Finite Group Databasebeta

← #13107135

Subgroups — #13107135

Order 1600.

1384 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 235 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×D4×C5⋊D5C212yesmaximal, complemented
2C52×#121C211yesmaximal, characteristic, complemented
3#11717259C212yesmaximal, complemented
4#11717257C212yesmaximal, complemented
5#11717250C211yesmaximal, characteristic, complemented
6C2×#87808C211yesmaximal, characteristic, complemented
7#11717214C212yesmaximal, complemented
8#11717203C212yesmaximal, complemented
9#11717202C212yesmaximal, complemented
10C23×C5⋊D5C2211yescharacteristic
11C23×C5⋊D5C2212yescomplemented
12C2×C52×D4C2212yescomplemented
13C2×C52×D4C2211yescharacteristic, complemented
14C22×C4×C52C2211yescharacteristic, complemented
15C2×C22⋊(C5⋊D5)C2212yes—
16C2×C22⋊(C5⋊D5)C2212yescomplemented
17C2×C22⋊(C5⋊D5)C2212yescomplemented
18C22×#1039C2211yescharacteristic, complemented
19D4×C5⋊D5—28no—
20C2×C4⋊(C5⋊D5)C2212yes—
21C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
22C2×C4×C5⋊D5C2212yescomplemented
23C52×C4⋊C4C2211yescharacteristic, complemented
24C52×C22⋊C4C2212yescomplemented
25C22⋊#1039C2212yescomplemented
26#87808C2212yescomplemented
27#87808C2212yescomplemented
28#87808C2212yescomplemented
29#87808C2212yescomplemented
30C4⋊#1039C2211yescharacteristic, complemented
31#87806C2212yescomplemented
32#65939—56nomaximal
33C23×C52C2312yes—
34C23×C52C2311yescharacteristic
35C22×C5⋊D5—24no—
36C22×C5⋊D5—41no—
37C22×C5⋊D5—41no—
38C22×C5⋊D5—42no—
39C22×C5⋊D5—42no—
40C22×C5⋊D5D414yescomplemented
41C22×C5⋊D5C2312yes—
42C22×C5⋊D5C2312yes—
43C52×D4—24no—
44C52×D4—41no—
45C2×C4×C52—41no—
46C2×C4×C52C2311yescharacteristic
47C2×C4×C52C2311yescharacteristic
48C2×C4×C52C2312yes—
49C22⋊(C5⋊D5)—24no—
50C22⋊(C5⋊D5)—24no—
51C22⋊(C5⋊D5)—42no—
52C2×#1039—41no—
53C2×#1039C2312yes—
54C2×#1039C2312yes—
55C4⋊(C5⋊D5)—24no—
56C4⋊(C5⋊D5)—41no—
57C4×C5⋊D5—24no—
58C2×D4×D5—512no—
59C5×#121D516yescomplemented
60#4085—512no—
61#4083—512no—
62#4076—56no—
63C2×#685—56no—
64#4040—512no—
65#4029—512no—
66#4028—512no—
67C22×C52—21no—
68C22×C52—21no—
69C22×C52—22no—
70C22×C52—22no—
71C22×C52—22no—
72C22×C52C2×D412yes—
73C22×C52C2411yescharacteristic
74C2×C5⋊D5—22no—
75C2×C5⋊D5—22no—
76C2×C5⋊D5—22no—
77C2×C5⋊D5—22no—
78C2×C5⋊D5—22no—
79C2×C5⋊D5—44no—
80C2×C5⋊D5—48no—
81C2×C5⋊D5C2×D414yes—
82C4×C52—22no—
83C4×C52—21no—
84C4×C52—21no—
85#1039—22no—
86#1039—22no—
87C23×D5—56no—
88C23×D5—512no—
89C2×C5×D4C2×D5112yescomplemented
90C2×C5×D4C2×D516yescomplemented
91C22×C4×C5C2×D516yescomplemented
92C2×C22⋊D5—512no—
93C2×C22⋊D5—512no—
94C2×C22⋊D5—512no—
95C22×Dic5—56no—
96D4×D5—1048no—
97C2×D20—512no—
98C2×D20—56no—
99C2×C4×D5—512no—
100C5×C4⋊C4C2×D516yescomplemented
101C5×C22⋊C4C2×D5112yescomplemented
102C22⋊Dic5—512no—
103#685—512no—
104#685—512no—
105#685—512no—
106#685—512no—
107C4⋊Dic5—56no—
108#683—512no—
109#534—251noSylow
110C2×C52—22no—
111C2×C52—42no—
112C2×C52C22×D411yescharacteristic
113C2×C52H(5,𝔽2)11yescharacteristic
114C2×C52H(5,𝔽2)11yescharacteristic
115C5⋊D5—24no—
116C5⋊D5—42no—
117C23×C5C22×D5112yes—
118C23×C5C22×D516yes—
119C22×D5—512no—
120C22×D5—524no—
121C22×D5—512no—
122C22×D5—1024no—
123C22×D5—206no—
124C22×D5—2012no—
125C22×D5—206no—
126C22×D5—2012no—
127C5×D4—224no—
128C5×D4—46no—
129C2×C4×C5—46no—
130C2×C4×C5C22×D5112yes—
131C2×C4×C5C22×D516yes—
132C2×C4×C5C22×D516yes—
133C22⋊D5—1024no—
134C22⋊D5—1024no—
135C22⋊D5—2012no—
136C2×Dic5—512no—
137C2×Dic5—512no—
138C2×Dic5—206no—
139D20—1024no—
140D20—206no—
141C4×D5—1024no—
