Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1860 conjugacy classes of subgroups; only the 593 normal subgroups are shown.
This group has 1860 conjugacy classes of subgroups — too many to draw. Only its 593 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 | C3×C5×#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11756459 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | #11756458 | C2 | 1 | 10 | yes | maximal, complemented |
| 4 | #11756450 | C2 | 1 | 10 | yes | maximal, complemented |
| 5 | #11756447 | C2 | 1 | 5 | yes | maximal, complemented |
| 6 | #10619040 | — | 3 | 1 | no | maximal, Hall |
| 7 | C3×C5×#125 | C22 | 1 | 5 | yes | — |
| 8 | C3×C5×#125 | C22 | 1 | 10 | yes | — |
| 9 | #91787 | C22 | 1 | 20 | yes | — |
| 10 | #91786 | C22 | 1 | 10 | yes | — |
| 11 | #91785 | C22 | 1 | 20 | yes | — |
| 12 | #91784 | C22 | 1 | 10 | yes | — |
| 13 | #91783 | C22 | 1 | 20 | yes | — |
| 14 | #91782 | C22 | 1 | 10 | yes | — |
| 15 | #91781 | C22 | 1 | 10 | yes | — |
| 16 | #91777 | C22 | 1 | 10 | yes | — |
| 17 | #91776 | C22 | 1 | 10 | yes | — |
| 18 | #91764 | C22 | 1 | 10 | yes | — |
| 19 | C42×D15 | C22 | 1 | 5 | yes | — |
| 20 | #91759 | C22 | 1 | 5 | yes | — |
| 21 | #87139 | — | 5 | 1 | no | maximal, Hall |
| 22 | C5×#564 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | #66034 | — | 3 | 5 | no | — |
| 24 | #66033 | — | 3 | 10 | no | — |
| 25 | #66025 | — | 3 | 10 | no | — |
| 26 | #66022 | — | 3 | 5 | no | — |
| 27 | C2×D60 | C23 | 1 | 10 | yes | — |
| 28 | C2×C4×D15 | C23 | 1 | 5 | yes | — |
| 29 | C2×C4×D15 | C23 | 1 | 10 | yes | — |
| 30 | C2×Dic30 | C23 | 1 | 10 | yes | — |
| 31 | C3×C5×C4⋊C4 | C23 | 1 | 10 | yes | — |
| 32 | C3×C5×C4⋊C4 | C23 | 1 | 10 | yes | — |
| 33 | C3×C5×C4⋊C4 | C23 | 1 | 10 | yes | — |
| 34 | C3×C42×C5 | C23 | 1 | 5 | yes | — |
| 35 | #6714 | C23 | 1 | 20 | yes | — |
| 36 | #6714 | C23 | 1 | 20 | yes | — |
| 37 | C4⋊Dic15 | C23 | 1 | 10 | yes | — |
| 38 | #6712 | C23 | 1 | 20 | yes | — |
| 39 | C4×Dic15 | C23 | 1 | 10 | yes | — |
| 40 | C4×Dic15 | C23 | 1 | 5 | yes | — |
| 41 | C3×#564 | D5 | 1 | 1 | yes | characteristic, complemented |
| 42 | #5700 | — | 5 | 5 | no | — |
| 43 | #5699 | — | 5 | 10 | no | — |
| 44 | #5691 | — | 5 | 10 | no | — |
| 45 | #5688 | — | 5 | 5 | no | — |
| 46 | C5×#125 | C2×D3 | 1 | 5 | yes | — |
| 47 | C5×#125 | C2×D3 | 1 | 10 | yes | — |
| 48 | #4044 | — | 3 | 20 | no | — |
| 49 | #4043 | — | 3 | 10 | no | — |
| 50 | #4042 | — | 3 | 20 | no | — |
