect |
| 8. C5×C7⋊GL(2,𝔽3) | C4 | semidirect |
| 10. Q8×C7⋊AGL(1,𝔽5) | D3 | semidirect |
| 22. C5×C7×SL(2,𝔽3) | C2×C4 | semidirect |
| 41. #91886 | D7 | semidirect |
| 53. C7⋊GL(2,𝔽3) | AGL(1,𝔽5) | semidirect |
| 60. C5×C7×Q8 | C4×D3 | semidirect |
| 89. C7×SL(2,𝔽3) | C2×AGL(1,𝔽5) | semidirect |
| 104. C7⋊AGL(1,𝔽5) | GL(2,𝔽3) | semidirect |
| 112. C5×SL(2,𝔽3) | C4×D7 | semidirect |
| 165. C7×Q8 | D3×AGL(1,𝔽5) | semidirect |
| 191. C5×Q8 | D21⋊C4 | semidirect |
| 200. C5×C7 | C4×GL(2,𝔽3) | semidirect |
| 227. SL(2,𝔽3) | D7×AGL(1,𝔽5) | semidirect |
| 272. Q8 | D21⋊AGL(1,𝔽5) | semidirect |
| 284. C7 | AGL(1,𝔽5)×GL(2,𝔽3) | semidirect |
| 290. C5 | #13057073 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#91886 | 2 |
| D5×C7⋊GL(2,𝔽3) | 2 |
| #72218287 | 3 |
| SL(2,𝔽3)×C7⋊AGL(1,𝔽5) | 2 |
| C2×D21⋊AGL(1,𝔽5) | 4 |
| #13057073 | 5 |
| AGL(1,𝔽5)×GL(2,𝔽3) | 7 |