Finite Group DatabasebetaFinite Group DatabasebetaOrder 26973.
318 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 80 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 |
| prime radical | C37×#6856 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C37 ◁ C32×C9×C37 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C37×#737 | C3 | semidirect |
| 2. C37×#6894 | C3 | semidirect |
| 3. C9×H(3,𝔽3)×C37 | C3 | semidirect |
| 4. C37×#6860 | C3 | semidirect |
| 5. C37×#6859 | C3 | semidirect |
| 6. C37×#6858 | C3 | semidirect |
| 7. C37×#6857 | C3 | semidirect |
| 8. C37×#6857 | C3 | semidirect |
| 50. #10624595 | C37 | direct (N × H) |
| 70. C3×#737 | C3×C37 | semidirect |
| 71. #6894 | C3×C37 | semidirect |
| 72. C9×H(3,𝔽3) | C3×C37 | semidirect |
| 73. #6860 | C3×C37 | semidirect |
| 74. #6859 | C3×C37 | semidirect |
| 75. #6858 | C3×C37 | semidirect |
| 76. #6857 | C3×C37 | semidirect |
| 77. #6857 | C3×C37 | semidirect |
| 104. C37 | #10624595 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C37×#737 | 3 |
| C37×#6894 | 3 |
| C9×H(3,𝔽3)×C37 | 3 |
| C37×#6860 | 3 |
| C37×#6859 | 3 |
| C37×#6858 | 3 |
| C37×#6857 | 3 |
| C37×#6856 | 3 |
| #10624595 | 37 |