| 51 | #4041 | — | 3 | 10 | no | — |
| 52 | #4040 | — | 3 | 20 | no | — |
| 53 | #4039 | — | 3 | 10 | no | — |
| 54 | #4038 | — | 3 | 10 | no | — |
| 55 | #4034 | — | 3 | 10 | no | — |
| 56 | #4033 | — | 3 | 10 | no | — |
| 57 | #4021 | — | 3 | 10 | no | — |
| 58 | C42×D5 | — | 3 | 5 | no | — |
| 59 | #4016 | — | 3 | 5 | no | — |
| 60 | #3531 | — | 15 | 1 | no | Sylow |
| 61 | C22×D15 | C24 | 1 | 5 | yes | — |
| 62 | C2×C3×C4×C5 | C24 | 1 | 10 | yes | — |
| 63 | C2×C3×C4×C5 | C24 | 1 | 5 | yes | — |
| 64 | C2×Dic15 | C24 | 1 | 10 | yes | — |
| 65 | C2×Dic15 | C24 | 1 | 1 | yes | characteristic |
| 66 | D60 | — | 4 | 10 | no | — |
| 67 | C4×D15 | — | 2 | 10 | no | — |
| 68 | C4×D15 | — | 4 | 10 | no | — |
| 69 | Dic30 | — | 4 | 10 | no | — |
| 70 | C3×#125 | C2×D5 | 1 | 5 | yes | — |
| 71 | C3×#125 | C2×D5 | 1 | 10 | yes | — |
| 72 | #898 | — | 5 | 20 | no | — |
| 73 | #897 | — | 5 | 10 | no | — |
| 74 | #896 | — | 5 | 20 | no | — |
| 75 | #895 | — | 5 | 10 | no | — |
| 76 | #894 | — | 5 | 20 | no | — |
| 77 | #893 | — | 5 | 10 | no | — |
| 78 | #892 | — | 5 | 10 | no | — |
| 79 | #888 | — | 5 | 10 | no | — |
| 80 | #887 | — | 5 | 10 | no | — |
| 81 | #875 | — | 5 | 10 | no | — |
| 82 | C42×D3 | — | 5 | 5 | no | — |
| 83 | #870 | — | 5 | 5 | no | — |
| 84 | C2×D20 | — | 3 | 10 | no | — |
| 85 | C2×C4×D5 | — | 3 | 5 | no | — |
| 86 | C2×C4×D5 | — | 3 | 10 | no | — |
| 87 | C2×Dic10 | — | 3 | 10 | no | — |
| 88 | C5×C4⋊C4 | C22×D3 | 1 | 10 | yes | — |
| 89 | C5×C4⋊C4 | C22×D3 | 1 | 10 | yes | — |
| 90 | C5×C4⋊C4 | C22×D3 | 1 | 10 | yes | — |
| 91 | C42×C5 | C22×D3 | 1 | 5 | yes | — |
| 92 | #685 | — | 3 | 20 | no | — |
| 93 | #685 | — | 3 | 20 | no | — |
| 94 | C4⋊Dic5 | — | 3 | 10 | no | — |
| 95 | #683 | — | 3 | 20 | no | — |
| 96 | C4×Dic5 | — | 3 | 10 | no | — |
| 97 | C4×Dic5 | — | 3 | 5 | no | — |
| 98 | #564 | D15 | 1 | 1 | yes | characteristic, complemented |
| 99 | #563 | — | 15 | 10 | no | — |
| 100 | #562 | — | 15 | 5 | no | — |
| 101 | #533 | — | 15 | 5 | no | — |
| 102 | #529 | — | 15 | 10 | no | — |
| 103 | C22×C3×C5 | C25 | 1 | 1 | yes | characteristic |
| 104 | C2×D15 | — | 2 | 5 | no | — |
| 105 | C2×D15 | — | 2 | 10 | no | — |
| 106 | C3×C4×C5 | — | 2 | 5 | no | — |
| 107 | C3×C4×C5 | — | 2 | 10 | no | — |
| 108 | Dic15 | — | 2 | 10 | no | — |
| 109 | Dic15 | 2−1+4 | 1 | 2 | yes | — |
| 110 | C2×D12 | — | 5 | 10 | no | — |
| 111 | C2×C4×D3 | — | 5 | 5 | no | — |
| 112 | C2×C4×D3 | — | 5 | 10 | no | — |
| 113 | C2×Dic6 | — | 5 | 10 | no | — |
