Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
2268 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 136 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 | #41×A5 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C2×D4×A5 | C2 | 1 | 9 | yes | maximal, complemented |
| 3 | C23×A5 | C22 | 1 | 6 | yes | complemented |
| 4 | Q8×A5 | C22 | 1 | 2 | yes | complemented |
| 5 | D4×A5 | C22 | 1 | 18 | yes | complemented |
| 6 | C2×C4×A5 | C22 | 1 | 9 | yes | complemented |
| 7 | A4×H(5,𝔽2) | — | 5 | 1 | no | maximal, Hall |
| 8 | D5×H(5,𝔽2) | — | 6 | 1 | no | maximal |
| 9 | C22×A5 | — | 4 | 6 | no | — |
| 10 | C22×A5 | C23 | 1 | 9 | yes | — |
| 11 | C4×A5 | C23 | 1 | 6 | yes | — |
| 12 | D3×H(5,𝔽2) | — | 10 | 1 | no | maximal |
| 13 | A4×#41 | — | 5 | 6 | no | — |
| 14 | C2×D4×A4 | — | 5 | 9 | no | — |
| 15 | C5×H(5,𝔽2) | — | 6 | 1 | no | — |
| 16 | #4149 | — | 6 | 6 | no | — |
| 17 | D5×#41 | — | 6 | 6 | no | — |
| 18 | #4144 | — | 6 | 9 | no | — |
| 19 | C2×D4×D5 | — | 6 | 9 | no | — |
| 20 | C22×H(5,𝔽2) | — | 5 | 1 | no | Sylow |
| 21 | C2×A5 | — | 2 | 9 | no | — |
| 22 | C2×A5 | C24 | 1 | 1 | yes | characteristic |
| 23 | C23×A4 | — | 5 | 6 | no | — |
| 24 | C3×H(5,𝔽2) | — | 10 | 1 | no | — |
| 25 | #1009 | — | 10 | 6 | no | — |
| 26 | D3×#41 | — | 10 | 6 | no | — |
| 27 | #1004 | — | 10 | 9 | no | — |
| 28 | C2×D3×D4 | — | 10 | 9 | no | — |
| 29 | Q8×A4 | — | 5 | 2 | no | — |
| 30 | D4×A4 | — | 5 | 18 | no | — |
| 31 | C2×C4×A4 | — | 5 | 9 | no | — |
| 32 | C23×D5 | — | 6 | 6 | no | — |
| 33 | C5×#41 | — | 6 | 6 | no | — |
| 34 | C2×C5×D4 | — | 6 | 9 | no | — |
| 35 | C2×C22⋊D5 | — | 6 | 18 | no | — |
| 36 | #713 | — | 6 | 6 | no | — |
| 37 | Q8×D5 | — | 6 | 2 | no | — |
| 38 | #711 | — | 6 | 18 | no | — |
| 39 | D4×D5 | — | 6 | 18 | no | — |
| 40 | D4×D5 | — | 6 | 36 | no | — |
| 41 | #709 | — | 6 | 18 | no | — |
| 42 | C2×D20 | — | 6 | 9 | no | — |
| 43 | C2×C4×D5 | — | 6 | 9 | no | — |
| 44 | C2×H(5,𝔽2) | — | 15 | 9 | no | — |
| 45 | C2×H(5,𝔽2) | — | 15 | 6 | no | — |
| 46 | C2×H(5,𝔽2) | — | 15 | 1 | no | — |
| 47 | C22×#41 | — | 5 | 6 | no | — |
| 48 | C23×D4 | — | 5 | 9 | no | — |
| 49 | A5 | H(5,𝔽2) | 1 | 1 | yes | characteristic, direct product factor |
| 50 | C23×D3 | — | 10 | 6 | no | — |
| 51 | C22×A4 | — | 5 | 9 | no | — |
| 52 | C22×A4 | — | 20 | 6 | no | — |
| 53 | C3×#41 | — | 10 | 6 | no | — |
