Finite Group Databasebeta

← C22×C5×Q8×D5 #13107862

Subgroups — C22×C5×Q8×D5 #13107862

Order 1600.

1856 conjugacy classes of subgroups; only the 1006 normal subgroups are shown.

normal subgroups only

This group has 1856 conjugacy classes of subgroups — too many to draw. Only its 1006 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
1C22×C52×Q8C211yesmaximal, characteristic, complemented
2C2×C5×Q8×D5C2124yesmaximal, direct product factor
3C22×C4×C5×D5C213yesmaximal
4C22×C5×Dic10C213yesmaximal, complemented
5C23×C5×D5C2211yescharacteristic
6C2×C52×Q8C22112yescomplemented
7C22×C4×C52C2213yes—
8C22×C5×Dic5C2213yes—
9C5×Q8×D5C22164yesdirect product factor
10C2×C4×C5×D5C22136yes—
11C2×C5×Dic10C22136yescomplemented
12C23×C5×Q8—51nomaximal
13C22×Q8×D5C511yesmaximal, characteristic, direct product factor
14C23×C52C2311yescharacteristic
15C22×C5×D5Q818yesdirect product factor
16C22×C5×D5C2316yes—
17C52×Q8C23116yescomplemented
18C2×C4×C52C23118yes—
19C2×C5×Dic5C23118yes—
20C4×C5×D5C23148yes—
21C5×Dic10C23148yescomplemented
22C22×C5×Q8—22no—
23C22×C5×Q8—524no—
24C22×C5×Q8—53no—
25C22×C5×Q8D511yescharacteristic, direct product factor
26C22×C5×Q8C2×C511yescharacteristic, complemented
27C23×C4×C5—53no—
28C2×Q8×D5C2×C5124yesdirect product factor
29C22×C4×D5C2×C513yes—
30C22×Dic10C2×C513yescomplemented
31C22×C52C2×Q814yescomplemented
32C22×C52C2413yes—
33C2×C5×D5C2×Q8124yesdirect product factor
34C2×C5×D5C2414yes—
35C4×C52C24112yes—
36C5×Dic5C24112yes—
37C24×C5—51no—
38C23×D5C22×C511yescharacteristic
39C2×C5×Q8—224no—
40C2×C5×Q8—564no—
41C2×C5×Q8—536no—
42C2×C5×Q8C2×D5112yesdirect product factor
43C2×C5×Q8C22×C5112yescomplemented
44C22×C4×C5—26no—
45C22×C4×C5—53no—
46C22×C4×C5—536no—
47C22×C4×C5C2×D513yes—
48C22×C4×C5C22×C513yes—
49C22×Dic5C22×C513yes—
50Q8×D5C22×C5164yesdirect product factor
51C2×C4×D5C22×C5136yes—
52C2×Dic10C22×C5136yescomplemented
53C23×Q8—51noSylow
54C2×C52C22×Q816yescomplemented
55C2×C52C2511yescharacteristic
56C5×D5C22×Q818yesdirect product factor
57C23×C5—22no—
58C23×C5—58no—
59C23×C5—56no—
60C23×C5C22×D511yescharacteristic
61C23×C5C23×C511yescharacteristic
62C22×D5C5×Q818yesdirect product factor
63C22×D5C23×C516yes—
64C5×Q8—232no—
65C5×Q8—548no—
66C5×Q8C22×D5116yesdirect product factor
67C5×Q8C23×C5116yescomplemented
68C2×C4×C5—236no—
69C2×C4×C5—518no—
70C2×C4×C5—548no—
71C2×C4×C5C22×D5118yes—
72C2×C4×C5C23×C5118yes—
73C2×Dic5C23×C5118yes—
74C4×D5C23×C5148yes—
75Dic10C23×C5148yescomplemented
76C22×Q8—524no—
77C22×Q8—53no—
78C22×Q8C5×D511yescharacteristic, direct product factor
79C23×C4—53no—
80C52C23×Q811yescharacteristic, complemented, Sylow
81C22×C5—28no—
82C22×C5—26no—
83C22×C5—524no—
84C22×C5—54no—
85C22×C5Q8×D514yesdirect product factor
