Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
440 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 65 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 | C2×#87877 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11717448 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11717435 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11716781 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C22×D52 | C4 | 1 | 2 | yes | — |
| 6 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C5×D20 | C4 | 1 | 2 | yes | — |
| 8 | C2×#6037 | C22 | 1 | 2 | yes | — |
| 9 | #87877 | — | 2 | 1 | no | — |
| 10 | #87877 | — | 2 | 1 | no | — |
| 11 | #87877 | — | 2 | 1 | no | — |
| 12 | #87877 | — | 2 | 1 | no | — |
| 13 | #87867 | C22 | 1 | 2 | yes | complemented |
| 14 | #87863 | — | 2 | 1 | no | — |
| 15 | #87862 | — | 2 | 1 | no | — |
| 16 | #87860 | C22 | 1 | 2 | yes | complemented |
| 17 | #87856 | — | 2 | 1 | no | — |
| 18 | #87855 | — | 2 | 1 | no | — |
| 19 | D5⋊#100 | — | 5 | 2 | no | maximal |
| 20 | C22×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C22×C5×D5 | C2×C4 | 1 | 4 | yes | — |
| 22 | C2×D52 | — | 2 | 2 | no | — |
| 23 | C2×D52 | — | 2 | 2 | no | — |
| 24 | C2×D52 | — | 4 | 4 | no | — |
| 25 | C2×C4×C52 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×#1039 | C23 | 1 | 1 | yes | characteristic |
| 27 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 28 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 29 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 30 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 31 | C5×D20 | — | 2 | 2 | no | — |
| 32 | C5×D20 | — | 2 | 2 | no | — |
| 33 | #6037 | — | 2 | 2 | no | — |
| 34 | #6037 | — | 2 | 2 | no | — |
| 35 | #6034 | — | 2 | 2 | no | — |
| 36 | #6033 | — | 2 | 2 | no | — |
| 37 | C2×D4×D5 | — | 5 | 2 | no | — |
| 38 | D5⋊M16 | — | 5 | 2 | no | — |
| 39 | D5⋊M16 | — | 5 | 2 | no | — |
| 40 | D5⋊M16 | — | 10 | 2 | no | — |
| 41 | #4013 | — | 5 | 8 | no | — |
| 42 | C22×C52 | C22×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 44 | C2×C5⋊D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C2×C5⋊D5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | C2×C5×D5 | — | 2 | 4 | no | — |
| 47 | C2×C5×D5 | — | 4 | 4 | no | — |
| 48 | D52 | — | 4 | 4 | no | — |
| 49 | D52 | — | 4 | 4 | no | — |
| 50 | C4×C52 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 51 | C4×C52 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 52 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 53 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C23×D5 | — | 5 | 4 | no | — |
| 55 | C2×C5×D4 | — | 5 | 2 | no | — |
| 56 | C2×C22⋊D5 | — | 5 | 4 | no | — |
| 57 | D4×D5 | — | 10 | 2 | no | — |
| 58 | D4×D5 | — | 10 | 2 | no | — |
| 59 | D4×D5 | — | 10 | 2 | no | — |
| 60 | D4×D5 | — | 10 | 2 | no | — |
