Finite Group Databasebeta

← #13107206

Subgroups — #13107206

Order 1600.

1336 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 237 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×#87904C211yesmaximal, characteristic, complemented
2C2×D4×C5⋊D5C211yesmaximal, characteristic, complemented
3C52×#124C211yesmaximal, characteristic, complemented
4#11717266C211yesmaximal, characteristic, complemented
5#11717258C211yesmaximal, characteristic, complemented
6#11717204C212yesmaximal, complemented
7#11717203C212yesmaximal, complemented
8#11717202C212yesmaximal, complemented
9C22⋊C4×C5⋊D5C212yesmaximal, complemented
10C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
11C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
12C23×C5⋊D5C2212yes—
13C2×C52×Q8C2211yescharacteristic, complemented
14C2×C52×D4C2211yescharacteristic
15C2×C22⋊(C5⋊D5)C2212yes—
16C2×C22⋊(C5⋊D5)C2212yescomplemented
17#87904—24no—
18D4×C5⋊D5—24no—
19C2×C4⋊(C5⋊D5)C2211yescharacteristic
20C2×C4⋊(C5⋊D5)C2211yescharacteristic
21C2×C4⋊(C5⋊D5)C2212yescomplemented
22C2×C4×C5⋊D5C2211yescharacteristic
23C2×C4×C5⋊D5C2211yescharacteristic, complemented
24C2×C4×C5⋊D5C2212yescomplemented
25C2×C4×C5⋊D5C2212yescomplemented
26C52×C22⋊C4C2212yescomplemented
27C52×C22⋊C4C2212yescomplemented
28C42×C52C2211yescharacteristic, complemented
29C22⋊#1039C2212yescomplemented
30#87808C2211yescharacteristic
31#87808C2212yes—
32#87808C2212yescomplemented
33#87808C2211yescharacteristic, complemented
34C4⋊#1039C2211yescharacteristic, complemented
35C4⋊#1039C2211yescharacteristic, complemented
36#87806C2212yescomplemented
37#66010—56nomaximal
38C23×C52C2312yes—
39C22×C5⋊D5—24no—
40C22×C5⋊D5—24no—
41C22×C5⋊D5—42no—
42C22×C5⋊D5C2311yescharacteristic
43C22×C5⋊D5C2312yes—
44C22×C5⋊D5C2311yescharacteristic
45C52×Q8—22no—
46C52×D4—22no—
47C2×C4×C52C2311yescharacteristic
48C2×C4×C52C2312yes—
49C2×C4×C52C2311yescharacteristic
50C2×C4×C52C2311yescharacteristic
51C22⋊(C5⋊D5)—24no—
52C22⋊(C5⋊D5)—42no—
53C2×#1039C2312yes—
54C2×#1039C2311yescharacteristic
55C2×#1039C2311yescharacteristic
56C4⋊(C5⋊D5)—24no—
57C4⋊(C5⋊D5)—22no—
58C4⋊(C5⋊D5)D414yescomplemented
59C4×C5⋊D5—24no—
60C4×C5⋊D5—22no—
61C4×C5⋊D5—22no—
62C4×C5⋊D5—42no—
63C2×#713—56no—
64C2×D4×D5—56no—
65C5×#124D516yescomplemented
66#4092—56no—
67#4084—56no—
68#4030—512no—
69#4029—512no—
70#4028—512no—
71D5×C22⋊C4—512no—
72C4×D20—56no—
73C4×D20—56no—
74C22×C52—22no—
75C22×C52—22no—
76C22×C52—22no—
77C22×C52C2411yescharacteristic
78C2×C5⋊D5—21no—
79C2×C5⋊D5—22no—
80C2×C5⋊D5—22no—
81C2×C5⋊D5—21no—
82C2×C5⋊D5—21no—
83C2×C5⋊D5—21no—
84C2×C5⋊D5—21no—
85C2×C5⋊D5—48no—
86C2×C5⋊D5C2×D414yes—
87C2×C5⋊D5#4112yescomplemented
88C4×C52—22no—
89C4×C52—21no—
90C4×C52—21no—
91C4×C52C2×D412yes—
92#1039—21no—
93#1039—21no—
94#1039—22no—
95C23×D5—512no—
96C2×C5×Q8C2×D516yescomplemented
97C2×C5×D4C2×D516yes—
98C2×C22⋊D5—512no—
99C2×C22⋊D5—512no—
100#713—1024no—
101D4×D5—1024no—
102C2×D20—56no—
103C2×D20—512no—
104C2×D20—56no—
105C2×C4×D5—56no—
106C2×C4×D5—56no—
107C2×C4×D5—512no—
108C2×C4×D5—512no—
109C5×C22⋊C4C2×D5112yescomplemented
110C5×C22⋊C4C2×D5112yescomplemented
111C42×C5C2×D516yescomplemented
112C22⋊Dic5—512no—
113#685—56no—
114#685—512no—
115#685—512no—
116#685—56no—
117C4⋊Dic5—56no—
118C4⋊Dic5—56no—
119#683—512no—
120#546—251noSylow
121C2×C52—42no—
122C2×C52C22×D411yescharacteristic
123C2×C52C2×#4111yescharacteristic
124C2×C52H(5,𝔽2)11yescharacteristic
125C5⋊D5—22no—
126C5⋊D5—24no—
127C5⋊D5—41no—
128C23×C5C22×D5112yes—
129C22×D5—56no—
130C22×D5—512no—
131C22×D5—56no—
