Finite Group DatabasebetaFinite Group DatabasebetaOrder 800.
154 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D52 |
| commutator, socle, cosocle | C22×C52 | C23 |
| Fitting | C2×C4×C52 | C22 |
| prime radical | C22×C5⋊D5 | C22 |
| hypercenter | C2×C4 | D52 |
| hypocenter | C52 | #126 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C5×#683 | C2 | semidirect |
| 3. C4×C5×Dic5 | C2 | semidirect |
| 4. #87781 | C2 | semidirect |
| 5. #87780 | C2 | semidirect |
| 40. #683 | D5 | semidirect |
| 41. C4×Dic5 | D5 | semidirect |
| 64. C52 | #126 | semidirect |
| 118. C5 | #4022 | semidirect |
| 119. C5 | #4044 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #87808 | 2 |
| C5×#683 | 2 |
| C4×C5×Dic5 | 2 |
| #87781 | 2 |
| #87780 | 2 |
| #87779 | 2 |
| #4044 | 5 |
| #4022 | 5 |