142C22×D4—252no—
143#123—252no—
144#123—252no—
145#121—252no—
146#121—251no—
147#121C5⋊D511yescharacteristic, complemented
148#120—252no—
149#120—252no—
150C2×C22⋊C4—251no—
151C52#53411yescharacteristic, complemented, Sylow
152C22×C5—26no—
153C22×C5—26no—
154C22×C5—212no—
155C22×C5—212no—
156C22×C5—212no—
157C22×C5D4×D5112yes—
158C22×C5C23×D516yes—
159C2×D5—524no—
160C2×D5—1012no—
161C2×D5—1012no—
162C2×D5—1012no—
163C2×D5—1012no—
164C2×D5—1012no—
165C2×D5—2024no—
166C2×D5—2048no—
167C4×C5—212no—
168C4×C5—26no—
169C4×C5—26no—
170Dic5—1012no—
171Dic5—1012no—
172C24—251no—
173C24—252no—
174C2×D4—252no—
175C2×D4—252no—
176C2×D4—252no—
177C2×D4—252no—
178C2×D4—251no—
179C2×D4—508no—
180C2×D4C2×C5⋊D512yescomplemented
181C2×D4C2×C5⋊D511yescharacteristic, complemented
182C22×C4—251no—
183C22×C4—252no—
184C22×C4C2×C5⋊D511yescharacteristic, complemented
185C4⋊C4—252no—
186C4⋊C4—251no—
187C4⋊C4C2×C5⋊D511yescharacteristic, complemented
188C22⋊C4—252no—
189C22⋊C4—252no—
190C22⋊C4—252no—
191C22⋊C4—252no—
192C22⋊C4—252no—
193C22⋊C4C2×C5⋊D512yescomplemented
194C2×C5—212no—
195C2×C5—412no—
196C2×C5C2×D4×D516yes—
197C2×C5#414416yes—
198C2×C5#414916yes—
199D5—1024no—
200D5—2012no—
201C23—252no—
202C23—254no—
203C23—252no—
204C23—504no—
205C23—1001no—
206C23—1001no—
207C23—1002no—
208C23—1002no—
209C23C22×C5⋊D512yes—
210C23C22×C5⋊D511yescharacteristic
211D4—24no—
212D4—41no—
213D4—504no—
214D4—504no—
215D4—504no—
216D4—1001no—
217D4—1002no—
218C2×C4—41no—
219C2×C4—252no—
220C2×C4—252no—
221C2×C4—504no—
222C2×C4—1001no—
223C2×C4C22×C5⋊D511yescharacteristic
224C2×C4C22×C5⋊D511yescharacteristic
225C2×C4C22×C5⋊D512yes—
226C5#6593916yescomplemented
227C22—21no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—254no—
233C22—502no—
234C22—502no—
235C22—502no—
236C22—502no—
237C22—502no—
238C22—1004no—
239C22—1008no—
240C22D4×C5⋊D512yes—
241C22C23×C5⋊D511yescharacteristic
242C4—22no—
243C4—21no—
244C4—21no—
245C4—502no—
246C4—502no—
247C2—22no—
248C2—42no—
249C2—504no—
250C2—1002no—
251C2C2×D4×C5⋊D511yescharacteristic
252C2#1171751711yescharacteristic
253C2#1171752211yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#121C2
hypercenter#121C5⋊D5
hypocenterC52#534
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×C5⋊D5C2semidirect
2. C52×#121C2semidirect
3. #11717259C2semidirect
4. #11717257C2semidirect
5. #11717250C2semidirect
6. C2×#87808C2semidirect
7. #11717214C2semidirect
8. #11717203C2semidirect
9. #11717202C2semidirect
11. C23×C5⋊D5C22semidirect
12. C2×C52×D4C22semidirect
13. C2×C52×D4C22semidirect
14. C22×C4×C52C22semidirect
16. C2×C22⋊(C5⋊D5)C22semidirect
17. C2×C22⋊(C5⋊D5)C22semidirect
18. C22×#1039C22semidirect
21. C2×C4⋊(C5⋊D5)C22semidirect
22. C2×C4×C5⋊D5C22semidirect
23. C52×C4⋊C4C22semidirect
24. C52×C22⋊C4C22semidirect
25. C22⋊#1039C22semidirect
26. #87808C22semidirect
27. #87808C22semidirect
28. #87808C22semidirect
29. #87808C22semidirect
30. C4⋊#1039C22semidirect
31. #87806C22semidirect
40. C22×C5⋊D5D4semidirect
59. C5×#121D5semidirect
89. C2×C5×D4C2×D5semidirect
90. C2×C5×D4C2×D5semidirect
91. C22×C4×C5C2×D5semidirect
100. C5×C4⋊C4C2×D5semidirect
101. C5×C22⋊C4C2×D5semidirect
147. #121C5⋊D5semidirect
151. C52#534semidirect
180. C2×D4C2×C5⋊D5semidirect
181. C2×D4C2×C5⋊D5semidirect
184. C22×C4C2×C5⋊D5semidirect
187. C4⋊C4C2×C5⋊D5semidirect
193. C22⋊C4C2×C5⋊D5semidirect
226. C5#65939semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C5⋊D52
C52×#1212
#117172592
#117172572
#117172502
C2×#878082
#117172142
#117172032
#117172022
#659395