| 61 | C2×D20 | AGL(1,𝔽5) | 1 | 2 | yes | — |
| 62 | C2×C4×D5 | — | 5 | 2 | no | — |
| 63 | C2×C4×D5 | — | 10 | 2 | no | — |
| 64 | #704 | — | 5 | 4 | no | — |
| 65 | #704 | — | 5 | 4 | no | — |
| 66 | #704 | — | 10 | 4 | no | — |
| 67 | #700 | — | 10 | 2 | no | — |
| 68 | #700 | — | 10 | 2 | no | — |
| 69 | #700 | — | 10 | 4 | no | — |
| 70 | D5⋊C8 | — | 10 | 2 | no | — |
| 71 | D5⋊C8 | — | 10 | 2 | no | — |
| 72 | D5⋊C8 | — | 10 | 4 | no | — |
| 73 | C2×#100 | — | 25 | 1 | no | Sylow |
| 74 | C2×C52 | — | 2 | 1 | no | — |
| 75 | C2×C52 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C5⋊D5 | — | 2 | 1 | no | — |
| 77 | C5⋊D5 | #100 | 1 | 2 | yes | complemented |
| 78 | C5×D5 | — | 4 | 4 | no | — |
| 79 | C23×C5 | — | 5 | 4 | no | — |
| 80 | C22×D5 | — | 5 | 2 | no | — |
| 81 | C22×D5 | — | 10 | 2 | no | — |
| 82 | C22×D5 | — | 10 | 4 | no | — |
| 83 | C22×D5 | — | 10 | 4 | no | — |
| 84 | C22×D5 | — | 20 | 8 | no | — |
| 85 | C22×D5 | C2×AGL(1,𝔽5) | 1 | 4 | yes | — |
| 86 | C5×D4 | — | 10 | 2 | no | — |
| 87 | C5×D4 | — | 10 | 2 | no | — |
| 88 | C2×C4×C5 | — | 2 | 2 | no | — |
| 89 | C2×C4×C5 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 90 | C22⋊D5 | — | 10 | 4 | no | — |
| 91 | C22⋊D5 | — | 10 | 4 | no | — |
| 92 | C2×Dic5 | — | 5 | 2 | no | — |
| 93 | C2×Dic5 | — | 10 | 2 | no | — |
| 94 | D20 | — | 2 | 2 | no | — |
| 95 | D20 | — | 2 | 2 | no | — |
| 96 | C4×D5 | — | 5 | 2 | no | — |
| 97 | C4×D5 | — | 5 | 2 | no | — |
| 98 | C4×D5 | — | 5 | 2 | no | — |
| 99 | C4×D5 | — | 5 | 2 | no | — |
| 100 | C4×D5 | — | 10 | 2 | no | — |
| 101 | C4×D5 | — | 10 | 2 | no | — |
| 102 | C4×D5 | — | 10 | 2 | no | — |
| 103 | C4×D5 | — | 10 | 2 | no | — |
| 104 | #170 | — | 10 | 4 | no | — |
| 105 | #170 | — | 10 | 4 | no | — |
| 106 | #170 | — | 10 | 8 | no | — |
| 107 | C22×D4 | — | 25 | 1 | no | — |
| 108 | C2×M16 | — | 25 | 1 | no | — |
| 109 | C2×M16 | — | 25 | 1 | no | — |
| 110 | #100 | — | 25 | 4 | no | — |
| 111 | C52 | C2×#100 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 112 | C22×C5 | — | 2 | 2 | no | — |
| 113 | C22×C5 | — | 10 | 4 | no | — |
| 114 | C22×C5 | — | 20 | 4 | no | — |
| 115 | C22×C5 | C22×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 116 | C2×D5 | — | 2 | 4 | no | — |
| 117 | C2×D5 | — | 4 | 4 | no | — |
| 118 | C2×D5 | — | 5 | 2 | no | — |
| 119 | C2×D5 | — | 5 | 2 | no | — |
| 120 | C2×D5 | — | 10 | 2 | no | — |
| 121 | C2×D5 | — | 10 | 2 | no | — |
| 122 | C2×D5 | — | 10 | 4 | no | — |
| 123 | C2×D5 | — | 20 | 4 | no | — |
| 124 | C2×D5 | — | 20 | 8 | no | — |
| 125 | C2×D5 | — | 20 | 8 | no | — |
| 126 | C4×C5 | — | 2 | 2 | no | — |
| 127 | C4×C5 | — | 2 | 2 | no | — |
| 128 | C4×C5 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 129 | C4×C5 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 130 | Dic5 | — | 5 | 2 | no | — |
| 131 | Dic5 | — | 5 | 2 | no | — |
