Finite Group DatabasebetaFinite Group DatabasebetaOrder 17552.
20 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×D1097 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22578588 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C8×C1097 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×D1097 | C4 | 1 | 1 | yes | characteristic |
| 5 | C4×C1097 | C22 | 1 | 1 | yes | characteristic |
| 6 | Dic1097 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×C1097 | C2×C4 | 1 | 1 | yes | characteristic |
| 8 | D1097 | — | 2 | 1 | no | — |
| 9 | C1097 | M16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | M16 | — | 1097 | 1 | no | maximal, Sylow |
| 11 | C2×C4 | — | 1097 | 1 | no | — |
| 12 | C8 | — | 1097 | 1 | no | — |
| 13 | C8 | D1097 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22 | — | 1097 | 1 | no | — |
| 15 | C4 | — | 1097 | 1 | no | — |
| 16 | C4 | C2×D1097 | 1 | 1 | yes | characteristic |
| 17 | C2 | — | 2194 | 1 | no | — |
| 18 | C2 | C4×D1097 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D1097 |
| commutator, socle | C2×C1097 | C2×C4 |
| Fitting | C8×C1097 | C2 |
| prime radical | C2×D1097 | C4 |
| cosocle | C4×C1097 | C22 |
| hypercenter | C8 | D1097 |
| hypocenter | C1097 | M16 |
| LCS | G ▷ C2×C1097 ▷ C1097 | |
| UCS | C1 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22578588 | C2 | semidirect |
| 3. C8×C1097 | C2 | semidirect |
| 9. C1097 | M16 | semidirect |
| 13. C8 | D1097 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D1097 | 2 |
| #22578588 | 2 |
| C8×C1097 | 2 |
| M16 | 1097 |