Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
794 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C83 |
| prime radical | C2×H(5,𝔽2)×C83 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C24×C83 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#123×C83 | C2 | semidirect |
| 3. #458×C83 | C2 | direct (N × H) |
| 4. C2×#99×C83 | C2 | semidirect |
| 10. C23×C4×C83 | C22 | semidirect |
| 12. #123×C83 | C22 | semidirect |
| 15. C2×C22⋊C4×C83 | C22 | semidirect |
| 16. #99×C83 | C22 | semidirect |
| 35. C22×C4×C83 | C23 | semidirect |
| 39. C22⋊C4×C83 | C23 | semidirect |
| 80. C2×C83 | #458 | direct (N × H) |
| 82. C2×#458 | C83 | direct (N × H) |
| 83. C83 | C2×#458 | direct (N × H) |
| 85. C2×#123 | C2×C83 | semidirect |
| 86. #458 | C2×C83 | direct (N × H) |
| 87. C2×#99 | C2×C83 | semidirect |
| 93. C23×C4 | C22×C83 | semidirect |
| 95. #123 | C22×C83 | semidirect |
| 98. C2×C22⋊C4 | C22×C83 | semidirect |
| 99. #99 | C22×C83 | semidirect |
| 118. C22×C4 | C23×C83 | semidirect |
| 122. C22⋊C4 | C23×C83 | semidirect |
| 163. C2 | #458×C83 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2)×C83 | 2 |
| C2×#123×C83 | 2 |
| #458×C83 | 2 |
| C2×#99×C83 | 2 |
| C2×#458 | 83 |