Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #20261859 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #15690900 | C4 | 1 | 1 | yes | characteristic |
| 3 | #13095514 | C8 | 1 | 1 | yes | characteristic |
| 4 | #11715895 | C16 | 1 | 1 | yes | characteristic |
| 5 | C4×C97 | C32 | 1 | 1 | yes | characteristic |
| 6 | C2×C97 | C64 | 1 | 1 | yes | characteristic |
| 7 | C128 | — | 97 | 1 | no | maximal, Sylow |
| 8 | C97 | C128 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C64 | — | 97 | 1 | no | — |
| 10 | C32 | — | 97 | 1 | no | — |
| 11 | C16 | — | 97 | 1 | no | — |
| 12 | C8 | — | 97 | 1 | no | — |
| 13 | C4 | #15690911 | 1 | 1 | yes | characteristic |
| 14 | C2 | #20261868 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #15690911 |
| commutator, hypocenter | C97 | C128 |
| Fitting | C4×C97 | C32 |
| socle, prime radical | C2×C97 | C64 |
| cosocle | #20261859 | C2 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C97 | C128 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261859 | 2 |
| C128 | 97 |