Finite Group DatabasebetaFinite Group DatabasebetaOrder 272.
38 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8×C17 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4×D17 | C2 | 1 | 3 | yes | maximal |
| 3 | Dic34 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×D17 | C22 | 1 | 1 | yes | characteristic |
| 5 | C4×C17 | C22 | 1 | 3 | yes | — |
| 6 | Dic17 | C22 | 1 | 3 | yes | — |
| 7 | C2×C17 | C23 | 1 | 1 | yes | characteristic |
| 8 | D17 | Q8 | 1 | 2 | yes | direct product factor |
| 9 | C17 | C2×Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C2×Q8 | — | 17 | 1 | no | maximal, Sylow |
| 11 | Q8 | — | 17 | 3 | no | — |
| 12 | Q8 | D17 | 1 | 1 | yes | characteristic, direct product factor |
| 13 | C2×C4 | — | 17 | 3 | no | — |
| 14 | C22 | — | 17 | 1 | no | — |
| 15 | C4 | — | 17 | 3 | no | — |
| 16 | C4 | C2×D17 | 1 | 3 | yes | — |
| 17 | C2 | — | 17 | 2 | no | — |
| 18 | C2 | C22×D17 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D17 |
| commutator, socle, cosocle | C2×C17 | C23 |
| Fitting | Q8×C17 | C2 |
| prime radical | C2×D17 | C22 |
| hypercenter | Q8 | D17 |
| hypocenter | C17 | C2×Q8 |
| LCS | G ▷ C2×C17 ▷ C17 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C17 | C2 | semidirect |
| 3. Dic34 | C2 | semidirect |
| 8. D17 | Q8 | direct (N × H) |
| 9. C17 | C2×Q8 | semidirect |
| 12. Q8 | D17 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C17 | 2 |
| C4×D17 | 2 |
| Dic34 | 2 |
| C2×Q8 | 17 |