Finite Group DatabasebetaFinite Group DatabasebetaOrder 8540.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×D305 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×C7×C61 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | C7⋊#6923 | — | 5 | 1 | no | maximal, Hall |
| 4 | #11927413 | — | 7 | 1 | no | maximal, Hall |
| 5 | C7×D61 | — | 5 | 1 | no | — |
| 6 | D305 | D7 | 1 | 1 | yes | characteristic |
| 7 | C7×C61 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Hall |
| 8 | C5×C61 | Dic7 | 1 | 1 | yes | characteristic, complemented, Hall |
| 9 | #6923 | — | 35 | 1 | no | Hall |
| 10 | C7⋊AGL(1,𝔽5) | — | 61 | 1 | no | maximal, Hall |
| 11 | D61 | — | 5 | 1 | no | — |
| 12 | C7×D5 | — | 61 | 1 | no | — |
| 13 | C61 | C7⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C5×C7 | #6923 | 1 | 1 | yes | characteristic, complemented, Hall |
| 15 | Dic7 | — | 305 | 1 | no | Hall |
| 16 | AGL(1,𝔽5) | — | 427 | 1 | no | Hall |
| 17 | C2×C7 | — | 305 | 1 | no | — |
| 18 | D5 | — | 61 | 1 | no | — |
| 19 | C7 | #11927413 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 20 | C5 | C7⋊#6923 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C4 | — | 2135 | 1 | no | Sylow |
| 22 | C2 | — | 305 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C7×C61 | C4 |
| LCS | G ▷ C5×C7×C61 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×C7×C61 | C4 | semidirect |
| 7. C7×C61 | AGL(1,𝔽5) | semidirect |
| 8. C5×C61 | Dic7 | semidirect |
| 13. C61 | C7⋊AGL(1,𝔽5) | semidirect |
| 14. C5×C7 | #6923 | semidirect |
| 19. C7 | #11927413 | semidirect |
| 20. C5 | C7⋊#6923 | semidirect |