Finite Group DatabasebetaFinite Group DatabasebetaOrder 12340.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D3085 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×C617 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | #14578674 | — | 5 | 1 | no | maximal, Hall |
| 4 | D617 | — | 5 | 1 | no | — |
| 5 | C617 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | AGL(1,𝔽5) | — | 617 | 1 | no | maximal, Hall |
| 7 | D5 | — | 617 | 1 | no | — |
| 8 | C5 | #14578674 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C4 | — | 3085 | 1 | no | Sylow |
| 10 | C2 | — | 3085 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C617 | C4 |
| prime radical, cosocle | D3085 | C2 |
| LCS | G ▷ C5×C617 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×C617 | C4 | semidirect |
| 5. C617 | AGL(1,𝔽5) | semidirect |
| 8. C5 | #14578674 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3085 | 2 |
| #14578674 | 5 |
| AGL(1,𝔽5) | 617 |