Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
852 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 248 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×C131 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C131 |
| prime radical | C2×#121×C131 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C131 | C2 | semidirect |
| 2. C2×#124×C131 | C2 | semidirect |
| 4. #393×C131 | C2 | semidirect |
| 5. #391×C131 | C2 | semidirect |
| 6. #386×C131 | C2 | semidirect |
| 7. C4×C22⋊C4×C131 | C2 | semidirect |
| 32. C2×Q8×C131 | D4 | semidirect |
| 85. C22×C131 | #124 | semidirect |
| 101. C131 | #2436 | direct (N × H) |
| 102. #2436 | C131 | direct (N × H) |
| 103. C23×Q8 | C2×C131 | semidirect |
| 104. C2×#124 | C2×C131 | semidirect |
| 106. #393 | C2×C131 | semidirect |
| 107. #391 | C2×C131 | semidirect |
| 108. #386 | C2×C131 | semidirect |
| 109. C4×C22⋊C4 | C2×C131 | semidirect |
| 134. C2×Q8 | D4×C131 | semidirect |
| 187. C22 | #124×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C131 | 2 |
| C2×#124×C131 | 2 |
| C2×#121×C131 | 2 |
| #393×C131 | 2 |
| #391×C131 | 2 |
| #386×C131 | 2 |
| C4×C22⋊C4×C131 | 2 |
| #2436 | 131 |