Finite Group DatabasebetaFinite Group DatabasebetaOrder 400.
160 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C52×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C52×D4 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C4×C52 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C22×C52 | C22 | 1 | 3 | yes | — |
| 5 | C4×C52 | C22 | 1 | 3 | yes | — |
| 6 | C4×C52 | C22 | 1 | 1 | yes | characteristic |
| 7 | C5×#41 | C5 | 1 | 6 | yes | maximal, direct product factor |
| 8 | C2×C52 | — | 2 | 3 | no | — |
| 9 | C2×C52 | C23 | 1 | 1 | yes | characteristic |
| 10 | C5×Q8 | C2×C5 | 1 | 6 | yes | complemented |
| 11 | C5×D4 | C2×C5 | 1 | 18 | yes | complemented |
| 12 | C2×C4×C5 | C2×C5 | 1 | 18 | yes | complemented |
| 13 | C52 | #41 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C22×C5 | C22×C5 | 1 | 18 | yes | — |
| 15 | C4×C5 | C22×C5 | 1 | 18 | yes | — |
| 16 | C4×C5 | C22×C5 | 1 | 6 | yes | — |
| 17 | #41 | C52 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 18 | C2×C5 | — | 2 | 18 | no | — |
| 19 | C2×C5 | C23×C5 | 1 | 6 | yes | — |
| 20 | Q8 | C2×C52 | 1 | 1 | yes | characteristic, complemented |
| 21 | D4 | C2×C52 | 1 | 3 | yes | complemented |
| 22 | C2×C4 | C2×C52 | 1 | 3 | yes | complemented |
| 23 | C5 | C5×#41 | 1 | 6 | yes | direct product factor |
| 24 | C22 | C22×C52 | 1 | 3 | yes | — |
| 25 | C4 | C22×C52 | 1 | 1 | yes | characteristic |
| 26 | C4 | C22×C52 | 1 | 3 | yes | — |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C23×C52 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4×C52 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C52 |
| socle | C2×C52 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C52 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C52×Q8 | C2 | semidirect |
| 2. C52×D4 | C2 | semidirect |
| 3. C2×C4×C52 | C2 | semidirect |
| 7. C5×#41 | C5 | direct (N × H) |
| 10. C5×Q8 | C2×C5 | semidirect |
| 11. C5×D4 | C2×C5 | semidirect |
| 12. C2×C4×C5 | C2×C5 | semidirect |
| 13. C52 | #41 | direct (N × H) |
| 17. #41 | C52 | direct (N × H) |
| 20. Q8 | C2×C52 | semidirect |
| 21. D4 | C2×C52 | semidirect |
| 22. C2×C4 | C2×C52 | semidirect |
| 23. C5 | C5×#41 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C52×Q8 | 2 |
| C52×D4 | 2 |
| C2×C4×C52 | 2 |
| C5×#41 | 5 |