Finite Group DatabasebetaFinite Group DatabasebetaOrder 7232.
228 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D113 |
| commutator, socle | C22×C113 | C24 |
| Fitting | C22⋊Q8×C113 | C2 |
| hypercenter | C22⋊Q8 | D113 |
| hypocenter | C113 | #554 |
| LCS | G ▷ C22×C113 ▷ C113 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #16813620 | C2 | semidirect |
| 2. #16813612 | C2 | semidirect |
| 3. #16813612 | C2 | semidirect |
| 4. #16813600 | C2 | semidirect |
| 5. #16813600 | C2 | semidirect |
| 6. #16813599 | C2 | semidirect |
| 7. #16813599 | C2 | semidirect |
| 8. #16813598 | C2 | semidirect |
| 9. #16813587 | C2 | semidirect |
| 13. C2×#14268626 | C2 | semidirect |
| 14. C4⋊C4×D113 | C2 | semidirect |
| 15. C22⋊Q8×C113 | C2 | semidirect |
| 61. C22⋊D113 | Q8 | semidirect |
| 101. C113 | #554 | semidirect |
| 111. C22⋊Q8 | D113 | semidirect |