Finite Group Databasebeta

← C5×#525 #66189

Subgroups — C5×#525 #66189

Order 320.

330 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#41C211yesmaximal, characteristic, complemented
2C23×C4×C5C211yesmaximal, characteristic, complemented
3C5×#123C212yesmaximal, complemented
4C5×C22⋊Q8C212yesmaximal, complemented
5C5×#121C212yesmaximal, complemented
6C5×#120C212yesmaximal, complemented
7C4×C5×D4C214yesmaximal, complemented
8C5×#117C211yesmaximal, characteristic, complemented
9C24×C5C2211yescharacteristic
10C5×#41—24no—
11C2×C5×Q8C2211yescharacteristic, complemented
12C2×C5×D4C2214yes—
13C2×C5×D4C2212yes—
14C2×C5×D4C2211yescharacteristic, complemented
15C22×C4×C5—24no—
16C22×C4×C5C2212yes—
17C22×C4×C5C2211yescharacteristic
18C22×C4×C5C2212yes—
19C22×C4×C5C2211yescharacteristic
20C22×C4×C5C2212yescomplemented
21C5×C4⋊C4C2212yescomplemented
22C5×C4⋊C4C2214yescomplemented
23C5×C22⋊C4C2212yes—
24C5×C22⋊C4C2214yes—
25C5×C22⋊C4C2214yescomplemented
26C42×C5C2212yescomplemented
27#525C511yesmaximal, characteristic, direct product factor, Sylow
28C23×C5—22no—
29C23×C5—24no—
30C23×C5C2312yes—
31C23×C5C2311yescharacteristic
32C23×C5C2312yes—
33C5×Q8—22no—
34C5×D4—24no—
35C5×D4—28no—
36C5×D4—22no—
37C2×C4×C5—22no—
38C2×C4×C5—24no—
39C2×C4×C5—24no—
40C2×C4×C5—24no—
41C2×C4×C5D414yescomplemented
42C2×C4×C5C2314yes—
43C2×C4×C5C2312yes—
44C2×C4×C5C2312yes—
45C2×C4×C5C2311yescharacteristic
46C2×C4×C5C2311yescharacteristic
47C2×#41C2×C511yescharacteristic, complemented
48C23×C4C2×C511yescharacteristic, complemented
49#123C2×C512yescomplemented
50C22⋊Q8C2×C512yescomplemented
51#121C2×C512yescomplemented
52#120C2×C512yescomplemented
53C4×D4C2×C514yescomplemented
54#117C2×C511yescharacteristic, complemented
55C22×C5—28no—
56C22×C5—22no—
57C22×C5—22no—
58C22×C5—22no—
59C22×C5—22no—
60C22×C5—24no—
61C22×C5C2×D412yes—
62C22×C5#4114yescomplemented
63C22×C5C2411yescharacteristic
64C4×C5—22no—
65C4×C5—22no—
66C4×C5—24no—
67C4×C5C2×D412yes—
68C4×C5C2×D412yes—
69C24C22×C511yescharacteristic
70#41—24no—
71C2×Q8C22×C511yescharacteristic, complemented
72C2×D4C22×C514yes—
73C2×D4C22×C512yes—
74C2×D4C22×C511yescharacteristic, complemented
75C22×C4—24no—
76C22×C4C22×C512yes—
77C22×C4C22×C512yes—
78C22×C4C22×C511yescharacteristic
79C22×C4C22×C511yescharacteristic
80C22×C4C22×C512yescomplemented
81C4⋊C4C22×C512yescomplemented
82C4⋊C4C22×C514yescomplemented
83C22⋊C4C22×C512yes—
84C22⋊C4C22×C514yes—
85C22⋊C4C22×C514yescomplemented
86C42C22×C512yescomplemented
87C2×C5—22no—
88C2×C5—24no—
89C2×C5—42no—
90C2×C5C22×D411yescharacteristic
91C2×C5C2×#4112yes—
92C23—22no—
93C23—24no—
94C23C23×C512yes—
95C23C23×C511yescharacteristic
96C23C23×C512yes—
97Q8—22no—
98D4—24no—
99D4—28no—
100D4—22no—
101C2×C4—22no—
102C2×C4—24no—
103C2×C4—24no—
104C2×C4—24no—
105C2×C4C5×D414yescomplemented
106C2×C4C23×C514yes—
107C2×C4C23×C512yes—
108C2×C4C23×C512yes—
109C2×C4C23×C511yescharacteristic
110C2×C4C23×C511yescharacteristic
111C5#52511yescharacteristic, direct product factor, Sylow
112C22—28no—
113C22—22no—
114C22—22no—
115C22—22no—
116C22—22no—
117C22—24no—
118C22C2×C5×D412yes—
119C22C5×#4114yescomplemented
120C22C24×C511yescharacteristic
121C4—22no—
122C4—22no—
123C4—24no—
124C4C2×C5×D412yes—
125C4C2×C5×D412yes—
126C2—22no—
127C2—24no—
128C2—42no—
129C2C22×C5×D411yescharacteristic
130C2C2×C5×#4112yes—
subgroupG/H
centerC2×C4×C5C23
commutator, Frattini, cosocleC22C24×C5
socleC22×C5C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C5×#41C2semidirect
2. C23×C4×C5C2semidirect
3. C5×#123C2semidirect
4. C5×C22⋊Q8C2semidirect
5. C5×#121C2semidirect
6. C5×#120C2semidirect
7. C4×C5×D4C2semidirect
8. C5×#117C2semidirect
11. C2×C5×Q8C22semidirect
14. C2×C5×D4C22semidirect
20. C22×C4×C5C22semidirect
21. C5×C4⋊C4C22semidirect
22. C5×C4⋊C4C22semidirect
25. C5×C22⋊C4C22semidirect
26. C42×C5C22semidirect
27. #525C5direct (N × H)
41. C2×C4×C5D4semidirect
47. C2×#41C2×C5semidirect
48. C23×C4C2×C5semidirect
49. #123C2×C5semidirect
50. C22⋊Q8C2×C5semidirect
51. #121C2×C5semidirect
52. #120C2×C5semidirect
53. C4×D4C2×C5semidirect
54. #117C2×C5semidirect
62. C22×C5#41semidirect
71. C2×Q8C22×C5semidirect
74. C2×D4C22×C5semidirect
80. C22×C4C22×C5semidirect
81. C4⋊C4C22×C5semidirect
82. C4⋊C4C22×C5semidirect
85. C22⋊C4C22×C5semidirect
86. C42C22×C5semidirect
105. C2×C4C5×D4semidirect
111. C5#525direct (N × H)
119. C22C5×#41semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C5×#412
C23×C4×C52
C5×#1232
C5×C22⋊Q82
C5×#1212
C5×#1202
C4×C5×D42
C5×#1172
#5255