Finite Group DatabasebetaFinite Group DatabasebetaOrder 34006.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C49×C347 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | C7×D347 | C7 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7×C347 | C2×C7 | 1 | 1 | yes | characteristic |
| 4 | D347 | C49 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 5 | C347 | C2×C49 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C2×C49 | — | 347 | 1 | no | maximal, Hall |
| 7 | C49 | D347 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 8 | C2×C7 | — | 347 | 1 | no | — |
| 9 | C7 | C7×D347 | 1 | 1 | yes | characteristic |
| 10 | C2 | — | 347 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C49 | D347 |
| commutator, hypocenter | C347 | C2×C49 |
| Frattini | C7 | C7×D347 |
| Fitting | C49×C347 | C2 |
| socle, cosocle | C7×C347 | C2×C7 |
| prime radical | C7×D347 | C7 |
| LCS | G ▷ C347 | |
| UCS | C1 ◁ C49 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C49×C347 | C2 | semidirect |
| 4. D347 | C49 | direct (N × H) |
| 5. C347 | C2×C49 | semidirect |
| 7. C49 | D347 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C49×C347 | 2 |
| C7×D347 | 7 |
| C2×C49 | 347 |