Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
224 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 47 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 | #11717451 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11717448 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#87862 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11716802 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×C4×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22⋊#1044 | C4 | 1 | 2 | yes | — |
| 7 | #87867 | — | 2 | 1 | no | — |
| 8 | C2×#6034 | — | 2 | 1 | no | — |
| 9 | C4⋊#1044 | C4 | 1 | 2 | yes | — |
| 10 | C4×#1044 | C22 | 1 | 2 | yes | — |
| 11 | #87863 | — | 2 | 1 | no | — |
| 12 | #87862 | — | 2 | 1 | no | — |
| 13 | #87862 | C22 | 1 | 2 | yes | — |
| 14 | #87862 | C22 | 1 | 2 | yes | — |
| 15 | C22×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C2×#1044 | C2×C4 | 1 | 4 | yes | — |
| 17 | C2×C4×C52 | Q8 | 1 | 1 | yes | characteristic |
| 18 | C2×#1039 | D4 | 1 | 1 | yes | characteristic |
| 19 | C4×C5⋊D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | C4×C5⋊D5 | D4 | 1 | 1 | yes | characteristic |
| 21 | C4×C5⋊D5 | Q8 | 1 | 1 | yes | characteristic |
| 22 | C4×C5⋊D5 | C23 | 1 | 1 | yes | characteristic |
| 23 | #6034 | — | 2 | 2 | no | — |
| 24 | #6034 | — | 2 | 2 | no | — |
| 25 | #4130 | — | 10 | 1 | no | — |
| 26 | D5⋊M16 | — | 10 | 1 | no | — |
| 27 | C2×D5⋊C8 | — | 10 | 1 | no | — |
| 28 | C22×C52 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C5⋊D5 | — | 2 | 2 | no | — |
| 30 | C2×C5⋊D5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 31 | C2×C5⋊D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 32 | #1044 | — | 2 | 4 | no | — |
| 33 | C4×C52 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 34 | C4×C52 | C2×Q8 | 1 | 1 | yes | characteristic |
| 35 | #1039 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 36 | #1039 | C2×D4 | 1 | 1 | yes | characteristic |
| 37 | C2×C4×D5 | — | 10 | 1 | no | — |
| 38 | C2×C4×D5 | — | 10 | 1 | no | — |
| 39 | C2×C4×D5 | — | 10 | 1 | no | — |
| 40 | C22⋊AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 41 | #704 | — | 10 | 2 | no | — |
| 42 | C2×#170 | — | 10 | 2 | no | — |
| 43 | C4⋊AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 44 | C4×AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 45 | #700 | — | 10 | 2 | no | — |
| 46 | D5⋊C8 | — | 10 | 2 | no | — |
| 47 | D5⋊C8 | — | 10 | 2 | no | — |
| 48 | D5⋊C8 | — | 10 | 2 | no | — |
| 49 | #424 | — | 25 | 1 | no | maximal, Sylow |
| 50 | C2×C52 | — | 2 | 1 | no | — |
| 51 | C2×C52 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 52 | C5⋊D5 | — | 2 | 1 | no | — |
| 53 | C5⋊D5 | #131 | 1 | 2 | yes | — |
| 54 | C22×D5 | — | 10 | 1 | no | — |
| 55 | C22×D5 | — | 10 | 1 | no | — |
| 56 | C22×D5 | — | 10 | 1 | no | — |
| 57 | C2×AGL(1,𝔽5) | — | 10 | 4 | no | — |
| 58 | C2×C4×C5 | — | 2 | 1 | no | — |
| 59 | C2×C4×C5 | — | 2 | 1 | no | — |
| 60 | C2×C4×C5 | — | 2 | 1 | no | — |
| 61 | C2×Dic5 | — | 10 | 1 | no | — |
| 62 | C2×Dic5 | — | 10 | 1 | no | — |
| 63 | C2×Dic5 | — | 10 | 1 | no | — |
| 64 | C4×D5 | — | 10 | 1 | no | — |
| 65 | C4×D5 | — | 10 | 1 | no | — |
| 66 | C4×D5 | — | 10 | 1 | no | — |
| 67 | C4×D5 | — | 10 | 1 | no | — |
| 68 | C4×D5 | — | 10 | 1 | no | — |
| 69 | C4×D5 | — | 10 | 1 | no | — |
| 70 | C4×D5 | — | 10 | 1 | no | — |
| 71 | C4×D5 | — | 10 | 1 | no | — |
