Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
1128 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C109 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C109 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C43×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#2867 | C2 | semidirect |
| 2. C109×#2860 | C2 | semidirect |
| 3. C109×#2041 | C2 | semidirect |
| 4. C109×#2011 | C2 | semidirect |
| 5. C109×#1839 | C2 | semidirect |
| 59. C42×C109 | D8 | semidirect |
| 96. #32796 | C109 | direct (N × H) |
| 102. #2867 | C2×C109 | semidirect |
| 103. #2860 | C2×C109 | semidirect |
| 104. #2041 | C2×C109 | semidirect |
| 105. #2011 | C2×C109 | semidirect |
| 106. #1839 | C2×C109 | semidirect |
| 107. C109 | #32796 | direct (N × H) |
| 161. C42 | D8×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2867 | 2 |
| C109×#2860 | 2 |
| C109×#2041 | 2 |
| C109×#2011 | 2 |
| C109×#1839 | 2 |
| #32796 | 109 |