Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
716 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 174 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 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C2×C52×Q8 | C22 |
| prime radical | C2×#6038 | C22 |
| hypercenter | C2×Q8 | D52 |
| hypocenter | C52 | D4×Q8 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×C5⋊D5 | C2 | semidirect |
| 2. C2×C5×Q8×D5 | C2 | semidirect |
| 3. C5×Q8×Dic5 | C2 | semidirect |
| 5. #11716990 | C2 | semidirect |
| 6. #11716983 | C2 | semidirect |
| 7. #11716975 | C2 | semidirect |
| 26. C52×Q8 | D4 | semidirect |
| 36. #6038 | Q8 | direct (N × H) |
| 41. C2×Q8×D5 | D5 | semidirect |
| 44. Q8×Dic5 | D5 | semidirect |
| 55. C2×C5⋊D5 | C2×Q8 | semidirect |
| 58. C2×C5×D5 | C2×Q8 | semidirect |
| 63. C5×Dic5 | C2×Q8 | semidirect |
| 109. C5×Q8 | D20 | semidirect |
| 110. C5×Q8 | C22⋊D5 | semidirect |
| 141. C52 | D4×Q8 | semidirect |
| 159. C2×D5 | Q8×D5 | semidirect |
| 171. Dic5 | Q8×D5 | semidirect |
| 204. Q8 | #6038 | direct (N × H) |
| 214. C5 | Q8×D20 | semidirect |
| 215. C5 | Q8×C22⋊D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×C5⋊D5 | 2 |
| C2×C5×Q8×D5 | 2 |
| C5×Q8×Dic5 | 2 |
| C4×#6038 | 2 |
| #11716990 | 2 |
| #11716983 | 2 |
| #11716975 | 2 |
| Q8×C22⋊D5 | 5 |
| Q8×D20 | 5 |