Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
182 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 16 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 | Q16×A5 | C2 | 1 | 2 | yes | maximal |
| 2 | C16×A5 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | Q8×A5 | — | 2 | 2 | no | — |
| 4 | C8×A5 | C22 | 1 | 1 | yes | characteristic |
| 5 | A4×Q32 | — | 5 | 1 | no | maximal, Hall |
| 6 | D5×Q32 | — | 6 | 1 | no | maximal |
| 7 | C4×A5 | — | 4 | 2 | no | — |
| 8 | C4×A5 | D4 | 1 | 1 | yes | characteristic |
| 9 | A4×Q16 | — | 5 | 2 | no | — |
| 10 | D3×Q32 | — | 10 | 1 | no | maximal |
| 11 | A4×C16 | — | 5 | 1 | no | — |
| 12 | D5×Q16 | — | 6 | 2 | no | — |
| 13 | C5×Q32 | — | 6 | 1 | no | — |
| 14 | #3961 | — | 6 | 2 | no | — |
| 15 | Dic40 | — | 6 | 1 | no | — |
| 16 | D5×C16 | — | 6 | 1 | no | — |
| 17 | C22×Q32 | — | 5 | 1 | no | Sylow |
| 18 | C2×A5 | D8 | 1 | 1 | yes | characteristic |
| 19 | Q8×A4 | — | 10 | 2 | no | — |
| 20 | D3×Q16 | — | 10 | 2 | no | — |
| 21 | C8×A4 | — | 5 | 1 | no | — |
| 22 | C3×Q32 | — | 10 | 1 | no | — |
| 23 | #829 | — | 10 | 2 | no | — |
| 24 | Dic24 | — | 10 | 1 | no | — |
| 25 | D3×C16 | — | 10 | 1 | no | — |
| 26 | Q8×D5 | — | 12 | 2 | no | — |
| 27 | C5×Q16 | — | 6 | 2 | no | — |
| 28 | #689 | — | 12 | 2 | no | — |
| 29 | Dic20 | — | 6 | 2 | no | — |
| 30 | C8×D5 | — | 6 | 1 | no | — |
| 31 | C5×C16 | — | 6 | 1 | no | — |
| 32 | #672 | — | 6 | 1 | no | — |
| 33 | C22×Q16 | — | 5 | 2 | no | — |
| 34 | C2×Q32 | — | 15 | 1 | no | — |
| 35 | C2×Q32 | — | 15 | 1 | no | — |
| 36 | C2×Q32 | — | 15 | 2 | no | — |
| 37 | C22×C16 | — | 5 | 1 | no | — |
| 38 | A5 | Q32 | 1 | 1 | yes | characteristic, direct product factor |
| 39 | D3×Q8 | — | 20 | 2 | no | — |
| 40 | C4×A4 | — | 5 | 1 | no | — |
| 41 | C4×A4 | — | 20 | 2 | no | — |
| 42 | C3×Q16 | — | 10 | 2 | no | — |
| 43 | #216 | — | 20 | 2 | no | — |
| 44 | Dic12 | — | 10 | 2 | no | — |
| 45 | D3×C8 | — | 10 | 1 | no | — |
| 46 | C3×C16 | — | 10 | 1 | no | — |
| 47 | C3⋊C16 | — | 10 | 1 | no | — |
| 48 | C5×Q8 | — | 12 | 2 | no | — |
| 49 | C4×D5 | — | 6 | 1 | no | — |
| 50 | C4×D5 | — | 24 | 2 | no | — |
| 51 | Dic10 | — | 12 | 2 | no | — |
| 52 | Dic10 | — | 24 | 2 | no | — |
| 53 | C5×C8 | — | 6 | 1 | no | — |
| 54 | #168 | — | 6 | 1 | no | — |
| 55 | C22×Q8 | — | 10 | 2 | no | — |
| 56 | C2×Q16 | — | 15 | 2 | no | — |
| 57 | C2×Q16 | — | 15 | 2 | no | — |
| 58 | C2×Q16 | — | 30 | 2 | no | — |
| 59 | C22×C8 | — | 5 | 1 | no | — |
| 60 | Q32 | — | 15 | 2 | no | — |
| 61 | Q32 | — | 15 | 2 | no | — |
| 62 | Q32 | — | 15 | 1 | no | — |
| 63 | Q32 | A5 | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 64 | C2×C16 | — | 15 | 1 | no | — |
| 65 | C2×C16 | — | 15 | 1 | no | — |