| 114 | C3×C4⋊C4 | C22×D5 | 1 | 10 | yes | — |
| 115 | C3×C4⋊C4 | C22×D5 | 1 | 10 | yes | — |
| 116 | C3×C4⋊C4 | C22×D5 | 1 | 10 | yes | — |
| 117 | C3×C42 | C22×D5 | 1 | 5 | yes | — |
| 118 | (C2×D3)⋊C4 | — | 5 | 20 | no | — |
| 119 | (C2×D3)⋊C4 | — | 5 | 20 | no | — |
| 120 | C4⋊Dic3 | — | 5 | 10 | no | — |
| 121 | Dic3⋊C4 | — | 5 | 20 | no | — |
| 122 | C4×Dic3 | — | 5 | 10 | no | — |
| 123 | C4×Dic3 | — | 5 | 5 | no | — |
| 124 | C22×D5 | — | 3 | 5 | no | — |
| 125 | C2×C4×C5 | C23×D3 | 1 | 10 | yes | — |
| 126 | C2×C4×C5 | C23×D3 | 1 | 5 | yes | — |
| 127 | C2×Dic5 | — | 3 | 1 | no | — |
| 128 | C2×Dic5 | — | 3 | 10 | no | — |
| 129 | D20 | — | 12 | 10 | no | — |
| 130 | C4×D5 | — | 6 | 10 | no | — |
| 131 | C4×D5 | — | 12 | 10 | no | — |
| 132 | Dic10 | — | 12 | 10 | no | — |
| 133 | C4⋊Q8 | — | 15 | 10 | no | — |
| 134 | #126 | — | 15 | 20 | no | — |
| 135 | #125 | — | 15 | 5 | no | — |
| 136 | #125 | C2×D15 | 1 | 10 | yes | — |
| 137 | #125 | C2×D15 | 1 | 5 | yes | — |
| 138 | #124 | — | 15 | 10 | no | — |
| 139 | #123 | — | 15 | 20 | no | — |
| 140 | C22⋊Q8 | — | 15 | 10 | no | — |
| 141 | C22⋊Q8 | — | 15 | 20 | no | — |
| 142 | #121 | — | 15 | 10 | no | — |
| 143 | C4×Q8 | — | 15 | 10 | no | — |
| 144 | C4×D4 | — | 15 | 10 | no | — |
| 145 | #117 | — | 15 | 10 | no | — |
| 146 | C2×C42 | — | 15 | 5 | no | — |
| 147 | C2×C3×C5 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C5 | #585 | 1 | 2 | yes | — |
| 149 | D15 | — | 4 | 5 | no | — |
| 150 | C22×D3 | — | 5 | 5 | no | — |
| 151 | C2×C3×C4 | C23×D5 | 1 | 10 | yes | — |
| 152 | C2×C3×C4 | C23×D5 | 1 | 5 | yes | — |
| 153 | C2×Dic3 | — | 5 | 1 | no | — |
| 154 | C2×Dic3 | — | 5 | 10 | no | — |
| 155 | D12 | — | 20 | 10 | no | — |
| 156 | C4×D3 | — | 10 | 10 | no | — |
| 157 | C4×D3 | — | 20 | 10 | no | — |
| 158 | Dic6 | — | 20 | 10 | no | — |
| 159 | C22×C5 | C24×D3 | 1 | 1 | yes | characteristic |
| 160 | C2×D5 | — | 6 | 5 | no | — |
| 161 | C2×D5 | — | 6 | 10 | no | — |
| 162 | C4×C5 | — | 2 | 5 | no | — |
| 163 | C4×C5 | — | 2 | 10 | no | — |
| 164 | Dic5 | — | 3 | 2 | no | — |
| 165 | Dic5 | — | 6 | 10 | no | — |
| 166 | C2×Q8 | — | 15 | 10 | no | — |
| 167 | C2×D4 | — | 15 | 10 | no | — |
| 168 | C22×C4 | — | 15 | 5 | no | — |
| 169 | C22×C4 | — | 15 | 10 | no | — |
| 170 | C4⋊C4 | — | 15 | 10 | no | — |
| 171 | C4⋊C4 | — | 15 | 20 | no | — |
| 172 | C4⋊C4 | C22×D15 | 1 | 10 | yes | — |
| 173 | C4⋊C4 | C22×D15 | 1 | 10 | yes | — |
| 174 | C4⋊C4 | C22×D15 | 1 | 10 | yes | — |
