Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
196 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #20260490 | C2 | 1 | 6 | yes | maximal |
| 2 | #376×C97 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×#11715895 | C22 | 1 | 4 | yes | — |
| 4 | C2×C42×C97 | C22 | 1 | 3 | yes | — |
| 5 | #95×C97 | C4 | 1 | 4 | yes | — |
| 6 | C2×#11715895 | — | 2 | 12 | no | — |
| 7 | C22×C4×C97 | C2×C4 | 1 | 6 | yes | — |
| 8 | C22×C4×C97 | C23 | 1 | 1 | yes | characteristic |
| 9 | C42×C97 | — | 2 | 6 | no | — |
| 10 | C23×C97 | C22×C4 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C97 | — | 2 | 6 | no | — |
| 12 | C2×C4×C97 | — | 2 | 12 | no | — |
| 13 | C2×C4×C97 | #41 | 1 | 6 | yes | — |
| 14 | #11715895 | — | 4 | 4 | no | — |
| 15 | C22×C97 | #117 | 1 | 3 | yes | — |
| 16 | C22×C97 | #131 | 1 | 4 | yes | — |
| 17 | C4×C97 | — | 4 | 6 | no | — |
| 18 | C4×C97 | #126 | 1 | 4 | yes | — |
| 19 | C2×C97 | #383 | 1 | 1 | yes | characteristic |
| 20 | C2×C97 | #433 | 1 | 6 | yes | — |
| 21 | #1840 | — | 97 | 1 | no | maximal, Sylow |
| 22 | C97 | #1840 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 23 | #376 | D97 | 1 | 1 | yes | characteristic |
| 24 | #336 | — | 97 | 6 | no | — |
| 25 | C22×C8 | — | 97 | 4 | no | — |
| 26 | C2×C42 | C2×D97 | 1 | 3 | yes | — |
| 27 | #95 | Dic97 | 1 | 4 | yes | — |
| 28 | C22×C4 | C2×Dic97 | 1 | 6 | yes | — |
| 29 | C22×C4 | C22×D97 | 1 | 1 | yes | characteristic |
| 30 | C2×C8 | — | 194 | 12 | no | — |
| 31 | C42 | — | 2 | 6 | no | — |
| 32 | C23 | C22×Dic97 | 1 | 1 | yes | characteristic |
| 33 | C2×C4 | — | 2 | 6 | no | — |
| 34 | C2×C4 | — | 2 | 12 | no | — |
| 35 | C2×C4 | #13095511 | 1 | 6 | yes | — |
| 36 | C8 | — | 388 | 4 | no | — |
| 37 | C22 | #15690769 | 1 | 3 | yes | — |
| 38 | C22 | #15690817 | 1 | 4 | yes | — |
| 39 | C4 | — | 4 | 6 | no | — |
| 40 | C4 | #15690804 | 1 | 4 | yes | — |
| 41 | C2 | #20260603 | 1 | 1 | yes | characteristic |
| 42 | C2 | #20260796 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D97 |
| commutator, socle, prime radical | C23×C97 | C22×C4 |
| Fitting | #376×C97 | C2 |
| cosocle | C22×C4×C97 | C23 |
| hypercenter | #376 | D97 |
| hypocenter | C97 | #1840 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 22. C97 | #1840 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20260490 | 2 |
| #376×C97 | 2 |
| #1840 | 97 |