Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
5650 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C25×C199 | C2 | 1 | 63 | yes | maximal, direct product factor |
| 2 | C24×C199 | C22 | 1 | 651 | yes | direct product factor |
| 3 | C23×C199 | C23 | 1 | 1395 | yes | direct product factor |
| 4 | C22×C199 | C24 | 1 | 651 | yes | direct product factor |
| 5 | C2×C199 | C25 | 1 | 63 | yes | direct product factor |
| 6 | C199 | C26 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 7 | C26 | C199 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 8 | C25 | C2×C199 | 1 | 63 | yes | direct product factor |
| 9 | C24 | C22×C199 | 1 | 651 | yes | direct product factor |
| 10 | C23 | C23×C199 | 1 | 1395 | yes | direct product factor |
| 11 | C22 | C24×C199 | 1 | 651 | yes | direct product factor |
| 12 | C2 | C25×C199 | 1 | 63 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×C199 | C2 | direct (N × H) |
| 2. C24×C199 | C22 | direct (N × H) |
| 3. C23×C199 | C23 | direct (N × H) |
| 4. C22×C199 | C24 | direct (N × H) |
| 5. C2×C199 | C25 | direct (N × H) |
| 6. C199 | C26 | direct (N × H) |
| 7. C26 | C199 | direct (N × H) |
| 8. C25 | C2×C199 | direct (N × H) |
| 9. C24 | C22×C199 | direct (N × H) |
| 10. C23 | C23×C199 | direct (N × H) |
| 11. C22 | C24×C199 | direct (N × H) |
| 12. C2 | C25×C199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C199 | 2 |
| C26 | 199 |