Finite Group DatabasebetaFinite Group DatabasebetaOrder 27712.
98 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#20294109 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 2 | C22×#16812365 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20294109 | C22 | 1 | 4 | yes | direct product factor |
| 4 | C2×#16812365 | C4 | 1 | 3 | yes | — |
| 5 | C2×#16812365 | C22 | 1 | 3 | yes | — |
| 6 | C22×Dic433 | C4 | 1 | 1 | yes | characteristic |
| 7 | #16812365 | C2×C4 | 1 | 3 | yes | — |
| 8 | #16812365 | C23 | 1 | 1 | yes | characteristic |
| 9 | C2×Dic433 | C8 | 1 | 3 | yes | — |
| 10 | C2×Dic433 | C2×C4 | 1 | 3 | yes | — |
| 11 | C23×C433 | C8 | 1 | 1 | yes | characteristic |
| 12 | C22×C433 | C16 | 1 | 4 | yes | complemented |
| 13 | C22×C433 | C2×C8 | 1 | 3 | yes | — |
| 14 | Dic433 | C2×C8 | 1 | 3 | yes | — |
| 15 | Dic433 | C22×C4 | 1 | 1 | yes | characteristic |
| 16 | C2×C433 | C2×C16 | 1 | 6 | yes | complemented |
| 17 | C2×C433 | C22×C8 | 1 | 1 | yes | characteristic |
| 18 | C433 | C22×C16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | C22×C16 | — | 433 | 1 | no | maximal, Sylow |
| 20 | C22×C8 | — | 433 | 1 | no | — |
| 21 | C2×C16 | — | 433 | 6 | no | — |
| 22 | C22×C4 | — | 433 | 1 | no | — |
| 23 | C2×C8 | — | 433 | 3 | no | — |
| 24 | C2×C8 | — | 433 | 3 | no | — |
| 25 | C16 | — | 433 | 4 | no | — |
| 26 | C23 | #16812367 | 1 | 1 | yes | characteristic |
| 27 | C2×C4 | — | 433 | 3 | no | — |
| 28 | C2×C4 | — | 433 | 3 | no | — |
| 29 | C8 | — | 433 | 1 | no | — |
| 30 | C8 | — | 433 | 3 | no | — |
| 31 | C22 | #20294109 | 1 | 4 | yes | direct product factor |
| 32 | C22 | C2×#16812367 | 1 | 3 | yes | — |
| 33 | C4 | — | 433 | 1 | no | — |
| 34 | C4 | — | 433 | 3 | no | — |
| 35 | C2 | C2×#20294109 | 1 | 6 | yes | direct product factor |
| 36 | C2 | C22×#16812367 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #16812367 |
| commutator, hypocenter | C433 | C22×C16 |
| Frattini | C2 | C22×#16812367 |
| Fitting, socle, prime radical | C23×C433 | C8 |
| cosocle | #16812365 | C23 |
| LCS | G ▷ C433 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#20294109 | C2 | direct (N × H) |
| 3. #20294109 | C22 | direct (N × H) |
| 12. C22×C433 | C16 | semidirect |
| 16. C2×C433 | C2×C16 | semidirect |
| 18. C433 | C22×C16 | semidirect |
| 31. C22 | #20294109 | direct (N × H) |
| 35. C2 | C2×#20294109 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20294109 | 2 |
| C22×#16812365 | 2 |
| C22×C16 | 433 |