Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
1238 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 32 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×#585⋊C5 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | H(7,𝔽2)⋊C5 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C3×2−1+4⋊C5 | C22 | 1 | 1 | yes | characteristic, complemented |
| 4 | C3×H(7,𝔽2) | C5 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | #585⋊C5 | C2×C3 | 1 | 3 | yes | complemented |
| 6 | C3×#585 | — | 5 | 3 | no | — |
| 7 | C3×#585 | — | 5 | 2 | no | — |
| 8 | C3×#585 | C2×C5 | 1 | 3 | yes | — |
| 9 | C2×C3×H(5,𝔽2) | — | 5 | 1 | no | — |
| 10 | C2×C3×H(5,𝔽2) | — | 5 | 6 | no | — |
| 11 | 2−1+4⋊C5 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 12 | H(7,𝔽2) | C3×C5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C3×C5×Q8 | — | 16 | 1 | no | maximal |
| 14 | C3×2−1+4 | — | 5 | 6 | no | — |
| 15 | C3×2−1+4 | — | 5 | 2 | no | — |
| 16 | C3×2−1+4 | — | 5 | 3 | no | — |
| 17 | C3×2−1+4 | C22×C5 | 1 | 1 | yes | characteristic |
| 18 | C3×H(5,𝔽2) | — | 5 | 2 | no | — |
| 19 | C3×H(5,𝔽2) | — | 5 | 6 | no | — |
| 20 | C3×H(5,𝔽2) | — | 5 | 12 | no | — |
| 21 | C3×H(5,𝔽2) | — | 5 | 12 | no | — |
| 22 | C3×H(5,𝔽2) | — | 5 | 12 | no | — |
| 23 | C3×H(5,𝔽2) | — | 5 | 6 | no | — |
| 24 | C3×H(5,𝔽2) | — | 5 | 6 | no | — |
| 25 | C2×C3×#41 | — | 5 | 3 | no | — |
| 26 | C2×C3×#41 | — | 5 | 2 | no | — |
| 27 | C2×C3×#41 | — | 5 | 12 | no | — |
| 28 | C2×C3×#41 | — | 5 | 1 | no | — |
| 29 | C2×C3×#41 | — | 5 | 6 | no | — |
| 30 | C2×C3×#41 | — | 5 | 12 | no | — |
| 31 | C2×C3×#41 | — | 5 | 6 | no | — |
| 32 | C22×C3×D4 | — | 5 | 12 | no | — |
| 33 | C22×C3×D4 | — | 5 | 3 | no | — |
| 34 | C22×C3×D4 | — | 5 | 6 | no | — |
| 35 | #585 | — | 5 | 3 | no | — |
| 36 | #585 | — | 5 | 2 | no | — |
| 37 | #585 | C2×C3×C5 | 1 | 3 | yes | — |
| 38 | C2×H(5,𝔽2) | — | 5 | 1 | no | — |
| 39 | C2×H(5,𝔽2) | — | 5 | 6 | no | — |
| 40 | C3×C4×C5 | — | 16 | 3 | no | — |
| 41 | C24×C3 | — | 5 | 6 | no | — |
| 42 | C3×#41 | — | 5 | 12 | no | — |
| 43 | C3×#41 | — | 5 | 12 | no | — |
| 44 | C3×#41 | — | 5 | 6 | no | — |
| 45 | C3×#41 | — | 5 | 24 | no | — |
| 46 | C3×#41 | — | 5 | 12 | no | — |
| 47 | C3×#41 | — | 5 | 12 | no | — |
| 48 | C3×#41 | — | 5 | 6 | no | — |
| 49 | C3×#41 | — | 5 | 12 | no | — |
| 50 | C3×#41 | — | 5 | 2 | no | — |
| 51 | C3×#41 | — | 5 | 6 | no | — |
| 52 | C3×#41 | — | 5 | 6 | no | — |
| 53 | C3×#41 | — | 5 | 2 | no | — |
| 54 | C2×C3×Q8 | — | 5 | 1 | no | — |
| 55 | C2×C3×Q8 | — | 5 | 6 | no | — |
| 56 | C2×C3×Q8 | — | 5 | 6 | no | — |
| 57 | C2×C3×Q8 | — | 5 | 1 | no | — |
| 58 | C2×C3×D4 | — | 5 | 6 | no | — |
| 59 | C2×C3×D4 | — | 5 | 12 | no | — |
| 60 | C2×C3×D4 | — | 5 | 24 | no | — |
| 61 | C2×C3×D4 | — | 5 | 12 | no | — |
| 62 | C2×C3×D4 | — | 5 | 12 | no | — |
| 63 | C2×C3×D4 | — | 5 | 24 | no | — |
| 64 | C2×C3×D4 | — | 5 | 3 | no | — |
