Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
688 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #2840 |
| commutator, Frattini, cosocle | C24 | C24×C109 |
| prime radical | C22×#41×C109 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C24×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#2032 | C2 | semidirect |
| 2. C109×#2030 | C2 | semidirect |
| 10. C22×Q8×C109 | Q8 | semidirect |
| 11. C22×D4×C109 | Q8 | semidirect |
| 13. C23×C4×C109 | Q8 | semidirect |
| 60. #32874 | C109 | direct (N × H) |
| 65. #2032 | C2×C109 | semidirect |
| 66. #2030 | C2×C109 | semidirect |
| 68. C109 | #32874 | direct (N × H) |
| 75. C22×Q8 | Q8×C109 | semidirect |
| 76. C22×D4 | Q8×C109 | semidirect |
| 78. C23×C4 | Q8×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2032 | 2 |
| C109×#2030 | 2 |
| C109×#1782 | 2 |
| #32874 | 109 |