Finite Group DatabasebetaFinite Group DatabasebetaOrder 13952.
1968 conjugacy classes of subgroups; only the 952 normal subgroups are shown.
This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 952 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C109 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C109 |
| prime radical | C23×D4×C109 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C24×C4×C109 | C2 | semidirect |
| 3. C2×#123×C109 | C2 | direct (N × H) |
| 4. C22×C4⋊C4×C109 | C2 | semidirect |
| 5. C22×C22⋊C4×C109 | C2 | semidirect |
| 6. C22×C22⋊C4×C109 | C2 | semidirect |
| 15. C23×C4×C109 | C22 | semidirect |
| 16. #123×C109 | C22 | direct (N × H) |
| 17. C2×C4⋊C4×C109 | C22 | semidirect |
| 18. C2×C22⋊C4×C109 | C22 | semidirect |
| 19. C2×C22⋊C4×C109 | C22 | semidirect |
| 36. C22×C4×C109 | C23 | semidirect |
| 37. C4⋊C4×C109 | C23 | semidirect |
| 38. C22⋊C4×C109 | C23 | semidirect |
| 39. C22⋊C4×C109 | C23 | semidirect |
| 64. C22×C109 | #123 | direct (N × H) |
| 74. C2×C109 | C2×#123 | direct (N × H) |
| 77. C22×#123 | C109 | direct (N × H) |
| 78. C109 | C22×#123 | direct (N × H) |
| 80. C24×C4 | C2×C109 | semidirect |
| 81. C2×#123 | C2×C109 | direct (N × H) |
| 82. C22×C4⋊C4 | C2×C109 | semidirect |
| 83. C22×C22⋊C4 | C2×C109 | semidirect |
| 84. C22×C22⋊C4 | C2×C109 | semidirect |
| 93. C23×C4 | C22×C109 | semidirect |
| 94. #123 | C22×C109 | direct (N × H) |
| 95. C2×C4⋊C4 | C22×C109 | semidirect |
| 96. C2×C22⋊C4 | C22×C109 | semidirect |
| 97. C2×C22⋊C4 | C22×C109 | semidirect |
| 114. C22×C4 | C23×C109 | semidirect |
| 115. C4⋊C4 | C23×C109 | semidirect |
| 116. C22⋊C4 | C23×C109 | semidirect |
| 117. C22⋊C4 | C23×C109 | semidirect |
| 142. C22 | #123×C109 | direct (N × H) |
| 152. C2 | C2×#123×C109 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4×C109 | 2 |
| C24×C4×C109 | 2 |
| C2×#123×C109 | 2 |
| C22×C4⋊C4×C109 | 2 |
| C22×C22⋊C4×C109 | 2 |
| C22×#123 | 109 |