Finite Group DatabasebetaFinite Group DatabasebetaOrder 23571.
50 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C32⋊C9×C97 | C3 | 1 | 4 | yes | maximal |
| 2 | C33×C97 | C32 | 1 | 1 | yes | characteristic |
| 3 | C3×C9×C97 | — | 3 | 4 | no | — |
| 4 | C32×C97 | — | 3 | 4 | no | — |
| 5 | C32×C97 | H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 6 | C9×C97 | — | 9 | 4 | no | — |
| 7 | C3×C97 | — | 9 | 1 | no | — |
| 8 | C3×C97 | #731 | 1 | 4 | yes | — |
| 9 | #6862 | C97 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 10 | C97 | #6862 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 11 | C32⋊C9 | C3×C97 | 1 | 4 | yes | — |
| 12 | C33 | C32×C97 | 1 | 1 | yes | characteristic |
| 13 | C3×C9 | — | 3 | 4 | no | — |
| 14 | C32 | — | 3 | 4 | no | — |
| 15 | C32 | H(3,𝔽3)×C97 | 1 | 1 | yes | characteristic |
| 16 | C9 | — | 9 | 4 | no | — |
| 17 | C3 | — | 9 | 1 | no | — |
| 18 | C3 | #731×C97 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C97 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C32×C97 |
| prime radical | C33×C97 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C97 ◁ C33×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #6862 | C97 | direct (N × H) |
| 10. C97 | #6862 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32⋊C9×C97 | 3 |
| #6862 | 97 |