Finite Group Databasebeta

← C2×#11717258 #13107330

Subgroups — C2×#11717258 #13107330

Order 1600.

1704 conjugacy classes of subgroups; only the 363 normal subgroups are shown.

normal subgroups only

This group has 1704 conjugacy classes of subgroups — too many to draw. Only its 363 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×D4C211yesmaximal, characteristic, complemented
2C22×C22⋊(C5⋊D5)C212yesmaximal, complemented
3C22×C4×C5⋊D5C211yesmaximal, characteristic, complemented
4C2×C22⋊#1039C212yesmaximal, complemented
5#11717258C218yesmaximal, direct product factor
6C2×C4⋊#1039C211yesmaximal, characteristic, complemented
7C24×C52C2212yes—
8C23×C5⋊D5C2211yescharacteristic
9C2×C52×D4—24no—
10C2×C52×D4C2214yescomplemented
11C22×C4×C52C2211yescharacteristic
12C2×C22⋊(C5⋊D5)—28no—
13C2×C22⋊(C5⋊D5)C2218yescomplemented
14C22×#1039C2212yes—
15C22×#1039C2211yescharacteristic
16C2×C4×C5⋊D5—24no—
17C2×C4×C5⋊D5C2214yescomplemented
18C22⋊#1039C2218yescomplemented
19C4⋊#1039C2214yescomplemented
20C2×#4084—56nomaximal
21C23×C52—24no—
22C23×C52—24no—
23C23×C52—24no—
24C23×C52C2314yes—
25C23×C52C2311yescharacteristic
26C22×C5⋊D5—22no—
27C22×C5⋊D5—22no—
28C22×C5⋊D5D414yescomplemented
29C22×C5⋊D5C2312yes—
30C52×D4—28no—
31C2×C4×C52D414yescomplemented
32C2×C4×C52C2312yes—
33C22⋊(C5⋊D5)—216no—
34C2×#1039—22no—
35C2×#1039—24no—
36C2×#1039C2312yes—
37C2×#1039C2314yes—
38C4×C5⋊D5—28no—
39C22×C5×D4D516yescomplemented
40C22×C22⋊D5—512no—
41C22×C4×D5—56no—
42C2×C22⋊Dic5—512no—
43#4084—548no—
44C2×C4⋊Dic5—56no—
45C22×C52—24no—
46C22×C52—24no—
47C22×C52—24no—
48C22×C52—48no—
49C22×C52C2×D412yes—
50C22×C52C2×D412yes—
51C22×C52#4112yes—
52C22×C52C2411yescharacteristic
53C2×C5⋊D5—28no—
54C2×C5⋊D5—22no—
55C2×C5⋊D5—22no—
56C2×C5⋊D5C2×D414yescomplemented
57C4×C52C2×D414yescomplemented
58#1039—22no—
59#1039—24no—
60C24×C5C2×D5112yes—
61C23×D5—56no—
62C2×C5×D4—224no—
63C2×C5×D4C2×D5124yescomplemented
64C22×C4×C5C2×D516yes—
65C2×C22⋊D5—548no—
66C2×C22⋊D5—1048no—
67C22×Dic5—56no—
68C22×Dic5—512no—
69C2×C4×D5—524no—
70C2×C4×D5—1024no—
71C22⋊Dic5—548no—
72C4⋊Dic5—524no—
73C2×#121—251noSylow
74C2×C52—44no—
75C2×C52#12114yescomplemented
76C2×C52C22×D411yescharacteristic
77C2×C52C22×D411yescharacteristic
78C2×C52C2×#4111yescharacteristic
79C5⋊D5—24no—
80C23×C5—224no—
81C23×C5—224no—
82C23×C5—224no—
83C23×C5C22×D5124yes—
84C23×C5C22×D516yes—
85C22×D5—524no—
86C22×D5—512no—
87C22×D5—1012no—
88C22×D5—1012no—
89C5×D4—248no—
90C2×C4×C5C22⋊D5124yescomplemented
91C2×C4×C5C22×D5112yes—
92C22⋊D5—1096no—
93C2×Dic5—524no—
94C2×Dic5—512no—
95C2×Dic5—1024no—
96C2×Dic5—1012no—
97C4×D5—1048no—
98C22×D4—252no—
99C22×D4C5⋊D511yescharacteristic, complemented
100C23×C4—251no—
101#121—258no—
