Finite Group DatabasebetaFinite Group DatabasebetaOrder 12393.
440 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C92×C17 | C3 | 1 | 4 | yes | maximal |
| 2 | C17×#6855 | C3 | 1 | 9 | yes | maximal |
| 3 | C32×C9×C17 | C32 | 1 | 12 | yes | — |
| 4 | C32×C9×C17 | C32 | 1 | 1 | yes | characteristic |
| 5 | C92×C17 | C9 | 1 | 36 | yes | complemented |
| 6 | #10624544 | C17 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C33×C17 | C33 | 1 | 1 | yes | characteristic |
| 8 | C3×C9×C17 | — | 3 | 36 | no | — |
| 9 | C3×C9×C17 | C3×C9 | 1 | 36 | yes | — |
| 10 | C3×C9×C17 | C3×C9 | 1 | 12 | yes | — |
| 11 | C3×C92 | C3×C17 | 1 | 4 | yes | — |
| 12 | #6855 | C3×C17 | 1 | 9 | yes | — |
| 13 | C32×C17 | C32×C9 | 1 | 4 | yes | — |
| 14 | C32×C17 | #737 | 1 | 9 | yes | — |
| 15 | C9×C17 | — | 3 | 36 | no | — |
| 16 | C9×C17 | C92 | 1 | 9 | yes | — |
| 17 | C32×C9 | C32×C17 | 1 | 12 | yes | — |
| 18 | C32×C9 | C32×C17 | 1 | 1 | yes | characteristic |
| 19 | C92 | C9×C17 | 1 | 36 | yes | complemented |
| 20 | C3×C17 | C3×C92 | 1 | 1 | yes | characteristic |
| 21 | C3×C17 | #6887 | 1 | 12 | yes | — |
| 22 | C33 | C33×C17 | 1 | 1 | yes | characteristic |
| 23 | C3×C9 | — | 3 | 36 | no | — |
| 24 | C3×C9 | C3×C9×C17 | 1 | 36 | yes | — |
| 25 | C3×C9 | C3×C9×C17 | 1 | 12 | yes | — |
| 26 | C17 | #10624544 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 27 | C32 | C32×C9×C17 | 1 | 4 | yes | — |
| 28 | C32 | C17×#737 | 1 | 9 | yes | — |
| 29 | C9 | — | 3 | 36 | no | — |
| 30 | C9 | C92×C17 | 1 | 9 | yes | — |
| 31 | C3 | C3×C92×C17 | 1 | 1 | yes | characteristic |
| 32 | C3 | C17×#6887 | 1 | 12 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9×C17 | C32 |
| commutator | C3 | C3×C92×C17 |
| Frattini, cosocle | C33 | C33×C17 |
| socle, prime radical | C33×C17 | C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9×C17 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C92×C17 | C9 | semidirect |
| 6. #10624544 | C17 | direct (N × H) |
| 19. C92 | C9×C17 | semidirect |
| 26. C17 | #10624544 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C92×C17 | 3 |
| C17×#6855 | 3 |
| #10624544 | 17 |