Finite Group DatabasebetaFinite Group DatabasebetaOrder 2400.
182 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 22 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×C5×SL(2,𝔽5) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C5×#6732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C5×SL(2,𝔽5) | C22 | 1 | 1 | yes | characteristic |
| 4 | C5×2−1+4⋊C3 | — | 5 | 1 | no | maximal |
| 5 | #91881 | C5 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 6 | C5×#711 | — | 6 | 1 | no | maximal |
| 7 | C5×#237 | — | 10 | 1 | no | maximal |
| 8 | C5×#41⋊C3 | — | 5 | 1 | no | — |
| 9 | C2×C5×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 10 | C2×SL(2,𝔽5) | C2×C5 | 1 | 2 | yes | complemented |
| 11 | #6732 | C2×C5 | 1 | 1 | yes | characteristic, complemented |
| 12 | C52×D4 | — | 6 | 1 | no | — |
| 13 | C5×C22⋊D5 | — | 6 | 2 | no | — |
| 14 | C2×C5×Dic5 | — | 6 | 2 | no | — |
| 15 | C4×C5×D5 | — | 6 | 1 | no | — |
| 16 | C5×Dic10 | — | 6 | 1 | no | — |
| 17 | C5×2−1+4 | — | 5 | 1 | no | — |
| 18 | C3×C5×D4 | — | 10 | 1 | no | — |
| 19 | C5×#67 | — | 10 | 2 | no | — |
| 20 | C2×C5×Dic3 | — | 10 | 2 | no | — |
| 21 | C4×C5×D3 | — | 10 | 1 | no | — |
| 22 | C5×Dic6 | — | 10 | 1 | no | — |
| 23 | C5×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 24 | SL(2,𝔽5) | C22×C5 | 1 | 1 | yes | characteristic |
| 25 | C22×C52 | — | 6 | 2 | no | — |
| 26 | C2×C5×D5 | — | 6 | 1 | no | — |
| 27 | C4×C52 | — | 6 | 1 | no | — |
| 28 | C5×Dic5 | — | 6 | 2 | no | — |
| 29 | C5×Dic5 | — | 6 | 1 | no | — |
| 30 | 2−1+4⋊C3 | — | 5 | 1 | no | Hall |
| 31 | C5×#41 | — | 5 | 1 | no | — |
| 32 | C5×#41 | — | 15 | 1 | no | — |
| 33 | C5×#41 | — | 15 | 2 | no | — |
| 34 | C2×C5×Q8 | — | 5 | 2 | no | — |
| 35 | C2×C5×Q8 | — | 15 | 1 | no | — |
| 36 | #711 | — | 6 | 1 | no | — |
| 37 | C22×C3×C5 | — | 10 | 2 | no | — |
| 38 | C2×C5×D3 | — | 10 | 1 | no | — |
| 39 | C3×C4×C5 | — | 10 | 1 | no | — |
| 40 | C5×Dic3 | — | 10 | 2 | no | — |
| 41 | C5×Dic3 | — | 10 | 1 | no | — |
| 42 | C2×C52 | — | 6 | 1 | no | — |
| 43 | C2×C52 | — | 12 | 2 | no | — |
| 44 | C5×D5 | — | 12 | 1 | no | — |
| 45 | #237 | — | 10 | 1 | no | — |
| 46 | #41⋊C3 | — | 5 | 1 | no | — |
| 47 | C2×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 48 | C5×Q8 | — | 5 | 1 | no | — |
| 49 | C5×Q8 | — | 15 | 2 | no | — |
| 50 | C5×Q8 | — | 15 | 1 | no | — |
| 51 | C5×D4 | — | 6 | 1 | no | — |
| 52 | C5×D4 | — | 12 | 2 | no | — |
| 53 | C5×D4 | — | 15 | 1 | no | — |
| 54 | C5×D4 | — | 15 | 2 | no | — |
| 55 | C5×D4 | A5 | 1 | 1 | yes | maximal normal, characteristic |
| 56 | C2×C4×C5 | — | 15 | 2 | no | — |
| 57 | C2×C4×C5 | — | 15 | 2 | no | — |
| 58 | C2×C4×C5 | — | 15 | 1 | no | — |
| 59 | C22⋊D5 | — | 6 | 2 | no | — |
| 60 | C2×Dic5 | — | 6 | 2 | no | — |
| 61 | C4×D5 | — | 6 | 1 | no | — |
| 62 | Dic10 | — | 6 | 1 | no | — |
| 63 | 2−1+4 | — | 5 | 1 | no | Sylow |
| 64 | C2×C3×C5 | — | 10 | 1 | no | — |
| 65 | C2×C3×C5 | — | 20 | 2 | no | — |
| 66 | C5×D3 | — | 20 | 1 | no | — |