| 54 | C2×C3×D4 | — | 10 | 9 | no | — |
| 55 | C2×#67 | — | 10 | 18 | no | — |
| 56 | #239 | — | 10 | 6 | no | — |
| 57 | D3×Q8 | — | 10 | 2 | no | — |
| 58 | #237 | — | 10 | 18 | no | — |
| 59 | D3×D4 | — | 10 | 18 | no | — |
| 60 | D3×D4 | — | 10 | 36 | no | — |
| 61 | #235 | — | 10 | 18 | no | — |
| 62 | C2×D12 | — | 10 | 9 | no | — |
| 63 | C2×C4×D3 | — | 10 | 9 | no | — |
| 64 | C4×A4 | — | 5 | 6 | no | — |
| 65 | C23×C5 | — | 6 | 6 | no | — |
| 66 | C22×D5 | — | 6 | 9 | no | — |
| 67 | C22×D5 | — | 6 | 18 | no | — |
| 68 | C22×D5 | — | 24 | 6 | no | — |
| 69 | C22×D5 | — | 24 | 6 | no | — |
| 70 | C5×Q8 | — | 6 | 2 | no | — |
| 71 | C5×D4 | — | 6 | 18 | no | — |
| 72 | C2×C4×C5 | — | 6 | 9 | no | — |
| 73 | C22⋊D5 | — | 6 | 36 | no | — |
| 74 | C2×Dic5 | — | 6 | 9 | no | — |
| 75 | D20 | — | 6 | 18 | no | — |
| 76 | C4×D5 | — | 6 | 6 | no | — |
| 77 | C4×D5 | — | 6 | 18 | no | — |
| 78 | Dic10 | — | 6 | 6 | no | — |
| 79 | C25 | — | 5 | 6 | no | — |
| 80 | H(5,𝔽2) | — | 15 | 4 | no | — |
| 81 | H(5,𝔽2) | — | 15 | 18 | no | — |
| 82 | H(5,𝔽2) | — | 15 | 36 | no | — |
| 83 | H(5,𝔽2) | — | 15 | 12 | no | — |
| 84 | H(5,𝔽2) | — | 15 | 6 | no | — |
| 85 | H(5,𝔽2) | — | 15 | 9 | no | — |
| 86 | H(5,𝔽2) | A5 | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 87 | C2×#41 | — | 15 | 18 | no | — |
| 88 | C2×#41 | — | 15 | 18 | no | — |
| 89 | C2×#41 | — | 15 | 6 | no | — |
| 90 | C2×#41 | — | 15 | 6 | no | — |
| 91 | C22×Q8 | — | 5 | 2 | no | — |
| 92 | C22×D4 | — | 5 | 18 | no | — |
| 93 | C22×D4 | — | 15 | 9 | no | — |
| 94 | C22×D4 | — | 15 | 9 | no | — |
| 95 | C22×D4 | — | 15 | 18 | no | — |
| 96 | C22×D4 | — | 15 | 36 | no | — |
| 97 | C23×C4 | — | 5 | 9 | no | — |
| 98 | C23×C3 | — | 10 | 6 | no | — |
| 99 | C22×D3 | — | 10 | 9 | no | — |
| 100 | C22×D3 | — | 10 | 18 | no | — |
| 101 | C22×D3 | — | 40 | 6 | no | — |
| 102 | C22×D3 | — | 40 | 6 | no | — |
| 103 | C2×A4 | — | 5 | 1 | no | — |
| 104 | C2×A4 | — | 10 | 9 | no | — |
| 105 | C3×Q8 | — | 10 | 2 | no | — |
| 106 | C3×D4 | — | 10 | 18 | no | — |
| 107 | C2×C3×C4 | — | 10 | 9 | no | — |
| 108 | #67 | — | 10 | 36 | no | — |
| 109 | C2×Dic3 | — | 10 | 9 | no | — |
| 110 | D12 | — | 10 | 18 | no | — |
| 111 | C4×D3 | — | 10 | 6 | no | — |
| 112 | C4×D3 | — | 10 | 18 | no | — |
| 113 | Dic6 | — | 10 | 6 | no | — |
| 114 | C22×C5 | — | 6 | 9 | no | — |
