Finite Group DatabasebetaFinite Group DatabasebetaOrder 26828.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C19×D353 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C19×C353 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | #13083443 | C19 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | D353 | C2×C19 | 1 | 1 | yes | characteristic |
| 5 | C353 | C4×C19 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C4×C19 | — | 353 | 1 | no | maximal, Hall |
| 7 | C2×C19 | — | 353 | 1 | no | — |
| 8 | C19 | #13083443 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C4 | — | 353 | 1 | no | Sylow |
| 10 | C2 | — | 353 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C19 | #13083443 |
| commutator, hypocenter | C353 | C4×C19 |
| Frattini | C1 | G |
| Fitting, socle | C19×C353 | C4 |
| prime radical | C19×D353 | C2 |
| cosocle | D353 | C2×C19 |
| LCS | G ▷ C353 | |
| UCS | C1 ◁ C19 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C19×C353 | C4 | semidirect |
| 3. #13083443 | C19 | direct (N × H) |
| 5. C353 | C4×C19 | semidirect |
| 8. C19 | #13083443 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C19×D353 | 2 |
| #13083443 | 19 |
| C4×C19 | 353 |