Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
712 conjugacy classes of subgroups; only the 518 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 518 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42×C109 | C24 |
| commutator | C22 | C22×C42×C109 |
| Frattini, cosocle | C42 | C24×C109 |
| socle | C22×C109 | C22×C42 |
| prime radical | C24×C109 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C109×#1453 | C2 | semidirect |
| 9. #322×C109 | C22 | semidirect |
| 10. C82×C109 | C22 | semidirect |
| 51. #17338 | C109 | direct (N × H) |
| 55. #1453 | C2×C109 | semidirect |
| 56. C109 | #17338 | direct (N × H) |
| 63. #322 | C22×C109 | semidirect |
| 64. C82 | C22×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2920 | 2 |
| C109×#1453 | 2 |
| #17338 | 109 |