| 132 | Dic5 | — | 10 | 2 | no | — |
| 133 | Dic5 | — | 10 | 2 | no | — |
| 134 | C24 | — | 25 | 2 | no | — |
| 135 | C2×D4 | — | 5 | 2 | no | — |
| 136 | C2×D4 | — | 25 | 2 | no | — |
| 137 | C2×D4 | — | 50 | 1 | no | — |
| 138 | C2×D4 | — | 50 | 1 | no | — |
| 139 | C2×D4 | — | 50 | 1 | no | — |
| 140 | C2×D4 | — | 50 | 1 | no | — |
| 141 | C22×C4 | — | 25 | 1 | no | — |
| 142 | M16 | — | 25 | 2 | no | — |
| 143 | M16 | — | 25 | 2 | no | — |
| 144 | M16 | — | 50 | 1 | no | — |
| 145 | M16 | — | 50 | 1 | no | — |
| 146 | C2×C8 | — | 50 | 1 | no | — |
| 147 | C2×C8 | — | 50 | 1 | no | — |
| 148 | C2×C5 | — | 2 | 2 | no | — |
| 149 | C2×C5 | — | 2 | 2 | no | — |
| 150 | C2×C5 | — | 4 | 2 | no | — |
| 151 | C2×C5 | — | 20 | 4 | no | — |
| 152 | C2×C5 | C2×C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 153 | D5 | — | 4 | 4 | no | — |
| 154 | D5 | — | 5 | 4 | no | — |
| 155 | D5 | — | 10 | 4 | no | — |
| 156 | D5 | — | 10 | 2 | no | — |
| 157 | D5 | — | 20 | 2 | no | — |
| 158 | C23 | — | 5 | 4 | no | — |
| 159 | C23 | — | 25 | 1 | no | — |
| 160 | C23 | — | 50 | 2 | no | — |
| 161 | C23 | — | 50 | 2 | no | — |
| 162 | C23 | — | 100 | 4 | no | — |
| 163 | D4 | — | 10 | 2 | no | — |
| 164 | D4 | — | 10 | 2 | no | — |
| 165 | D4 | — | 50 | 2 | no | — |
| 166 | D4 | — | 50 | 2 | no | — |
| 167 | C2×C4 | — | 25 | 1 | no | — |
| 168 | C2×C4 | — | 25 | 1 | no | — |
| 169 | C2×C4 | — | 25 | 1 | no | — |
| 170 | C2×C4 | — | 25 | 1 | no | — |
| 171 | C2×C4 | — | 25 | 1 | no | — |
| 172 | C2×C4 | D5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 173 | C8 | — | 50 | 2 | no | — |
| 174 | C8 | — | 50 | 2 | no | — |
| 175 | C5 | — | 2 | 2 | no | — |
| 176 | C5 | D5⋊#100 | 1 | 2 | yes | complemented |
| 177 | C22 | — | 10 | 4 | no | — |
| 178 | C22 | — | 20 | 4 | no | — |
| 179 | C22 | — | 25 | 1 | no | — |
| 180 | C22 | — | 25 | 1 | no | — |
| 181 | C22 | — | 50 | 2 | no | — |
| 182 | C22 | — | 100 | 4 | no | — |
| 183 | C22 | — | 100 | 4 | no | — |
| 184 | C22 | C2×D5⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 185 | C4 | — | 25 | 1 | no | — |
| 186 | C4 | — | 25 | 1 | no | — |
| 187 | C4 | #87831 | 1 | 1 | yes | characteristic |
| 188 | C4 | #87831 | 1 | 1 | yes | characteristic |
| 189 | C2 | — | 2 | 1 | no | — |
| 190 | C2 | — | 20 | 4 | no | — |
| 191 | C2 | — | 25 | 2 | no | — |
| 192 | C2 | — | 50 | 1 | no | — |
| 193 | C2 | C2×#87831 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11717448 | C2 | semidirect |
| 3. #11717435 | C2 | semidirect |
| 4. #11716781 | C2 | semidirect |
| 13. #87867 | C22 | semidirect |
| 16. #87860 | C22 | semidirect |
| 77. C5⋊D5 | #100 | semidirect |
| 111. C52 | C2×#100 | semidirect |
| 176. C5 | D5⋊#100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#87877 | 2 |
| #11717448 | 2 |
| #11717435 | 2 |
| #11716781 | 2 |
| D5⋊#100 | 5 |