Finite Group DatabasebetaFinite Group Databasebeta← GL(2,𝔽3)×C7⋊AGL(1,𝔽5) #72218823
Order 6720.
352 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 41 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. #72218633 | C2 | semidirect |
| 4. SL(2,𝔽3)×C7⋊AGL(1,𝔽5) | C2 | semidirect |
| 7. C7×C5⋊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 |
| 53. C7×GL(2,𝔽3) | AGL(1,𝔽5) | semidirect |
| 60. C5×C7×Q8 | C4×D3 | semidirect |
| 71. C5⋊GL(2,𝔽3) | Dic7 | semidirect |
| 72. C5×GL(2,𝔽3) | Dic7 | semidirect |
| 89. C7×SL(2,𝔽3) | C2×AGL(1,𝔽5) | semidirect |
| 105. C7⋊AGL(1,𝔽5) | GL(2,𝔽3) | direct (N × H) |
| 112. C5×SL(2,𝔽3) | C2×Dic7 | semidirect |
| 165. C7×Q8 | D3×AGL(1,𝔽5) | semidirect |
| 180. GL(2,𝔽3) | C7⋊AGL(1,𝔽5) | direct (N × H) |
| 191. C5×Q8 | D3×Dic7 | semidirect |
| 200. C5×C7 | C4×GL(2,𝔽3) | semidirect |
| 227. SL(2,𝔽3) | C2×C7⋊AGL(1,𝔽5) | semidirect |
| 272. Q8 | D3×C7⋊AGL(1,𝔽5) | semidirect |
| 284. C7 | AGL(1,𝔽5)×GL(2,𝔽3) | semidirect |
| 290. C5 | Dic7×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72218633 | 2 |
| SD16×C7⋊AGL(1,𝔽5) | 3 |
| C7×D5×GL(2,𝔽3) | 2 |
| SL(2,𝔽3)×C7⋊AGL(1,𝔽5) | 2 |
| C2×D3×C7⋊AGL(1,𝔽5) | 4 |
| Dic7×GL(2,𝔽3) | 5 |
| AGL(1,𝔽5)×GL(2,𝔽3) | 7 |