Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
760 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #1790 |
| prime radical | C109×#3529 | C2 |
| 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 |
|---|---|---|
| 2. C109×#2126 | C2 | semidirect |
| 3. C109×#1884 | C2 | semidirect |
| 4. #568×C109 | C22 | semidirect |
| 5. #563×C109 | C4 | semidirect |
| 7. H(3,𝔽4)×C109 | C4 | semidirect |
| 13. #410×C109 | C22 | semidirect |
| 122. #12772 | C109 | direct (N × H) |
| 128. #2126 | C2×C109 | semidirect |
| 129. #1884 | C2×C109 | semidirect |
| 130. C109 | #12772 | direct (N × H) |
| 131. #568 | C22×C109 | semidirect |
| 132. #563 | C4×C109 | semidirect |
| 134. H(3,𝔽4) | C4×C109 | semidirect |
| 140. #410 | C22×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#3529 | 2 |
| C109×#2126 | 2 |
| C109×#1884 | 2 |
| #12772 | 109 |