Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
332 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C37×C32⋊C9 | C3 | 1 | 4 | yes | maximal |
| 2 | C37×#6855 | C3 | 1 | 9 | yes | maximal, complemented |
| 3 | C34×C37 | C32 | 1 | 1 | yes | characteristic |
| 4 | C32×C9×C37 | C32 | 1 | 12 | yes | — |
| 5 | C37×C32⋊C9 | — | 3 | 12 | no | — |
| 6 | C33×C37 | — | 3 | 12 | no | — |
| 7 | C33×C37 | H(3,𝔽3) | 1 | 3 | yes | — |
| 8 | C33×C37 | C33 | 1 | 1 | yes | characteristic |
| 9 | C3×C9×C37 | — | 3 | 12 | no | — |
| 10 | C3×C9×C37 | — | 3 | 36 | no | — |
| 11 | #10624568 | C37 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 12 | C32×C37 | — | 3 | 12 | no | — |
| 13 | C32×C37 | — | 9 | 9 | no | — |
| 14 | C32×C37 | C3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 15 | C32×C37 | #737 | 1 | 12 | yes | — |
| 16 | C9×C37 | — | 9 | 12 | no | — |
| 17 | C3×C32⋊C9 | C3×C37 | 1 | 4 | yes | — |
| 18 | #6855 | C3×C37 | 1 | 9 | yes | complemented |
| 19 | C3×C37 | — | 9 | 3 | no | — |
| 20 | C3×C37 | #6893 | 1 | 4 | yes | — |
| 21 | C3×C37 | #6897 | 1 | 9 | yes | — |
| 22 | C34 | C32×C37 | 1 | 1 | yes | characteristic |
| 23 | C32×C9 | C32×C37 | 1 | 12 | yes | — |
| 24 | C32⋊C9 | — | 3 | 12 | no | — |
| 25 | C37 | #10624568 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 26 | C33 | — | 3 | 12 | no | — |
| 27 | C33 | H(3,𝔽3)×C37 | 1 | 3 | yes | — |
| 28 | C33 | C33×C37 | 1 | 1 | yes | characteristic |
| 29 | C3×C9 | — | 3 | 12 | no | — |
| 30 | C3×C9 | — | 3 | 36 | no | — |
| 31 | C32 | — | 3 | 12 | no | — |
| 32 | C32 | — | 9 | 9 | no | — |
| 33 | C32 | C3×H(3,𝔽3)×C37 | 1 | 1 | yes | characteristic |
| 34 | C32 | C37×#737 | 1 | 12 | yes | — |
| 35 | C9 | — | 9 | 12 | no | — |
| 36 | C3 | — | 9 | 3 | no | — |
| 37 | C3 | C37×#6893 | 1 | 4 | yes | — |
| 38 | C3 | C37×#6897 | 1 | 9 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C37 | C33 |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| prime radical | C34×C37 | C32 |
| LCS | G ▷ C33 ▷ C1 | |
| UCS | C1 ◁ C33×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C37×#6855 | C3 | semidirect |
| 11. #10624568 | C37 | direct (N × H) |
| 18. #6855 | C3×C37 | semidirect |
| 25. C37 | #10624568 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C37×C32⋊C9 | 3 |
| C37×#6855 | 3 |
| #10624568 | 37 |