Finite Group DatabasebetaFinite Group DatabasebetaOrder 2000.
280 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #11758758 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D5×#1039 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C5×#6037 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×C52×D5 | C22 | 1 | 3 | yes | — |
| 5 | #92553 | C22 | 1 | 1 | yes | characteristic |
| 6 | C5×#1039 | C22 | 1 | 3 | yes | — |
| 7 | #87880 | — | 5 | 3 | no | maximal |
| 8 | C2×C53 | C23 | 1 | 1 | yes | characteristic |
| 9 | C52×D5 | — | 2 | 3 | no | — |
| 10 | C2×#1039 | — | 5 | 3 | no | — |
| 11 | C5×C22⋊D5 | — | 5 | 6 | no | — |
| 12 | #6039 | — | 5 | 3 | no | — |
| 13 | #6037 | D5 | 1 | 3 | yes | complemented |
| 14 | D5×Dic5 | — | 5 | 6 | no | — |
| 15 | D5×Dic5 | — | 10 | 6 | no | — |
| 16 | C53 | #41 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | C22×C52 | — | 5 | 3 | no | — |
| 18 | C2×C5×D5 | — | 2 | 6 | no | — |
| 19 | C2×C5×D5 | C2×D5 | 1 | 6 | yes | — |
| 20 | #1039 | — | 5 | 3 | no | — |
| 21 | #1039 | — | 10 | 6 | no | — |
| 22 | #1039 | — | 20 | 4 | no | — |
| 23 | #1039 | C2×D5 | 1 | 3 | yes | — |
| 24 | C5×Dic5 | — | 5 | 6 | no | — |
| 25 | C5×Dic5 | — | 10 | 6 | no | — |
| 26 | #711 | — | 25 | 3 | no | — |
| 27 | C2×C52 | — | 2 | 6 | no | — |
| 28 | C2×C52 | — | 4 | 4 | no | — |
| 29 | C2×C52 | — | 10 | 3 | no | — |
| 30 | C2×C52 | C22×D5 | 1 | 3 | yes | — |
| 31 | C5×D5 | — | 2 | 12 | no | — |
| 32 | C5×D5 | — | 2 | 6 | no | — |
| 33 | C5×D4 | — | 25 | 3 | no | — |
| 34 | C22⋊D5 | — | 5 | 6 | no | — |
| 35 | C2×Dic5 | — | 25 | 6 | no | — |
| 36 | C2×Dic5 | — | 50 | 6 | no | — |
| 37 | C4×D5 | — | 25 | 3 | no | — |
| 38 | Dic10 | — | 25 | 3 | no | — |
| 39 | C52 | — | 2 | 6 | no | — |
| 40 | C52 | — | 4 | 4 | no | — |
| 41 | C52 | #711 | 1 | 3 | yes | complemented |
| 42 | C22×C5 | — | 5 | 6 | no | — |
| 43 | C22×C5 | — | 10 | 6 | no | — |
| 44 | C2×D5 | D52 | 1 | 3 | yes | — |
| 45 | C4×C5 | — | 25 | 3 | no | — |
| 46 | Dic5 | — | 5 | 6 | no | — |
| 47 | Dic5 | — | 10 | 6 | no | — |
| 48 | Dic5 | — | 25 | 3 | no | — |
| 49 | Dic5 | — | 50 | 6 | no | — |
| 50 | Dic5 | — | 100 | 4 | no | — |
| 51 | #41 | — | 125 | 1 | no | Sylow |
| 52 | C2×C5 | — | 2 | 6 | no | — |
| 53 | C2×C5 | — | 4 | 4 | no | — |
| 54 | C2×C5 | — | 10 | 12 | no | — |
| 55 | C2×C5 | — | 10 | 6 | no | — |
| 56 | C2×C5 | C2×D52 | 1 | 3 | yes | — |
| 57 | D5 | — | 2 | 3 | no | — |
| 58 | Q8 | — | 125 | 1 | no | — |
| 59 | D4 | — | 25 | 3 | no | — |
| 60 | C2×C4 | — | 125 | 3 | no | — |
| 61 | C5 | — | 2 | 6 | no | — |
| 62 | C5 | — | 4 | 4 | no | — |
| 63 | C5 | #87880 | 1 | 3 | yes | complemented |
| 64 | C22 | — | 5 | 3 | no | — |
| 65 | C4 | — | 25 | 3 | no | — |
| 66 | C4 | — | 125 | 1 | no | — |
| 67 | C2 | — | 10 | 3 | no | — |
| 68 | C2 | D53 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D53 |
| commutator, Fitting, socle, cosocle | C2×C53 | C23 |
| hypocenter | C53 | #41 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11758758 | C2 | semidirect |
| 2. D5×#1039 | C2 | semidirect |
| 3. C5×#6037 | C2 | semidirect |
| 13. #6037 | D5 | semidirect |
| 16. C53 | #41 | semidirect |
| 41. C52 | #711 | semidirect |
| 63. C5 | #87880 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11758758 | 2 |
| D5×#1039 | 2 |
| C5×#6037 | 2 |
| #87880 | 5 |