Finite Group DatabasebetaFinite Group DatabasebetaOrder 19780.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D5×C23×C43 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×C23×C43 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | C4×C23×C43 | — | 5 | 1 | no | maximal, Hall |
| 4 | C2×C23×C43 | — | 5 | 1 | no | — |
| 5 | C23×C43 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 6 | AGL(1,𝔽5)×C43 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 7 | AGL(1,𝔽5)×C23 | C43 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 8 | D5×C43 | C2×C23 | 1 | 1 | yes | characteristic |
| 9 | D5×C23 | C2×C43 | 1 | 1 | yes | characteristic |
| 10 | C5×C43 | C4×C23 | 1 | 1 | yes | characteristic, complemented, Hall |
| 11 | C4×C43 | — | 5 | 1 | no | Hall |
| 12 | C5×C23 | C4×C43 | 1 | 1 | yes | characteristic, complemented, Hall |
| 13 | C4×C23 | — | 5 | 1 | no | Hall |
| 14 | C2×C43 | — | 5 | 1 | no | — |
| 15 | C2×C23 | — | 5 | 1 | no | — |
| 16 | C43 | AGL(1,𝔽5)×C23 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 17 | C23 | AGL(1,𝔽5)×C43 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 18 | AGL(1,𝔽5) | C23×C43 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 19 | D5 | C2×C23×C43 | 1 | 1 | yes | characteristic |
| 20 | C5 | C4×C23×C43 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 21 | C4 | — | 5 | 1 | no | Sylow |
| 22 | C2 | — | 5 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C43 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C4×C23×C43 |
| Frattini | C1 | G |
| Fitting, socle | C5×C23×C43 | C4 |
| prime radical | D5×C23×C43 | C2 |
| cosocle | D5 | C2×C23×C43 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C43 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×C23×C43 | C4 | semidirect |
| 5. C23×C43 | AGL(1,𝔽5) | direct (N × H) |
| 6. AGL(1,𝔽5)×C43 | C23 | direct (N × H) |
| 7. AGL(1,𝔽5)×C23 | C43 | direct (N × H) |
| 10. C5×C43 | C4×C23 | semidirect |
| 12. C5×C23 | C4×C43 | semidirect |
| 16. C43 | AGL(1,𝔽5)×C23 | direct (N × H) |
| 17. C23 | AGL(1,𝔽5)×C43 | direct (N × H) |
| 18. AGL(1,𝔽5) | C23×C43 | direct (N × H) |
| 20. C5 | C4×C23×C43 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5×C23×C43 | 2 |
| C4×C23×C43 | 5 |
| AGL(1,𝔽5)×C43 | 23 |
| AGL(1,𝔽5)×C23 | 43 |