Finite Group DatabasebetaFinite Group DatabasebetaOrder 26848.
62 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #41×C839 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | M16×C839 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C8×C839 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | Q8×C839 | C4 | 1 | 1 | yes | characteristic |
| 5 | D4×C839 | C4 | 1 | 3 | yes | — |
| 6 | C2×C4×C839 | C22 | 1 | 3 | yes | — |
| 7 | C8×C839 | C22 | 1 | 3 | yes | — |
| 8 | C8×C839 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C839 | C2×C4 | 1 | 3 | yes | — |
| 10 | C4×C839 | C2×C4 | 1 | 3 | yes | — |
| 11 | C4×C839 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C839 | — | 2 | 3 | no | — |
| 13 | C2×C839 | C22×C4 | 1 | 1 | yes | characteristic |
| 14 | C839 | #131 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 15 | #131 | C839 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 16 | #41 | C2×C839 | 1 | 1 | yes | characteristic |
| 17 | M16 | C2×C839 | 1 | 3 | yes | complemented |
| 18 | C2×C8 | C2×C839 | 1 | 3 | yes | complemented |
| 19 | Q8 | C4×C839 | 1 | 1 | yes | characteristic |
| 20 | D4 | C4×C839 | 1 | 3 | yes | — |
| 21 | C2×C4 | C22×C839 | 1 | 3 | yes | — |
| 22 | C8 | C22×C839 | 1 | 1 | yes | characteristic |
| 23 | C8 | C22×C839 | 1 | 3 | yes | — |
| 24 | C22 | C2×C4×C839 | 1 | 3 | yes | — |
| 25 | C4 | C2×C4×C839 | 1 | 3 | yes | — |
| 26 | C4 | C23×C839 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C22×C4×C839 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8×C839 | C22 |
| commutator | C2 | C22×C4×C839 |
| Frattini, cosocle | C4 | C23×C839 |
| socle | C2×C839 | C22×C4 |
| prime radical | #41×C839 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C839 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. M16×C839 | C2 | semidirect |
| 3. C2×C8×C839 | C2 | semidirect |
| 14. C839 | #131 | direct (N × H) |
| 15. #131 | C839 | direct (N × H) |
| 17. M16 | C2×C839 | semidirect |
| 18. C2×C8 | C2×C839 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×C839 | 2 |
| M16×C839 | 2 |
| C2×C8×C839 | 2 |
| #131 | 839 |