| 115 | C22×C5 | — | 24 | 6 | no | — |
| 116 | C2×D5 | — | 6 | 1 | no | — |
| 117 | C2×D5 | — | 6 | 9 | no | — |
| 118 | C2×D5 | — | 12 | 9 | no | — |
| 119 | C2×D5 | — | 12 | 9 | no | — |
| 120 | C2×D5 | — | 24 | 18 | no | — |
| 121 | C4×C5 | — | 6 | 6 | no | — |
| 122 | Dic5 | — | 6 | 6 | no | — |
| 123 | C24 | — | 5 | 9 | no | — |
| 124 | C24 | — | 15 | 18 | no | — |
| 125 | C24 | — | 15 | 6 | no | — |
| 126 | C24 | — | 20 | 6 | no | — |
| 127 | C24 | — | 60 | 6 | no | — |
| 128 | #41 | — | 15 | 36 | no | — |
| 129 | #41 | — | 15 | 36 | no | — |
| 130 | #41 | — | 15 | 12 | no | — |
| 131 | #41 | — | 15 | 6 | no | — |
| 132 | #41 | — | 15 | 18 | no | — |
| 133 | #41 | — | 15 | 18 | no | — |
| 134 | #41 | C2×A5 | 1 | 6 | yes | complemented |
| 135 | C2×Q8 | — | 15 | 6 | no | — |
| 136 | C2×Q8 | — | 15 | 2 | no | — |
| 137 | C2×D4 | — | 15 | 72 | no | — |
| 138 | C2×D4 | — | 15 | 18 | no | — |
| 139 | C2×D4 | — | 15 | 36 | no | — |
| 140 | C2×D4 | — | 15 | 18 | no | — |
| 141 | C2×D4 | — | 15 | 18 | no | — |
| 142 | C2×D4 | — | 15 | 36 | no | — |
| 143 | C2×D4 | — | 15 | 18 | no | — |
| 144 | C2×D4 | — | 15 | 36 | no | — |
| 145 | C2×D4 | — | 15 | 9 | no | — |
| 146 | C2×D4 | C2×A5 | 1 | 9 | yes | complemented |
| 147 | C22×C4 | — | 5 | 6 | no | — |
| 148 | C22×C4 | — | 15 | 9 | no | — |
| 149 | C22×C4 | — | 15 | 9 | no | — |
| 150 | C22×C4 | — | 15 | 18 | no | — |
| 151 | C22×C3 | — | 10 | 9 | no | — |
| 152 | C22×C3 | — | 40 | 6 | no | — |
| 153 | C2×D3 | — | 10 | 1 | no | — |
| 154 | C2×D3 | — | 10 | 9 | no | — |
| 155 | C2×D3 | — | 20 | 9 | no | — |
| 156 | C2×D3 | — | 20 | 9 | no | — |
| 157 | C2×D3 | — | 40 | 18 | no | — |
| 158 | A4 | — | 5 | 1 | no | — |
| 159 | C3×C4 | — | 10 | 6 | no | — |
| 160 | Dic3 | — | 10 | 6 | no | — |
| 161 | C2×C5 | — | 6 | 1 | no | — |
| 162 | C2×C5 | — | 12 | 9 | no | — |
| 163 | D5 | — | 6 | 1 | no | — |
| 164 | D5 | — | 6 | 1 | no | — |
| 165 | D5 | — | 12 | 9 | no | — |
| 166 | C23 | — | 5 | 1 | no | — |
| 167 | C23 | — | 10 | 9 | no | — |
| 168 | C23 | — | 15 | 12 | no | — |
| 169 | C23 | — | 15 | 9 | no | — |
| 170 | C23 | — | 15 | 9 | no | — |
| 171 | C23 | — | 15 | 18 | no | — |
| 172 | C23 | — | 30 | 9 | no | — |
| 173 | C23 | — | 60 | 6 | no | — |
| 174 | C23 | — | 60 | 6 | no | — |
| 175 | C23 | — | 60 | 18 | no | — |
| 176 | C23 | — | 60 | 18 | no | — |