102C2×C4⋊C4—251no—
103C2×C22⋊C4—252no—
104C52C2×#12111yescharacteristic, complemented, Sylow
105C22×C5—224no—
106C22×C5—224no—
107C22×C5—224no—
108C22×C5—448no—
109C22×C5D4×D5112yes—
110C22×C5#711112yes—
111C22×C5C2×C22⋊D5112yes—
112C22×C5C23×D516yes—
113C2×D5—524no—
114C2×D5—1012no—
115C2×D5—1012no—
116C2×D5—1048no—
117C4×C5C2×C22⋊D5124yescomplemented
118Dic5—1024no—
119Dic5—1012no—
120C24—251no—
121C24C2×C5⋊D512yes—
122C2×D4—24no—
123C2×D4—258no—
124C2×D4—508no—
125C2×D4C2×C5⋊D514yescomplemented
126C22×C4—251no—
127C22×C4—252no—
128C22×C4—254no—
129C22×C4—504no—
130C22×C4C2×C5⋊D511yescharacteristic
131C4⋊C4—254no—
132C22⋊C4—258no—
133C2×C5—424no—
134C2×C5#4084124yescomplemented
135C2×C5C2×D4×D516yes—
136C2×C5C2×#71116yes—
137C2×C5C22×C22⋊D516yes—
138D5—1024no—
139C23—24no—
140C23—24no—
141C23—24no—
142C23—254no—
143C23—252no—
144C23—502no—
145C23—502no—
146C23C22×C5⋊D514yes—
147C23C22×C5⋊D511yescharacteristic
148D4—28no—
149D4—5016no—
150C2×C4—252no—
151C2×C4—254no—
152C2×C4—504no—
153C2×C4—502no—
154C2×C4—508no—
155C2×C4C22⋊(C5⋊D5)14yescomplemented
156C2×C4C22×C5⋊D512yes—
157C5C2×#408416yescomplemented
158C22—24no—
159C22—24no—
160C22—24no—
161C22—48no—
162C22—254no—
163C22—508no—
164C22—502no—
165C22—502no—
166C22D4×C5⋊D512yes—
167C22#8790212yes—
168C22C2×C22⋊(C5⋊D5)12yes—
169C22C23×C5⋊D511yescharacteristic
170C4—504no—
171C4—502no—
172C4C2×C22⋊(C5⋊D5)14yescomplemented
173C2—44no—
174C2—504no—
175C2#1171725814yesdirect product factor
176C2C2×D4×C5⋊D511yescharacteristic
177C2C2×#8790211yescharacteristic
178C2C22×C22⋊(C5⋊D5)11yescharacteristic
subgroupG/H
centerC23C22×C5⋊D5
commutator, cosocleC22×C52C24
FrattiniC22C23×C5⋊D5
FittingC22×C52×D4C2
socleC23×C52C23
hypercenterC22×D4C5⋊D5
hypocenterC52C2×#121
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C52×D4C2semidirect
2. C22×C22⋊(C5⋊D5)C2semidirect
3. C22×C4×C5⋊D5C2semidirect
4. C2×C22⋊#1039C2semidirect
5. #11717258C2direct (N × H)
6. C2×C4⋊#1039C2semidirect
10. C2×C52×D4C22semidirect
13. C2×C22⋊(C5⋊D5)C22semidirect
17. C2×C4×C5⋊D5C22semidirect
18. C22⋊#1039C22semidirect
19. C4⋊#1039C22semidirect
28. C22×C5⋊D5D4semidirect
31. C2×C4×C52D4semidirect
39. C22×C5×D4D5semidirect
56. C2×C5⋊D5C2×D4semidirect
57. C4×C52C2×D4semidirect
63. C2×C5×D4C2×D5semidirect
75. C2×C52#121semidirect
90. C2×C4×C5C22⋊D5semidirect
99. C22×D4C5⋊D5semidirect
104. C52C2×#121semidirect
117. C4×C5C2×C22⋊D5semidirect
125. C2×D4C2×C5⋊D5semidirect
134. C2×C5#4084semidirect
155. C2×C4C22⋊(C5⋊D5)semidirect
157. C5C2×#4084semidirect
172. C4C2×C22⋊(C5⋊D5)semidirect
175. C2#11717258direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C52×D42
C22×C22⋊(C5⋊D5)2
C22×C4×C5⋊D52
C2×C22⋊#10392
#117172582
C2×C4⋊#10392
C2×#40845