Finite Group DatabasebetaFinite Group DatabasebetaOrder 26588.
18 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C23×C289 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C23×C289 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | C23×Dic17 | — | 17 | 1 | no | maximal |
| 4 | Dic289 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 5 | C2×C17×C23 | D17 | 1 | 1 | yes | characteristic |
| 6 | C2×C289 | C2×C23 | 1 | 1 | yes | characteristic |
| 7 | C17×C23 | Dic17 | 1 | 1 | yes | characteristic |
| 8 | C289 | C4×C23 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C4×C23 | — | 289 | 1 | no | Hall |
| 10 | Dic17 | — | 17 | 1 | no | — |
| 11 | C2×C23 | D289 | 1 | 1 | yes | characteristic |
| 12 | C2×C17 | C23×D17 | 1 | 1 | yes | characteristic |
| 13 | C23 | Dic289 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C17 | C23×Dic17 | 1 | 1 | yes | characteristic |
| 15 | C4 | — | 289 | 1 | no | Sylow |
| 16 | C2 | C23×D289 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C23 | D289 |
| commutator, hypocenter | C289 | C4×C23 |
| Frattini | C2×C17 | C23×D17 |
| Fitting | C2×C23×C289 | C2 |
| socle, prime radical | C2×C17×C23 | D17 |
| cosocle | C2×C289 | C2×C23 |
| LCS | G ▷ C289 | |
| UCS | C1 ◁ C2×C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C289 | C4 | semidirect |
| 4. Dic289 | C23 | direct (N × H) |
| 8. C289 | C4×C23 | semidirect |
| 13. C23 | Dic289 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C23×C289 | 2 |
| C23×Dic17 | 17 |
| Dic289 | 23 |