| 177 | C23 | C22×A5 | 1 | 6 | yes | complemented |
| 178 | Q8 | — | 15 | 4 | no | — |
| 179 | Q8 | — | 15 | 6 | no | — |
| 180 | Q8 | C22×A5 | 1 | 2 | yes | complemented |
| 181 | D4 | — | 15 | 36 | no | — |
| 182 | D4 | — | 15 | 36 | no | — |
| 183 | D4 | — | 15 | 18 | no | — |
| 184 | D4 | C22×A5 | 1 | 18 | yes | complemented |
| 185 | C2×C4 | — | 15 | 18 | no | — |
| 186 | C2×C4 | — | 15 | 9 | no | — |
| 187 | C2×C4 | — | 15 | 6 | no | — |
| 188 | C2×C4 | — | 15 | 18 | no | — |
| 189 | C2×C4 | — | 15 | 6 | no | — |
| 190 | C2×C4 | C22×A5 | 1 | 9 | yes | complemented |
| 191 | C2×C3 | — | 10 | 1 | no | — |
| 192 | C2×C3 | — | 20 | 9 | no | — |
| 193 | D3 | — | 10 | 1 | no | — |
| 194 | D3 | — | 10 | 1 | no | — |
| 195 | D3 | — | 20 | 9 | no | — |
| 196 | C5 | — | 6 | 1 | no | Sylow |
| 197 | C22 | — | 4 | 6 | no | — |
| 198 | C22 | — | 5 | 1 | no | — |
| 199 | C22 | — | 15 | 1 | no | — |
| 200 | C22 | — | 15 | 1 | no | — |
| 201 | C22 | — | 15 | 9 | no | — |
| 202 | C22 | — | 30 | 9 | no | — |
| 203 | C22 | — | 30 | 9 | no | — |
| 204 | C22 | — | 30 | 9 | no | — |
| 205 | C22 | — | 30 | 9 | no | — |
| 206 | C22 | — | 60 | 18 | no | — |
| 207 | C22 | — | 60 | 12 | no | — |
| 208 | C22 | C23×A5 | 1 | 9 | yes | — |
| 209 | C4 | — | 15 | 6 | no | — |
| 210 | C4 | C23×A5 | 1 | 6 | yes | — |
| 211 | C3 | — | 10 | 1 | no | Sylow |
| 212 | C2 | — | 2 | 9 | no | — |
| 213 | C2 | — | 15 | 1 | no | — |
| 214 | C2 | — | 15 | 1 | no | — |
| 215 | C2 | — | 30 | 9 | no | — |
| 216 | C2 | C24×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2 | C24×A5 |
| commutator, socle | C2×A5 | C24 |
| Fitting, hypercenter | H(5,𝔽2) | A5 |
| hypocenter | A5 | H(5,𝔽2) |
| LCS | G ▷ C2×A5 ▷ A5 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #41×A5 | C2 | semidirect |
| 2. C2×D4×A5 | C2 | semidirect |
| 3. C23×A5 | C22 | semidirect |
| 4. Q8×A5 | C22 | semidirect |
| 5. D4×A5 | C22 | semidirect |
| 6. C2×C4×A5 | C22 | semidirect |
| 49. A5 | H(5,𝔽2) | direct (N × H) |
| 86. H(5,𝔽2) | A5 | direct (N × H) |
| 134. #41 | C2×A5 | semidirect |
| 146. C2×D4 | C2×A5 | semidirect |
| 177. C23 | C22×A5 | semidirect |
| 180. Q8 | C22×A5 | semidirect |
| 184. D4 | C22×A5 | semidirect |
| 190. C2×C4 | C22×A5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×A5 | 2 |
| C2×D4×A5 | 2 |
| A4×H(5,𝔽2) | 5 |
| D5×H(5,𝔽2) | 6 |
| D3×H(5,𝔽2) | 10 |