Finite Group DatabasebetaFinite Group DatabasebetaOrder 19698.
24 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #23754415 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D3283 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | C49×C67 | C2×C3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 4 | #14584203 | — | 7 | 1 | no | maximal |
| 5 | #13064105 | — | 7 | 1 | no | — |
| 6 | D469 | — | 21 | 1 | no | — |
| 7 | C7×C67 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic |
| 8 | D67⋊C3 | — | 49 | 1 | no | Hall |
| 9 | D49⋊C3 | — | 67 | 1 | no | maximal, Hall |
| 10 | C67⋊C3 | — | 49 | 1 | no | Hall |
| 11 | C49⋊C3 | — | 67 | 1 | no | Hall |
| 12 | D67 | — | 147 | 1 | no | Hall |
| 13 | D49 | — | 201 | 1 | no | Hall |
| 14 | C67 | D49⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C49 | D67⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 16 | AGL(1,𝔽7) | — | 469 | 1 | no | — |
| 17 | ΓL(1,𝔽8) | — | 469 | 1 | no | — |
| 18 | D7 | — | 1407 | 1 | no | — |
| 19 | C7 | #14584203 | 1 | 1 | yes | characteristic |
| 20 | C2×C3 | — | 3283 | 1 | no | Hall |
| 21 | C3 | — | 3283 | 1 | no | Sylow |
| 22 | C2 | — | 9849 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C67 | C2×C3 |
| Frattini | C7 | #14584203 |
| socle | C7×C67 | AGL(1,𝔽7) |
| LCS | G ▷ C49×C67 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23754415 | C2 | semidirect |
| 2. D3283 | C3 | semidirect |
| 3. C49×C67 | C2×C3 | semidirect |
| 14. C67 | D49⋊C3 | semidirect |
| 15. C49 | D67⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23754415 | 2 |
| D3283 | 3 |
| #14584203 | 7 |
| D49⋊C3 | 67 |