Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
125 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #91881 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #91872 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | GL(2,𝔽5) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×SL(2,𝔽5) | — | 2 | 1 | no | — |
| 5 | #6732 | C22 | 1 | 1 | yes | characteristic |
| 6 | #6729 | — | 2 | 1 | no | — |
| 7 | #5431 | — | 5 | 1 | no | maximal, Hall |
| 8 | #4008 | — | 6 | 1 | no | maximal |
| 9 | SL(2,𝔽5) | D4 | 1 | 1 | yes | characteristic |
| 10 | 2−1+4⋊C3 | — | 5 | 1 | no | — |
| 11 | #986 | — | 5 | 1 | no | — |
| 12 | #911 | — | 10 | 1 | no | maximal |
| 13 | GU(2,𝔽3) | — | 5 | 1 | no | — |
| 14 | #711 | — | 6 | 1 | no | — |
| 15 | C4×AGL(1,𝔽5) | — | 6 | 1 | no | — |
| 16 | #700 | — | 6 | 1 | no | — |
| 17 | #454 | — | 15 | 1 | no | Sylow |
| 18 | #237 | — | 10 | 1 | no | — |
| 19 | D3×D4 | — | 10 | 1 | no | — |
| 20 | #41⋊C3 | — | 5 | 1 | no | — |
| 21 | C2×SL(2,𝔽3) | — | 10 | 1 | no | — |
| 22 | GL(2,𝔽3) | — | 10 | 1 | no | — |
| 23 | C3×D8 | — | 10 | 1 | no | — |
| 24 | #214 | — | 10 | 1 | no | — |
| 25 | #213 | — | 10 | 1 | no | — |
| 26 | #204 | — | 10 | 1 | no | — |
| 27 | ΓL(1,𝔽25) | — | 10 | 1 | no | — |
| 28 | C2×AGL(1,𝔽5) | — | 12 | 1 | no | — |
| 29 | C5×D4 | — | 6 | 1 | no | — |
| 30 | C22⋊D5 | — | 12 | 1 | no | — |
| 31 | C2×Dic5 | — | 12 | 1 | no | — |
| 32 | C4×D5 | — | 6 | 1 | no | — |
| 33 | Dic10 | — | 6 | 1 | no | — |
| 34 | #170 | — | 12 | 1 | no | — |
| 35 | 2−1+4 | — | 5 | 1 | no | — |
| 36 | AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 37 | AGL(1,ℤ/8) | — | 15 | 1 | no | — |
| 38 | #124 | — | 15 | 1 | no | — |
| 39 | #104 | — | 15 | 1 | no | — |
| 40 | #104 | — | 15 | 1 | no | — |
| 41 | #101 | — | 15 | 1 | no | — |
| 42 | C22×D3 | — | 20 | 1 | no | — |
| 43 | C3×D4 | — | 10 | 1 | no | — |
| 44 | C3×D4 | — | 10 | 1 | no | — |
| 45 | #67 | — | 20 | 1 | no | — |
| 46 | #67 | — | 20 | 1 | no | — |
| 47 | C2×Dic3 | — | 20 | 1 | no | — |
| 48 | D12 | — | 10 | 1 | no | — |
| 49 | C4×D3 | — | 10 | 1 | no | — |
| 50 | Dic6 | — | 10 | 1 | no | — |
| 51 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 52 | C3×C8 | — | 10 | 1 | no | — |
| 53 | C3⋊C8 | — | 10 | 1 | no | — |
| 54 | C22×C5 | — | 12 | 1 | no | — |
| 55 | C2×D5 | — | 6 | 1 | no | — |
| 56 | AGL(1,𝔽5) | — | 12 | 2 | no | — |
| 57 | C4×C5 | — | 6 | 1 | no | — |
| 58 | Dic5 | — | 6 | 1 | no | — |
| 59 | Dic5 | — | 12 | 1 | no | — |
| 60 | #41 | — | 5 | 1 | no | — |
| 61 | #41 | — | 15 | 1 | no | — |