| 67 | C52 | — | 6 | 1 | no | Sylow |
| 68 | C3×D4 | — | 10 | 1 | no | — |
| 69 | #67 | — | 10 | 2 | no | — |
| 70 | C2×Dic3 | — | 10 | 2 | no | — |
| 71 | C4×D3 | — | 10 | 1 | no | — |
| 72 | Dic6 | — | 10 | 1 | no | — |
| 73 | SL(2,𝔽3) | — | 5 | 1 | no | — |
| 74 | C22×C5 | — | 6 | 2 | no | — |
| 75 | C22×C5 | — | 12 | 4 | no | — |
| 76 | C22×C5 | — | 15 | 1 | no | — |
| 77 | C22×C5 | C2×A5 | 1 | 2 | yes | — |
| 78 | C2×D5 | — | 6 | 1 | no | — |
| 79 | C4×C5 | — | 6 | 1 | no | — |
| 80 | C4×C5 | — | 12 | 2 | no | — |
| 81 | C4×C5 | — | 15 | 1 | no | — |
| 82 | C4×C5 | — | 15 | 2 | no | — |
| 83 | C4×C5 | C2×A5 | 1 | 1 | yes | characteristic |
| 84 | Dic5 | — | 6 | 2 | no | — |
| 85 | Dic5 | — | 6 | 1 | no | — |
| 86 | #41 | — | 5 | 1 | no | — |
| 87 | #41 | — | 15 | 1 | no | — |
| 88 | #41 | — | 15 | 2 | no | — |
| 89 | C2×Q8 | — | 5 | 2 | no | — |
| 90 | C2×Q8 | — | 15 | 1 | no | — |
| 91 | C3×C5 | — | 10 | 1 | no | — |
| 92 | C22×C3 | — | 10 | 2 | no | — |
| 93 | C2×D3 | — | 10 | 1 | no | — |
| 94 | C3×C4 | — | 10 | 1 | no | — |
| 95 | Dic3 | — | 10 | 2 | no | — |
| 96 | Dic3 | — | 10 | 1 | no | — |
| 97 | C2×C5 | — | 2 | 2 | no | — |
| 98 | C2×C5 | — | 6 | 1 | no | — |
| 99 | C2×C5 | — | 12 | 2 | no | — |
| 100 | C2×C5 | — | 12 | 2 | no | — |
| 101 | C2×C5 | — | 24 | 4 | no | — |
| 102 | C2×C5 | — | 30 | 1 | no | — |
| 103 | C2×C5 | C22×A5 | 1 | 1 | yes | characteristic |
| 104 | D5 | — | 12 | 1 | no | — |
| 105 | Q8 | — | 5 | 1 | no | — |
| 106 | Q8 | — | 15 | 2 | no | — |
| 107 | Q8 | — | 15 | 1 | no | — |
| 108 | D4 | — | 15 | 1 | no | — |
| 109 | D4 | — | 15 | 2 | no | — |
| 110 | D4 | C5×A5 | 1 | 1 | yes | characteristic |
| 111 | C2×C4 | — | 15 | 2 | no | — |
| 112 | C2×C4 | — | 15 | 2 | no | — |
| 113 | C2×C4 | — | 15 | 1 | no | — |
| 114 | C2×C3 | — | 10 | 1 | no | — |
| 115 | C2×C3 | — | 20 | 2 | no | — |
| 116 | D3 | — | 20 | 1 | no | — |
| 117 | C5 | — | 6 | 1 | no | — |
| 118 | C5 | — | 12 | 2 | no | — |
| 119 | C5 | #91881 | 1 | 1 | yes | characteristic, direct product factor |
| 120 | C22 | — | 15 | 1 | no | — |
| 121 | C22 | C2×C5×A5 | 1 | 2 | yes | — |
| 122 | C4 | — | 15 | 1 | no | — |
| 123 | C4 | — | 15 | 2 | no | — |
| 124 | C4 | C2×C5×A5 | 1 | 1 | yes | characteristic |
| 125 | C3 | — | 10 | 1 | no | Sylow |
| 126 | C2 | — | 2 | 2 | no | — |
| 127 | C2 | — | 30 | 1 | no | — |
| 128 | C2 | C22×C5×A5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×A5 |
| commutator, hypocenter | SL(2,𝔽5) | C22×C5 |
| Frattini, cosocle | C2 | C22×C5×A5 |
| Fitting, hypercenter | C5×D4 | A5 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C5 ◁ C5×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×SL(2,𝔽5) | C2 | semidirect |
| 2. C5×#6732 | C2 | semidirect |
| 5. #91881 | C5 | direct (N × H) |
| 10. C2×SL(2,𝔽5) | C2×C5 | semidirect |
| 11. #6732 | C2×C5 | semidirect |
| 119. C5 | #91881 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×SL(2,𝔽5) | 2 |
| C5×#6732 | 2 |
| C5×2−1+4⋊C3 | 5 |
| #91881 | 5 |
| C5×#711 | 6 |
| C5×#237 | 10 |