| 175 | C22⋊C4 | — | 15 | 20 | no | — |
| 176 | C22⋊C4 | — | 15 | 20 | no | — |
| 177 | C42 | — | 15 | 10 | no | — |
| 178 | C42 | — | 15 | 5 | no | — |
| 179 | C42 | C22×D15 | 1 | 5 | yes | — |
| 180 | C3×C5 | #3531 | 1 | 1 | yes | characteristic, complemented, Hall |
| 181 | C22×C3 | C24×D5 | 1 | 1 | yes | characteristic |
| 182 | C2×D3 | — | 10 | 5 | no | — |
| 183 | C2×D3 | — | 10 | 10 | no | — |
| 184 | C3×C4 | — | 2 | 5 | no | — |
| 185 | C3×C4 | — | 2 | 10 | no | — |
| 186 | Dic3 | — | 5 | 2 | no | — |
| 187 | Dic3 | — | 10 | 10 | no | — |
| 188 | C2×C5 | D3×2−1+4 | 1 | 1 | yes | characteristic |
| 189 | C2×C5 | #5968 | 1 | 2 | yes | — |
| 190 | D5 | — | 12 | 5 | no | — |
| 191 | C23 | — | 15 | 5 | no | — |
| 192 | Q8 | — | 60 | 10 | no | — |
| 193 | D4 | — | 60 | 10 | no | — |
| 194 | C2×C4 | — | 15 | 1 | no | — |
| 195 | C2×C4 | — | 15 | 10 | no | — |
| 196 | C2×C4 | — | 30 | 10 | no | — |
| 197 | C2×C4 | — | 60 | 10 | no | — |
| 198 | C2×C4 | C23×D15 | 1 | 5 | yes | — |
| 199 | C2×C4 | C23×D15 | 1 | 10 | yes | — |
| 200 | C2×C3 | D5×2−1+4 | 1 | 1 | yes | characteristic |
| 201 | C2×C3 | #66287 | 1 | 2 | yes | — |
| 202 | D3 | — | 20 | 5 | no | — |
| 203 | C5 | #87139 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 204 | C22 | — | 30 | 5 | no | — |
| 205 | C22 | — | 30 | 10 | no | — |
| 206 | C22 | C24×D15 | 1 | 1 | yes | characteristic |
| 207 | C4 | — | 2 | 5 | no | — |
| 208 | C4 | — | 2 | 10 | no | — |
| 209 | C4 | — | 15 | 2 | no | — |
| 210 | C4 | — | 30 | 10 | no | — |
| 211 | C3 | #10619040 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 212 | C2 | — | 60 | 5 | no | — |
| 213 | C2 | D15×2−1+4 | 1 | 1 | yes | characteristic |
| 214 | C2 | #11757150 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D15 |
| commutator, socle, cosocle | C22×C3×C5 | C25 |
| Fitting | C3×C5×#564 | C2 |
| hypercenter | #564 | D15 |
| hypocenter | C3×C5 | #3531 |
| LCS | G ▷ C22×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×#564 | C2 | semidirect |
| 2. #11756459 | C2 | semidirect |
| 3. #11756458 | C2 | semidirect |
| 4. #11756450 | C2 | semidirect |
| 5. #11756447 | C2 | semidirect |
| 22. C5×#564 | D3 | semidirect |
| 41. C3×#564 | D5 | semidirect |
| 98. #564 | D15 | semidirect |
| 180. C3×C5 | #3531 | semidirect |
| 203. C5 | #87139 | semidirect |
| 211. C3 | #10619040 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×#564 | 2 |
| #11756459 | 2 |
| #11756458 | 2 |
| #11756450 | 2 |
| #11756447 | 2 |
| #10619040 | 3 |
| #87139 | 5 |