Finite Group DatabasebetaFinite Group DatabasebetaOrder 23148.
28 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22⋊(C643⋊C3) | C3 | 1 | 3 | yes | maximal, direct product factor |
| 2 | C22×C3×C643 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×C643⋊C3 | — | 4 | 1 | no | maximal, Hall |
| 4 | C2×C3×C643 | — | 3 | 1 | no | — |
| 5 | C22×C643 | C32 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C3×C643 | A4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C643⋊C3 | — | 4 | 3 | no | — |
| 8 | C2×C643 | — | 3 | 1 | no | — |
| 9 | C643 | C3×A4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C3×A4 | — | 643 | 1 | no | maximal, Hall |
| 11 | C22×C3 | C643⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 12 | A4 | — | 643 | 3 | no | — |
| 13 | C32 | — | 2572 | 1 | no | Sylow |
| 14 | C2×C3 | — | 3 | 1 | no | — |
| 15 | C22 | C3×C643⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | C3 | — | 2572 | 3 | no | — |
| 17 | C3 | C22⋊(C643⋊C3) | 1 | 1 | yes | characteristic, direct product factor |
| 18 | C2 | — | 3 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C22⋊(C643⋊C3) |
| commutator, cosocle, hypocenter | C22×C643 | C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C643 | C3 |
| LCS | G ▷ C22×C643 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22⋊(C643⋊C3) | C3 | direct (N × H) |
| 2. C22×C3×C643 | C3 | semidirect |
| 5. C22×C643 | C32 | semidirect |
| 6. C3×C643 | A4 | semidirect |
| 9. C643 | C3×A4 | semidirect |
| 11. C22×C3 | C643⋊C3 | semidirect |
| 15. C22 | C3×C643⋊C3 | semidirect |
| 17. C3 | C22⋊(C643⋊C3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22⋊(C643⋊C3) | 3 |
| C22×C3×C643 | 3 |
| C3×C643⋊C3 | 4 |
| C3×A4 | 643 |