Finite Group DatabasebetaFinite Group DatabasebetaOrder 7136.
30 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #16813154 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C223⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | M16×C223 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C223⋊C8 | — | 2 | 1 | no | — |
| 5 | C223⋊C8 | C4 | 1 | 2 | yes | — |
| 6 | C2×C4×C223 | C22 | 1 | 1 | yes | characteristic |
| 7 | C8×C223 | — | 2 | 1 | no | — |
| 8 | C22×C223 | Q8 | 1 | 1 | yes | characteristic |
| 9 | C4×C223 | C2×C4 | 1 | 1 | yes | characteristic |
| 10 | C4×C223 | D4 | 1 | 1 | yes | characteristic |
| 11 | C2×C223 | — | 2 | 1 | no | — |
| 12 | C2×C223 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 13 | C223 | #108 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | #108 | — | 223 | 1 | no | maximal, Sylow |
| 15 | M16 | — | 223 | 1 | no | — |
| 16 | M16 | D223 | 1 | 1 | yes | characteristic |
| 17 | C2×C8 | — | 223 | 1 | no | — |
| 18 | C2×C4 | C2×D223 | 1 | 1 | yes | characteristic |
| 19 | C8 | — | 2 | 1 | no | — |
| 20 | C8 | — | 446 | 1 | no | — |
| 21 | C8 | — | 223 | 2 | no | — |
| 22 | C22 | Dic446 | 1 | 1 | yes | characteristic |
| 23 | C4 | C4×D223 | 1 | 1 | yes | characteristic |
| 24 | C4 | C22⋊D223 | 1 | 1 | yes | characteristic |
| 25 | C2 | — | 2 | 1 | no | — |
| 26 | C2 | Dic223⋊C4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊D223 |
| commutator | C4×C223 | C2×C4 |
| Frattini | C2×C4 | C2×D223 |
| Fitting | M16×C223 | C2 |
| socle | C2×C223 | C4⋊C4 |
| hypercenter | M16 | D223 |
| hypocenter | C223 | #108 |
| LCS | G ▷ C4×C223 ▷ C2×C223 ▷ C223 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. C223 | #108 | semidirect |