Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
416 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4×C109 |
| Frattini, cosocle | C2×#41 | C23×C109 |
| socle | C2×C109 | #1867 |
| prime radical | #532×C109 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C2×C4×C109 ◁ C2×#41×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C109×#1858 | C2 | semidirect |
| 5. C109×#1395 | C2 | semidirect |
| 17. #356×C109 | C4 | semidirect |
| 18. #355×C109 | C4 | semidirect |
| 131. #12975 | C109 | direct (N × H) |
| 140. #1858 | C2×C109 | semidirect |
| 141. #1395 | C2×C109 | semidirect |
| 142. C109 | #12975 | direct (N × H) |
| 154. #356 | C4×C109 | semidirect |
| 155. #355 | C4×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2979 | 2 |
| C109×#2132 | 2 |
| C109×#1884 | 2 |
| C109×#1858 | 2 |
| C109×#1395 | 2 |
| #12975 | 109 |