| 66 | C2×A4 | — | 5 | 1 | no | — |
| 67 | C3×Q8 | — | 20 | 2 | no | — |
| 68 | C4×D3 | — | 10 | 1 | no | — |
| 69 | C4×D3 | — | 40 | 2 | no | — |
| 70 | Dic6 | — | 20 | 2 | no | — |
| 71 | Dic6 | — | 40 | 2 | no | — |
| 72 | C3×C8 | — | 10 | 1 | no | — |
| 73 | C3⋊C8 | — | 10 | 1 | no | — |
| 74 | C2×D5 | — | 6 | 1 | no | — |
| 75 | C4×C5 | — | 6 | 1 | no | — |
| 76 | C4×C5 | — | 24 | 2 | no | — |
| 77 | Dic5 | — | 6 | 1 | no | — |
| 78 | Dic5 | — | 24 | 2 | no | — |
| 79 | C2×Q8 | — | 30 | 2 | no | — |
| 80 | C2×Q8 | — | 30 | 2 | no | — |
| 81 | C2×Q8 | — | 60 | 2 | no | — |
| 82 | C22×C4 | — | 5 | 1 | no | — |
| 83 | C22×C4 | — | 20 | 2 | no | — |
| 84 | Q16 | — | 15 | 2 | no | — |
| 85 | Q16 | — | 30 | 2 | no | — |
| 86 | Q16 | — | 30 | 2 | no | — |
| 87 | Q16 | C2×A5 | 1 | 2 | yes | — |
| 88 | C2×C8 | — | 15 | 1 | no | — |
| 89 | C2×C8 | — | 15 | 1 | no | — |
| 90 | C16 | — | 15 | 1 | no | — |
| 91 | C16 | C2×A5 | 1 | 1 | yes | characteristic |
| 92 | C2×D3 | — | 10 | 1 | no | — |
| 93 | A4 | — | 5 | 1 | no | — |
| 94 | C3×C4 | — | 10 | 1 | no | — |
| 95 | C3×C4 | — | 40 | 2 | no | — |
| 96 | Dic3 | — | 10 | 1 | no | — |
| 97 | Dic3 | — | 40 | 2 | no | — |
| 98 | C2×C5 | — | 6 | 1 | no | — |
| 99 | D5 | — | 6 | 1 | no | — |
| 100 | D5 | — | 6 | 1 | no | — |
| 101 | C23 | — | 5 | 1 | no | — |
| 102 | Q8 | — | 2 | 2 | no | — |
| 103 | Q8 | — | 30 | 2 | no | — |
| 104 | Q8 | — | 60 | 2 | no | — |
| 105 | Q8 | — | 60 | 2 | no | — |
| 106 | C2×C4 | — | 15 | 1 | no | — |
| 107 | C2×C4 | — | 15 | 1 | no | — |
| 108 | C2×C4 | — | 60 | 2 | no | — |
| 109 | C2×C4 | — | 60 | 2 | no | — |
| 110 | C8 | — | 15 | 1 | no | — |
| 111 | C8 | C22×A5 | 1 | 1 | yes | characteristic |
| 112 | C2×C3 | — | 10 | 1 | no | — |
| 113 | D3 | — | 10 | 1 | no | — |
| 114 | D3 | — | 10 | 1 | no | — |
| 115 | C5 | — | 6 | 1 | no | Sylow |
| 116 | C22 | — | 5 | 1 | no | — |
| 117 | C22 | — | 15 | 1 | no | — |
| 118 | C22 | — | 15 | 1 | no | — |
| 119 | C4 | — | 4 | 2 | no | — |
| 120 | C4 | — | 15 | 1 | no | — |
| 121 | C4 | — | 60 | 2 | no | — |
| 122 | C4 | D4×A5 | 1 | 1 | yes | characteristic |
| 123 | C3 | — | 10 | 1 | no | Sylow |
| 124 | C2 | — | 15 | 1 | no | — |
| 125 | C2 | — | 15 | 1 | no | — |
| 126 | C2 | D8×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×A5 |
| commutator | C8×A5 | C22 |
| Frattini, cosocle | C8 | C22×A5 |
| Fitting, hypercenter | Q32 | A5 |
| socle, prime radical | C2×A5 | D8 |
| hypocenter | A5 | Q32 |
| LCS | G ▷ C8×A5 ▷ C4×A5 ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 38. A5 | Q32 | direct (N × H) |
| 63. Q32 | A5 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q16×A5 | 2 |
| C16×A5 | 2 |
| A4×Q32 | 5 |
| D5×Q32 | 6 |
| D3×Q32 | 10 |