Finite Group DatabasebetaFinite Group DatabasebetaOrder 10624.
788 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 788 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D83 |
| commutator, socle, cosocle | C23×C83 | C24 |
| Fitting | #384×C83 | C2 |
| hypercenter | #384 | D83 |
| hypocenter | C83 | #2537 |
| LCS | G ▷ C23×C83 ▷ C83 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#14581767 | C2 | semidirect |
| 2. C2×C4×D332 | C2 | semidirect |
| 3. #19127618 | C2 | semidirect |
| 4. #19127558 | C2 | semidirect |
| 5. #384×C83 | C2 | semidirect |
| 23. C2×D332 | D4 | semidirect |
| 36. C42×C83 | D4 | semidirect |
| 73. C4×C83 | #121 | semidirect |
| 82. C83 | #2537 | semidirect |
| 86. #384 | D83 | semidirect |
| 118. C42 | D332 | semidirect |
| 156. C4 | #14581767 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#14581767 | 2 |
| C2×C4×D332 | 2 |
| #19127618 | 2 |
| #19127558 | 2 |
| #384×C83 | 2 |
| #2537 | 83 |