| 72 | C4×D5 | — | 10 | 1 | no | — |
| 73 | C4×D5 | — | 10 | 1 | no | — |
| 74 | C4×D5 | — | 10 | 1 | no | — |
| 75 | C4×D5 | — | 10 | 1 | no | — |
| 76 | #170 | — | 10 | 4 | no | — |
| 77 | #170 | — | 10 | 4 | no | — |
| 78 | C2×M16 | — | 25 | 1 | no | — |
| 79 | C22×C8 | — | 25 | 1 | no | — |
| 80 | #117 | — | 25 | 1 | no | — |
| 81 | C4⋊C8 | — | 25 | 4 | no | — |
| 82 | C52 | #424 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 83 | C22×C5 | — | 2 | 1 | no | — |
| 84 | C22×C5 | — | 2 | 1 | no | — |
| 85 | C22×C5 | — | 2 | 1 | no | — |
| 86 | C2×D5 | — | 10 | 1 | no | — |
| 87 | C2×D5 | — | 10 | 1 | no | — |
| 88 | C2×D5 | — | 10 | 1 | no | — |
| 89 | C2×D5 | — | 10 | 1 | no | — |
| 90 | C2×D5 | — | 10 | 1 | no | — |
| 91 | C2×D5 | — | 10 | 1 | no | — |
| 92 | C2×D5 | — | 10 | 4 | no | — |
| 93 | C2×D5 | — | 20 | 2 | no | — |
| 94 | C2×D5 | — | 20 | 2 | no | — |
| 95 | AGL(1,𝔽5) | — | 20 | 4 | no | — |
| 96 | C4×C5 | — | 2 | 1 | no | — |
| 97 | C4×C5 | — | 2 | 1 | no | — |
| 98 | C4×C5 | — | 2 | 1 | no | — |
| 99 | C4×C5 | — | 2 | 1 | no | — |
| 100 | C4×C5 | — | 2 | 1 | no | — |
| 101 | C4×C5 | — | 2 | 1 | no | — |
| 102 | Dic5 | — | 10 | 1 | no | — |
| 103 | Dic5 | — | 10 | 1 | no | — |
| 104 | Dic5 | — | 10 | 1 | no | — |
| 105 | Dic5 | — | 10 | 1 | no | — |
| 106 | Dic5 | — | 10 | 1 | no | — |
| 107 | Dic5 | — | 10 | 1 | no | — |
| 108 | C22×C4 | — | 25 | 1 | no | — |
| 109 | M16 | — | 50 | 1 | no | — |
| 110 | M16 | — | 50 | 1 | no | — |
| 111 | C2×C8 | — | 25 | 2 | no | — |
| 112 | C2×C8 | — | 25 | 2 | no | — |
| 113 | C2×C8 | — | 50 | 1 | no | — |
| 114 | C2×C8 | — | 50 | 1 | no | — |
| 115 | C4⋊C4 | — | 25 | 2 | no | — |
| 116 | C22⋊C4 | — | 25 | 2 | no | — |
| 117 | C42 | — | 25 | 2 | no | — |
| 118 | C2×C5 | — | 2 | 2 | no | — |
| 119 | C2×C5 | — | 2 | 1 | no | — |
| 120 | C2×C5 | — | 2 | 1 | no | — |
| 121 | C2×C5 | — | 2 | 1 | no | — |
| 122 | C2×C5 | — | 4 | 1 | no | — |
| 123 | C2×C5 | — | 4 | 1 | no | — |
| 124 | D5 | — | 10 | 2 | no | — |
| 125 | D5 | — | 10 | 2 | no | — |
| 126 | D5 | — | 10 | 2 | no | — |
| 127 | D5 | — | 10 | 2 | no | — |
| 128 | D5 | — | 20 | 1 | no | — |
| 129 | D5 | — | 20 | 1 | no | — |
| 130 | C23 | — | 25 | 1 | no | — |
| 131 | C2×C4 | — | 25 | 1 | no | — |
| 132 | C2×C4 | — | 25 | 1 | no | — |
| 133 | C2×C4 | — | 25 | 1 | no | — |
| 134 | C2×C4 | — | 25 | 1 | no | — |
| 135 | C2×C4 | — | 25 | 1 | no | — |
| 136 | C2×C4 | — | 25 | 4 | no | — |
| 137 | C2×C4 | #6057 | 1 | 1 | yes | characteristic |
| 138 | C8 | — | 50 | 2 | no | — |
| 139 | C8 | — | 50 | 2 | no | — |
| 140 | C5 | — | 2 | 1 | no | — |
| 141 | C5 | — | 2 | 1 | no | — |
| 142 | C5 | — | 2 | 1 | no | — |
| 143 | C22 | — | 25 | 1 | no | — |
| 144 | C22 | — | 25 | 1 | no | — |
| 145 | C22 | — | 50 | 2 | no | — |
| 146 | C22 | #87840 | 1 | 1 | yes | characteristic |
| 147 | C4 | — | 25 | 1 | no | — |
| 148 | C4 | — | 25 | 1 | no | — |
| 149 | C4 | — | 50 | 4 | no | — |
| 150 | C4 | #87840 | 1 | 1 | yes | characteristic |
| 151 | C4 | C2×#6057 | 1 | 1 | yes | characteristic |
| 152 | C2 | — | 2 | 1 | no | — |
| 153 | C2 | — | 25 | 2 | no | — |
| 154 | C2 | — | 50 | 1 | no | — |
| 155 | C2 | C2×#87840 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #11716802 | C2 | semidirect |
| 82. C52 | #424 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11717451 | 2 |
| #11717448 | 2 |
| C2×#87862 | 2 |
| #11716802 | 2 |
| #424 | 25 |