Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
122 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #432×C251 | C2 | 1 | 3 | yes | maximal |
| 2 | C2×C42×C251 | C22 | 1 | 1 | yes | characteristic |
| 3 | C4⋊C8×C251 | — | 2 | 3 | no | — |
| 4 | C22⋊C8×C251 | C4 | 1 | 6 | yes | — |
| 5 | #97×C251 | — | 2 | 3 | no | — |
| 6 | C22×C4×C251 | C2×C4 | 1 | 3 | yes | — |
| 7 | C2×C8×C251 | — | 2 | 6 | no | — |
| 8 | C42×C251 | D4 | 1 | 3 | yes | — |
| 9 | C42×C251 | Q8 | 1 | 1 | yes | characteristic |
| 10 | C23×C251 | C42 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C251 | — | 4 | 3 | no | — |
| 12 | C2×C4×C251 | C22⋊C4 | 1 | 3 | yes | — |
| 13 | C2×C4×C251 | C4⋊C4 | 1 | 3 | yes | — |
| 14 | C8×C251 | — | 4 | 6 | no | — |
| 15 | C22×C251 | — | 2 | 3 | no | — |
| 16 | C22×C251 | #95 | 1 | 1 | yes | characteristic |
| 17 | C4×C251 | — | 2 | 3 | no | — |
| 18 | C4×C251 | — | 2 | 3 | no | — |
| 19 | C2×C251 | — | 4 | 1 | no | — |
| 20 | C2×C251 | #344 | 1 | 3 | yes | — |
| 21 | C251 | #1308 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 22 | #1308 | C251 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 23 | #432 | C2×C251 | 1 | 3 | yes | — |
| 24 | C2×C42 | C22×C251 | 1 | 1 | yes | characteristic |
| 25 | C4⋊C8 | — | 2 | 3 | no | — |
| 26 | C22⋊C8 | C4×C251 | 1 | 6 | yes | — |
| 27 | #97 | — | 2 | 3 | no | — |
| 28 | C22×C4 | C2×C4×C251 | 1 | 3 | yes | — |
| 29 | C2×C8 | — | 2 | 6 | no | — |
| 30 | C42 | D4×C251 | 1 | 3 | yes | — |
| 31 | C42 | Q8×C251 | 1 | 1 | yes | characteristic |
| 32 | C23 | C42×C251 | 1 | 1 | yes | characteristic |
| 33 | C2×C4 | — | 4 | 3 | no | — |
| 34 | C2×C4 | C22⋊C4×C251 | 1 | 3 | yes | — |
| 35 | C2×C4 | C4⋊C4×C251 | 1 | 3 | yes | — |
| 36 | C8 | — | 4 | 6 | no | — |
| 37 | C22 | — | 2 | 3 | no | — |
| 38 | C22 | #95×C251 | 1 | 1 | yes | characteristic |
| 39 | C4 | — | 2 | 3 | no | — |
| 40 | C4 | — | 2 | 3 | no | — |
| 41 | C2 | — | 4 | 1 | no | — |
| 42 | C2 | #344×C251 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C251 | #95 |
| commutator | C23 | C42×C251 |
| Frattini, cosocle | C2×C42 | C22×C251 |
| prime radical | C23×C251 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C251 ◁ C2×C42×C251 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. C251 | #1308 | direct (N × H) |
| 22. #1308 | C251 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #432×C251 | 2 |
| #1308 | 251 |