Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
272 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C109 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C109 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C42×C109 ◁ #446×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 66. Q16×C109 | Q16 | semidirect |
| 126. #12415 | C109 | direct (N × H) |
| 137. C109 | #12415 | direct (N × H) |
| 196. Q16 | Q16×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#3390 | 2 |
| C109×#2255 | 2 |
| C109×#2227 | 2 |
| C109×#1694 | 2 |
| C109×#1363 | 2 |
| C109×#1338 | 2 |
| C109×#1334 | 2 |
| #12415 | 109 |