| 65 | C2×C3×D4 | — | 5 | 3 | no | — |
| 66 | C2×C3×D4 | — | 5 | 12 | no | — |
| 67 | C2×C3×D4 | — | 5 | 6 | no | — |
| 68 | C2×C3×D4 | — | 5 | 6 | no | — |
| 69 | C2×C3×D4 | — | 5 | 6 | no | — |
| 70 | C22×C3×C4 | — | 5 | 12 | no | — |
| 71 | C22×C3×C4 | — | 5 | 3 | no | — |
| 72 | C22×C3×C4 | — | 5 | 6 | no | — |
| 73 | C5×Q8 | — | 16 | 1 | no | — |
| 74 | 2−1+4 | — | 5 | 6 | no | — |
| 75 | 2−1+4 | — | 5 | 2 | no | — |
| 76 | 2−1+4 | — | 5 | 3 | no | — |
| 77 | 2−1+4 | C22×C3×C5 | 1 | 1 | yes | characteristic |
| 78 | H(5,𝔽2) | — | 5 | 2 | no | — |
| 79 | H(5,𝔽2) | — | 5 | 6 | no | — |
| 80 | H(5,𝔽2) | — | 5 | 12 | no | — |
| 81 | H(5,𝔽2) | — | 5 | 12 | no | — |
| 82 | H(5,𝔽2) | — | 5 | 12 | no | — |
| 83 | H(5,𝔽2) | — | 5 | 6 | no | — |
| 84 | H(5,𝔽2) | — | 5 | 6 | no | — |
| 85 | C2×#41 | — | 5 | 3 | no | — |
| 86 | C2×#41 | — | 5 | 2 | no | — |
| 87 | C2×#41 | — | 5 | 12 | no | — |
| 88 | C2×#41 | — | 5 | 1 | no | — |
| 89 | C2×#41 | — | 5 | 6 | no | — |
| 90 | C2×#41 | — | 5 | 12 | no | — |
| 91 | C2×#41 | — | 5 | 6 | no | — |
| 92 | C22×D4 | — | 5 | 12 | no | — |
| 93 | C22×D4 | — | 5 | 3 | no | — |
| 94 | C22×D4 | — | 5 | 6 | no | — |
| 95 | C2×C3×C5 | — | 16 | 1 | no | — |
| 96 | C23×C3 | — | 5 | 12 | no | — |
| 97 | C23×C3 | — | 5 | 3 | no | — |
| 98 | C23×C3 | — | 5 | 6 | no | — |
| 99 | C23×C3 | — | 40 | 6 | no | — |
| 100 | C3×Q8 | — | 5 | 6 | no | — |
| 101 | C3×Q8 | — | 5 | 2 | no | — |
| 102 | C3×Q8 | — | 5 | 3 | no | — |
| 103 | C3×Q8 | C24⋊C5 | 1 | 1 | yes | characteristic |
| 104 | C3×D4 | — | 5 | 2 | no | — |
| 105 | C3×D4 | — | 5 | 6 | no | — |
| 106 | C3×D4 | — | 5 | 6 | no | — |
| 107 | C3×D4 | — | 5 | 12 | no | — |
| 108 | C3×D4 | — | 5 | 12 | no | — |
| 109 | C3×D4 | — | 5 | 12 | no | — |
| 110 | C3×D4 | — | 5 | 6 | no | — |
| 111 | C2×C3×C4 | — | 5 | 12 | no | — |
| 112 | C2×C3×C4 | — | 5 | 6 | no | — |
| 113 | C2×C3×C4 | — | 5 | 12 | no | — |
| 114 | C2×C3×C4 | — | 5 | 1 | no | — |
| 115 | C2×C3×C4 | — | 5 | 3 | no | — |
| 116 | C2×C3×C4 | — | 5 | 6 | no | — |
| 117 | C2×C3×C4 | — | 5 | 2 | no | — |
| 118 | C4×C5 | — | 16 | 3 | no | — |
| 119 | C24 | — | 5 | 6 | no | — |
| 120 | #41 | — | 5 | 12 | no | — |
| 121 | #41 | — | 5 | 12 | no | — |
| 122 | #41 | — | 5 | 6 | no | — |
| 123 | #41 | — | 5 | 24 | no | — |
| 124 | #41 | — | 5 | 12 | no | — |
| 125 | #41 | — | 5 | 12 | no | — |
| 126 | #41 | — | 5 | 6 | no | — |
| 127 | #41 | — | 5 | 12 | no | — |
| 128 | #41 | — | 5 | 2 | no | — |
| 129 | #41 | — | 5 | 6 | no | — |
| 130 | #41 | — | 5 | 6 | no | — |
| 131 | #41 | — | 5 | 2 | no | — |
| 132 | C2×Q8 | — | 5 | 1 | no | — |
| 133 | C2×Q8 | — | 5 | 6 | no | — |
| 134 | C2×Q8 | — | 5 | 6 | no | — |
| 135 | C2×Q8 | — | 5 | 1 | no | — |
| 136 | C2×D4 | — | 5 | 6 | no | — |
| 137 | C2×D4 | — | 5 | 12 | no | — |
| 138 | C2×D4 | — | 5 | 24 | no | — |
| 139 | C2×D4 | — | 5 | 12 | no | — |
| 140 | C2×D4 | — | 5 | 12 | no | — |
