Finite Group DatabasebetaFinite Group DatabasebetaOrder 8228.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C17×C121 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C17×C121 | C4 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 3 | C4×C11×C17 | C11 | 1 | 1 | yes | maximal, characteristic |
| 4 | C4×C121 | C17 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C2×C11×C17 | C2×C11 | 1 | 1 | yes | characteristic |
| 6 | C2×C121 | C2×C17 | 1 | 1 | yes | characteristic |
| 7 | C11×C17 | C4×C11 | 1 | 1 | yes | characteristic |
| 8 | C121 | C4×C17 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C4×C17 | C121 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 10 | C4×C11 | C11×C17 | 1 | 1 | yes | characteristic |
| 11 | C2×C17 | C2×C121 | 1 | 1 | yes | characteristic |
| 12 | C2×C11 | C2×C11×C17 | 1 | 1 | yes | characteristic |
| 13 | C17 | C4×C121 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C11 | C4×C11×C17 | 1 | 1 | yes | characteristic |
| 15 | C4 | C17×C121 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 16 | C2 | C2×C17×C121 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini | C2×C11 | C2×C11×C17 |
| socle | C2×C11×C17 | C2×C11 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C17×C121 | C4 | direct (N × H) |
| 4. C4×C121 | C17 | direct (N × H) |
| 8. C121 | C4×C17 | direct (N × H) |
| 9. C4×C17 | C121 | direct (N × H) |
| 13. C17 | C4×C121 | direct (N × H) |
| 15. C4 | C17×C121 | direct (N × H) |