Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
58424 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C25×D173 | C2 | 1 | 126 | yes | maximal, direct product factor |
| 2 | C26×C173 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C24×D173 | C22 | 1 | 2604 | yes | direct product factor |
| 4 | C25×C173 | C22 | 1 | 63 | yes | complemented |
| 5 | C23×D173 | C23 | 1 | 11160 | yes | direct product factor |
| 6 | C24×C173 | C23 | 1 | 651 | yes | complemented |
| 7 | C22×D173 | C24 | 1 | 10416 | yes | direct product factor |
| 8 | C23×C173 | C24 | 1 | 1395 | yes | complemented |
| 9 | C22×C173 | C25 | 1 | 651 | yes | complemented |
| 10 | C2×D173 | C25 | 1 | 2016 | yes | direct product factor |
| 11 | C2×C173 | C26 | 1 | 63 | yes | complemented |
| 12 | D173 | C26 | 1 | 64 | yes | direct product factor |
| 13 | C173 | C27 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C27 | — | 173 | 1 | no | maximal, Sylow |
| 15 | C26 | — | 173 | 126 | no | — |
| 16 | C26 | D173 | 1 | 1 | yes | characteristic, direct product factor |
| 17 | C25 | — | 173 | 2604 | no | — |
| 18 | C25 | C2×D173 | 1 | 63 | yes | direct product factor |
| 19 | C24 | — | 173 | 11160 | no | — |
| 20 | C24 | C22×D173 | 1 | 651 | yes | direct product factor |
| 21 | C23 | — | 173 | 10416 | no | — |
| 22 | C23 | C23×D173 | 1 | 1395 | yes | direct product factor |
| 23 | C22 | — | 173 | 2016 | no | — |
| 24 | C22 | C24×D173 | 1 | 651 | yes | direct product factor |
| 25 | C2 | — | 173 | 64 | no | — |
| 26 | C2 | C25×D173 | 1 | 63 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C26 | D173 |
| commutator, cosocle, hypocenter | C173 | C27 |
| Frattini | C1 | G |
| Fitting, socle | C26×C173 | C2 |
| LCS | G ▷ C173 | |
| UCS | C1 ◁ C26 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×D173 | C2 | direct (N × H) |
| 2. C26×C173 | C2 | semidirect |
| 3. C24×D173 | C22 | direct (N × H) |
| 4. C25×C173 | C22 | semidirect |
| 5. C23×D173 | C23 | direct (N × H) |
| 6. C24×C173 | C23 | semidirect |
| 7. C22×D173 | C24 | direct (N × H) |
| 8. C23×C173 | C24 | semidirect |
| 9. C22×C173 | C25 | semidirect |
| 10. C2×D173 | C25 | direct (N × H) |
| 11. C2×C173 | C26 | semidirect |
| 12. D173 | C26 | direct (N × H) |
| 13. C173 | C27 | semidirect |
| 16. C26 | D173 | direct (N × H) |
| 18. C25 | C2×D173 | direct (N × H) |
| 20. C24 | C22×D173 | direct (N × H) |
| 22. C23 | C23×D173 | direct (N × H) |
| 24. C22 | C24×D173 | direct (N × H) |
| 26. C2 | C25×D173 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×D173 | 2 |
| C26×C173 | 2 |
| C27 | 173 |