| 141 | C2×D4 | — | 5 | 24 | no | — |
| 142 | C2×D4 | — | 5 | 3 | no | — |
| 143 | C2×D4 | — | 5 | 3 | no | — |
| 144 | C2×D4 | — | 5 | 12 | no | — |
| 145 | C2×D4 | — | 5 | 6 | no | — |
| 146 | C2×D4 | — | 5 | 6 | no | — |
| 147 | C2×D4 | — | 5 | 6 | no | — |
| 148 | C22×C4 | — | 5 | 12 | no | — |
| 149 | C22×C4 | — | 5 | 3 | no | — |
| 150 | C22×C4 | — | 5 | 6 | no | — |
| 151 | C3×C5 | — | 16 | 1 | no | Hall |
| 152 | C22×C3 | — | 5 | 6 | no | — |
| 153 | C22×C3 | — | 5 | 1 | no | — |
| 154 | C22×C3 | — | 20 | 6 | no | — |
| 155 | C22×C3 | — | 20 | 12 | no | — |
| 156 | C22×C3 | — | 20 | 3 | no | — |
| 157 | C3×C4 | — | 5 | 3 | no | — |
| 158 | C3×C4 | — | 5 | 2 | no | — |
| 159 | C3×C4 | C2×C24⋊C5 | 1 | 3 | yes | — |
| 160 | C2×C5 | — | 16 | 1 | no | — |
| 161 | C23 | — | 5 | 12 | no | — |
| 162 | C23 | — | 5 | 3 | no | — |
| 163 | C23 | — | 5 | 6 | no | — |
| 164 | C23 | — | 40 | 6 | no | — |
| 165 | Q8 | — | 5 | 6 | no | — |
| 166 | Q8 | — | 5 | 2 | no | — |
| 167 | Q8 | — | 5 | 3 | no | — |
| 168 | Q8 | C3×C24⋊C5 | 1 | 1 | yes | characteristic |
| 169 | D4 | — | 5 | 2 | no | — |
| 170 | D4 | — | 5 | 6 | no | — |
| 171 | D4 | — | 5 | 6 | no | — |
| 172 | D4 | — | 5 | 12 | no | — |
| 173 | D4 | — | 5 | 12 | no | — |
| 174 | D4 | — | 5 | 12 | no | — |
| 175 | D4 | — | 5 | 6 | no | — |
| 176 | C2×C4 | — | 5 | 12 | no | — |
| 177 | C2×C4 | — | 5 | 6 | no | — |
| 178 | C2×C4 | — | 5 | 12 | no | — |
| 179 | C2×C4 | — | 5 | 1 | no | — |
| 180 | C2×C4 | — | 5 | 3 | no | — |
| 181 | C2×C4 | — | 5 | 6 | no | — |
| 182 | C2×C4 | — | 5 | 2 | no | — |
| 183 | C2×C3 | — | 10 | 6 | no | — |
| 184 | C2×C3 | — | 10 | 1 | no | — |
| 185 | C2×C3 | C22×C24⋊C5 | 1 | 1 | yes | characteristic |
| 186 | C5 | — | 16 | 1 | no | Sylow |
| 187 | C22 | — | 5 | 6 | no | — |
| 188 | C22 | — | 5 | 1 | no | — |
| 189 | C22 | — | 20 | 6 | no | — |
| 190 | C22 | — | 20 | 3 | no | — |
| 191 | C22 | — | 20 | 12 | no | — |
| 192 | C4 | — | 5 | 3 | no | — |
| 193 | C4 | — | 5 | 2 | no | — |
| 194 | C4 | C2×C3×C24⋊C5 | 1 | 3 | yes | — |
| 195 | C3 | H(7,𝔽2)⋊C5 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 196 | C2 | — | 10 | 1 | no | — |
| 197 | C2 | — | 10 | 6 | no | — |
| 198 | C2 | C22×C3×C24⋊C5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×C24⋊C5 |
| commutator, cosocle, hypocenter | 2−1+4 | C22×C3×C5 |
| Frattini | C2 | C22×C3×C24⋊C5 |
| Fitting | C3×H(7,𝔽2) | C5 |
| hypercenter | C3×Q8 | C24⋊C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#585⋊C5 | C2 | semidirect |
| 2. H(7,𝔽2)⋊C5 | C3 | direct (N × H) |
| 3. C3×2−1+4⋊C5 | C22 | semidirect |
| 4. C3×H(7,𝔽2) | C5 | semidirect |
| 5. #585⋊C5 | C2×C3 | semidirect |
| 11. 2−1+4⋊C5 | C22×C3 | semidirect |
| 12. H(7,𝔽2) | C3×C5 | semidirect |
| 195. C3 | H(7,𝔽2)⋊C5 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#585⋊C5 | 2 |
| H(7,𝔽2)⋊C5 | 3 |
| C3×H(7,𝔽2) | 5 |
| C3×C5×Q8 | 16 |