Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
506 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 506 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C37 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C37 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C34×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×H(3,𝔽3)×C37 | C3 | semidirect |
| 2. C37×#6891 | C3 | semidirect |
| 3. C37×C32⋊H(3,𝔽3) | C3 | semidirect |
| 4. C3×C37×C32⋊C9 | C3 | semidirect |
| 5. C37×#6856 | C3 | semidirect |
| 16. C3×H(3,𝔽3)×C37 | C32 | semidirect |
| 22. C37×C32⋊C9 | C32 | semidirect |
| 23. C37×C32⋊C9 | C32 | semidirect |
| 49. #10624596 | C37 | direct (N × H) |
| 79. C32×H(3,𝔽3) | C3×C37 | semidirect |
| 80. #6891 | C3×C37 | semidirect |
| 81. C32⋊H(3,𝔽3) | C3×C37 | semidirect |
| 82. C3×C32⋊C9 | C3×C37 | semidirect |
| 83. #6856 | C3×C37 | semidirect |
| 104. C3×H(3,𝔽3) | C32×C37 | semidirect |
| 110. C32⋊C9 | C32×C37 | semidirect |
| 111. C32⋊C9 | C32×C37 | semidirect |
| 112. C37 | #10624596 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×H(3,𝔽3)×C37 | 3 |
| C37×#6891 | 3 |
| C37×C32⋊H(3,𝔽3) | 3 |
| C3×C37×C32⋊C9 | 3 |
| C37×#6856 | 3 |
| #10624596 | 37 |