Finite Group DatabasebetaFinite Group DatabasebetaOrder 32108.
12 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×D349 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C23×C349 | C4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 3 | #13063703 | C23 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 4 | D349 | C2×C23 | 1 | 1 | yes | characteristic |
| 5 | C349 | C4×C23 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C4×C23 | — | 349 | 1 | no | maximal, Hall |
| 7 | C2×C23 | — | 349 | 1 | no | — |
| 8 | C23 | #13063703 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 9 | C4 | — | 349 | 1 | no | Sylow |
| 10 | C2 | — | 349 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #13063703 |
| commutator, hypocenter | C349 | C4×C23 |
| Frattini | C1 | G |
| Fitting, socle | C23×C349 | C4 |
| prime radical | C23×D349 | C2 |
| cosocle | D349 | C2×C23 |
| LCS | G ▷ C349 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C349 | C4 | semidirect |
| 3. #13063703 | C23 | direct (N × H) |
| 5. C349 | C4×C23 | semidirect |
| 8. C23 | #13063703 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D349 | 2 |
| #13063703 | 23 |
| C4×C23 | 349 |