| 62 | #41 | — | 30 | 1 | no | — |
| 63 | C2×Q8 | — | 10 | 1 | no | — |
| 64 | C2×Q8 | — | 15 | 1 | no | — |
| 65 | C2×D4 | — | 15 | 1 | no | — |
| 66 | SD16 | — | 30 | 1 | no | — |
| 67 | SD16 | — | 30 | 1 | no | — |
| 68 | D8 | — | 10 | 1 | no | — |
| 69 | D8 | — | 30 | 1 | no | — |
| 70 | M16 | — | 15 | 1 | no | — |
| 71 | M16 | — | 15 | 1 | no | — |
| 72 | C22⋊C4 | — | 30 | 2 | no | — |
| 73 | C42 | — | 15 | 1 | no | — |
| 74 | C22×C3 | — | 20 | 1 | no | — |
| 75 | C22×C3 | — | 20 | 1 | no | — |
| 76 | C2×D3 | — | 10 | 1 | no | — |
| 77 | C2×D3 | — | 20 | 1 | no | — |
| 78 | C2×D3 | — | 40 | 2 | no | — |
| 79 | C3×C4 | — | 10 | 1 | no | — |
| 80 | Dic3 | — | 10 | 1 | no | — |
| 81 | Dic3 | — | 20 | 1 | no | — |
| 82 | C2×C5 | — | 6 | 1 | no | — |
| 83 | C2×C5 | — | 24 | 1 | no | — |
| 84 | D5 | — | 12 | 1 | no | — |
| 85 | C23 | — | 30 | 1 | no | — |
| 86 | Q8 | — | 5 | 1 | no | — |
| 87 | Q8 | — | 15 | 1 | no | — |
| 88 | Q8 | — | 30 | 1 | no | — |
| 89 | D4 | — | 10 | 1 | no | — |
| 90 | D4 | — | 15 | 1 | no | — |
| 91 | D4 | — | 30 | 1 | no | — |
| 92 | D4 | — | 30 | 1 | no | — |
| 93 | D4 | S5 | 1 | 1 | yes | characteristic |
| 94 | C2×C4 | — | 15 | 1 | no | — |
| 95 | C2×C4 | — | 15 | 2 | no | — |
| 96 | C2×C4 | — | 30 | 1 | no | — |
| 97 | C2×C4 | — | 30 | 1 | no | — |
| 98 | C8 | — | 10 | 1 | no | — |
| 99 | C8 | — | 30 | 1 | no | — |
| 100 | C2×C3 | — | 10 | 1 | no | — |
| 101 | C2×C3 | — | 40 | 1 | no | — |
| 102 | C2×C3 | — | 40 | 1 | no | — |
| 103 | D3 | — | 20 | 1 | no | — |
| 104 | D3 | — | 40 | 1 | no | — |
| 105 | C5 | — | 6 | 1 | no | Sylow |
| 106 | C22 | — | 2 | 1 | no | — |
| 107 | C22 | — | 15 | 1 | no | — |
| 108 | C22 | — | 20 | 1 | no | — |
| 109 | C22 | — | 60 | 2 | no | — |
| 110 | C4 | — | 15 | 1 | no | — |
| 111 | C4 | — | 30 | 1 | no | — |
| 112 | C4 | — | 60 | 1 | no | — |
| 113 | C4 | C2×S5 | 1 | 1 | yes | characteristic |
| 114 | C3 | — | 10 | 1 | no | Sylow |
| 115 | C2 | — | 4 | 1 | no | — |
| 116 | C2 | — | 30 | 1 | no | — |
| 117 | C2 | — | 40 | 1 | no | — |
| 118 | C2 | C22⋊S5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊S5 |
| commutator, cosocle | #6732 | C22 |
| Frattini | C4 | C2×S5 |
| Fitting, hypercenter | D4 | S5 |
| hypocenter | SL(2,𝔽5) | D4 |
| LCS | G ▷ #6732 ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #91881 | C2 | semidirect |
| 2. #91872 | C2 | semidirect |
| 3. GL(2,𝔽5) | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #91881 | 2 |
| #91872 | 2 |
| GL(2,𝔽5) | 2 |
| #5431 | 5 |
| #4008 | 6 |
| #911 | 10 |