Finite Group DatabasebetaFinite Group DatabasebetaOrder 17504.
146 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #22576777 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #22576776 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | Q8×D547 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C2×Q8×C547 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | Dic1094 | C22 | 1 | 6 | yes | — |
| 6 | D2188 | C22 | 1 | 6 | yes | — |
| 7 | C22⋊D547 | C22 | 1 | 4 | yes | — |
| 8 | C4×D547 | C22 | 1 | 12 | yes | — |
| 9 | Q8×C547 | C22 | 1 | 4 | yes | — |
| 10 | C2×C4×C547 | C22 | 1 | 3 | yes | — |
| 11 | C22×C547 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×D547 | C23 | 1 | 4 | yes | — |
| 13 | C4×C547 | C23 | 1 | 6 | yes | — |
| 14 | Dic547 | C23 | 1 | 4 | yes | — |
| 15 | C2×C547 | — | 2 | 1 | no | — |
| 16 | C2×C547 | C24 | 1 | 1 | yes | characteristic |
| 17 | D547 | — | 2 | 4 | no | — |
| 18 | C547 | 2−1+4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | 2−1+4 | — | 547 | 1 | no | maximal, Sylow |
| 20 | #41 | — | 547 | 4 | no | — |
| 21 | #41 | — | 547 | 6 | no | — |
| 22 | C2×Q8 | — | 547 | 4 | no | — |
| 23 | C2×Q8 | D547 | 1 | 1 | yes | characteristic, complemented |
| 24 | Q8 | — | 547 | 6 | no | — |
| 25 | Q8 | C2×D547 | 1 | 4 | yes | — |
| 26 | D4 | — | 547 | 4 | no | — |
| 27 | D4 | — | 547 | 6 | no | — |
| 28 | C2×C4 | — | 547 | 12 | no | — |
| 29 | C2×C4 | C2×D547 | 1 | 3 | yes | — |
| 30 | C22 | — | 547 | 4 | no | — |
| 31 | C22 | C22×D547 | 1 | 1 | yes | characteristic |
| 32 | C4 | — | 547 | 4 | no | — |
| 33 | C4 | C22×D547 | 1 | 6 | yes | — |
| 34 | C2 | — | 2 | 1 | no | — |
| 35 | C2 | — | 1094 | 4 | no | — |
| 36 | C2 | C23×D547 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D547 |
| commutator, socle, cosocle | C2×C547 | C24 |
| Fitting | C2×Q8×C547 | C2 |
| hypercenter | C2×Q8 | D547 |
| hypocenter | C547 | 2−1+4 |
| LCS | G ▷ C2×C547 ▷ C547 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22576777 | C2 | semidirect |
| 2. #22576776 | C2 | semidirect |
| 3. Q8×D547 | C2 | semidirect |
| 4. C2×Q8×C547 | C2 | semidirect |
| 18. C547 | 2−1+4 | semidirect |
| 23. C2×Q8 | D547 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22576777 | 2 |
| #22576776 | 2 |
| Q8×D547 | 2 |
| C2×Q8×C547 | 2 |
| 2−1+4 | 547 |