86C22×C5C2×C5×Q814yescomplemented
87C22×C5C23×D513yes—
88C22×C5C24×C513yes—
89C2×D5C2×C5×Q8124yesdirect product factor
90C2×D5C24×C514yes—
91C4×C5—224no—
92C4×C5—512no—
93C4×C5C23×D5112yes—
94C4×C5C24×C5112yes—
95Dic5C24×C5112yes—
96C24—51no—
97C2×Q8—564no—
98C2×Q8—536no—
99C2×Q8C2×C5×D5112yesdirect product factor
100C22×C4—53no—
101C22×C4—536no—
102C22×C4C2×C5×D513yes—
103C2×C5—212no—
104C2×C5—22no—
105C2×C5—58no—
106C2×C5C2×Q8×D516yesdirect product factor
107C2×C5C22×C5×Q816yescomplemented
108C2×C5C24×D511yescharacteristic
109C2×C5C25×C511yescharacteristic
110D5C22×C5×Q818yesdirect product factor
111C23—58no—
112C23—56no—
113C23C22×C5×D511yescharacteristic
114Q8—548no—
115Q8C22×C5×D5116yesdirect product factor
116C2×C4—518no—
117C2×C4—548no—
118C2×C4C22×C5×D5118yes—
119C5—22no—
120C5C22×Q8×D511yescharacteristic, direct product factor
121C5C23×C5×Q811yescharacteristic, complemented
122C22—524no—
123C22—54no—
124C22C5×Q8×D514yesdirect product factor
125C22C23×C5×D513yes—
126C4—512no—
127C4C23×C5×D5112yes—
128C2—58no—
129C2C2×C5×Q8×D516yesdirect product factor
130C2C24×C5×D511yescharacteristic
subgroupG/H
centerC23×C5C22×D5
commutator, cosocleC2×C5C25×C5
FrattiniC2C24×C5×D5
FittingC22×C52×Q8C2
socleC23×C52C23
prime radicalC23×C5×D5C22
hypercenterC22×C5×Q8D5
hypocenterC5C23×C5×Q8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C5 ◁ C22×C5×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C52×Q8C2semidirect
2. C2×C5×Q8×D5C2direct (N × H)
4. C22×C5×Dic10C2semidirect
6. C2×C52×Q8C22semidirect
9. C5×Q8×D5C22direct (N × H)
11. C2×C5×Dic10C22semidirect
13. C22×Q8×D5C5direct (N × H)
15. C22×C5×D5Q8direct (N × H)
17. C52×Q8C23semidirect
21. C5×Dic10C23semidirect
25. C22×C5×Q8D5direct (N × H)
26. C22×C5×Q8C2×C5semidirect
28. C2×Q8×D5C2×C5direct (N × H)
30. C22×Dic10C2×C5semidirect
31. C22×C52C2×Q8semidirect
33. C2×C5×D5C2×Q8direct (N × H)
42. C2×C5×Q8C2×D5direct (N × H)
43. C2×C5×Q8C22×C5semidirect
50. Q8×D5C22×C5direct (N × H)
52. C2×Dic10C22×C5semidirect
54. C2×C52C22×Q8semidirect
56. C5×D5C22×Q8direct (N × H)
62. C22×D5C5×Q8direct (N × H)
66. C5×Q8C22×D5direct (N × H)
67. C5×Q8C23×C5semidirect
75. Dic10C23×C5semidirect
78. C22×Q8C5×D5direct (N × H)
80. C52C23×Q8semidirect
85. C22×C5Q8×D5direct (N × H)
86. C22×C5C2×C5×Q8semidirect
89. C2×D5C2×C5×Q8direct (N × H)
99. C2×Q8C2×C5×D5direct (N × H)
106. C2×C5C2×Q8×D5direct (N × H)
107. C2×C5C22×C5×Q8semidirect
110. D5C22×C5×Q8direct (N × H)
115. Q8C22×C5×D5direct (N × H)
120. C5C22×Q8×D5direct (N × H)
121. C5C23×C5×Q8semidirect
124. C22C5×Q8×D5direct (N × H)
129. C2C2×C5×Q8×D5direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C52×Q82
C2×C5×Q8×D52
C22×C4×C5×D52
C22×C5×Dic102
C23×C5×Q85
C22×Q8×D55