Finite Group DatabasebetaFinite Group DatabasebetaOrder 1744.
10 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C8×C109 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×C109 | C4 | 1 | 1 | yes | characteristic |
| 3 | C2×C109 | C8 | 1 | 1 | yes | characteristic |
| 4 | C109 | C16 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 5 | C16 | C109 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 6 | C8 | C2×C109 | 1 | 1 | yes | characteristic |
| 7 | C4 | C4×C109 | 1 | 1 | yes | characteristic |
| 8 | C2 | C8×C109 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8 | C2×C109 |
| socle, prime radical | C2×C109 | C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C109 | C16 | direct (N × H) |
| 5. C16 | C109 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×C109 | 2 |
| C16 | 109 |