132C22×D5—1024no—
133C22×D5—1024no—
134C22×D5—2012no—
135C5×Q8—212no—
136C5×D4—212no—
137C2×C4×C5C22×D5112yes—
138C2×C4×C5C22×D516yes—
139C2×C4×C5C22×D516yes—
140C2×C4×C5C22×D516yes—
141C22⋊D5—1024no—
142C22⋊D5—2012no—
143C2×Dic5—56no—
144C2×Dic5—512no—
145C2×Dic5—56no—
146D20—524no—
147D20—1012no—
148D20—1024no—
149C4×D5—1024no—
150C4×D5—1012no—
151C4×D5—1012no—
152C4×D5—2012no—
153C2×#41—251no—
154C22×D4—251no—
155#124C5⋊D511yescharacteristic, complemented
156#123—252no—
157C22⋊Q8—251no—
158#121—251no—
159#121—252no—
160#120—252no—
161C4×D4—251no—
162C4×D4—251no—
163C2×C22⋊C4—252no—
164C52#54611yescharacteristic, complemented, Sylow
165C22×C5—212no—
166C22×C5—212no—
167C22×C5—212no—
168C22×C5C23×D516yes—
169C2×D5—524no—
170C2×D5—512no—
171C2×D5—106no—
172C2×D5—1012no—
173C2×D5—1012no—
174C2×D5—106no—
175C2×D5—106no—
176C2×D5—106no—
177C2×D5—106no—
178C2×D5—2048no—
179C4×C5—212no—
180C4×C5—26no—
181C4×C5—26no—
182C4×C5D4×D5112yes—
183Dic5—106no—
184Dic5—106no—
185Dic5—1012no—
186C24—252no—
187#41—504no—
188C2×Q8C2×C5⋊D511yescharacteristic, complemented
189C2×D4—252no—
190C2×D4—252no—
191C2×D4—251no—
192C2×D4—251no—
193C2×D4—252no—
194C2×D4—504no—
195C2×D4C2×C5⋊D511yescharacteristic
196C22×C4—252no—
197C22×C4—251no—
198C22×C4—251no—
199C22×C4—252no—
200C4⋊C4—251no—
201C4⋊C4—251no—
202C4⋊C4—252no—
203C22⋊C4—252no—
204C22⋊C4—251no—
205C22⋊C4—252no—
206C22⋊C4—252no—
207C22⋊C4—251no—
208C22⋊C4C2×C5⋊D512yescomplemented
209C22⋊C4C2×C5⋊D512yescomplemented
210C42C2×C5⋊D511yescharacteristic, complemented
211C2×C5—412no—
212C2×C5C2×D4×D516yes—
213C2×C5D5×#4116yes—
214C2×C5#414916yes—
215D5—1024no—
216D5—1012no—
217D5—206no—
218C23—251no—
219C23—252no—
220C23—251no—
221C23—504no—
222C23—504no—
223C23—1002no—
224C23C22×C5⋊D512yes—
225Q8—22no—
226D4—22no—
227D4—254no—
228D4—504no—
229D4—502no—
230D4—504no—
231D4—1002no—
232C2×C4—252no—
233C2×C4—251no—
234C2×C4—251no—
235C2×C4—504no—
236C2×C4—502no—
237C2×C4—502no—
238C2×C4—1002no—
239C2×C4C22×C5⋊D511yescharacteristic
240C2×C4C22×C5⋊D511yescharacteristic
241C2×C4C22×C5⋊D512yes—
242C2×C4C22×C5⋊D511yescharacteristic
243C5#6601016yescomplemented
244C22—22no—
245C22—22no—
246C22—22no—
247C22—254no—
248C22—252no—
249C22—501no—
250C22—502no—
251C22—502no—
252C22—501no—
253C22—501no—
254C22—501no—
255C22—501no—
256C22—1008no—
257C22C23×C5⋊D511yescharacteristic
258C4—22no—
259C4—21no—
260C4—21no—
261C4—501no—
262C4—501no—
263C4—502no—
264C4D4×C5⋊D512yes—
265C2—42no—
266C2—502no—
267C2—504no—
268C2—1001no—
269C2C2×D4×C5⋊D511yescharacteristic
270C2#41×C5⋊D511yescharacteristic
271C2#1171752211yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#124C2
hypercenter#124C5⋊D5
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#87904C2semidirect
2. C2×D4×C5⋊D5C2semidirect
3. C52×#124C2semidirect
4. #11717266C2semidirect
5. #11717258C2semidirect
6. #11717204C2semidirect
7. #11717203C2semidirect
8. #11717202C2semidirect
9. C22⋊C4×C5⋊D5C2semidirect
10. C4×C4⋊(C5⋊D5)C2semidirect
11. C4×C4⋊(C5⋊D5)C2semidirect
13. C2×C52×Q8C22semidirect
16. C2×C22⋊(C5⋊D5)C22semidirect
21. C2×C4⋊(C5⋊D5)C22semidirect
23. C2×C4×C5⋊D5C22semidirect
24. C2×C4×C5⋊D5C22semidirect
25. C2×C4×C5⋊D5C22semidirect
26. C52×C22⋊C4C22semidirect
27. C52×C22⋊C4C22semidirect
28. C42×C52C22semidirect
29. C22⋊#1039C22semidirect
32. #87808C22semidirect
33. #87808C22semidirect
34. C4⋊#1039C22semidirect
35. C4⋊#1039C22semidirect
36. #87806C22semidirect
58. C4⋊(C5⋊D5)D4semidirect
65. C5×#124D5semidirect
87. C2×C5⋊D5#41semidirect
96. C2×C5×Q8C2×D5semidirect
109. C5×C22⋊C4C2×D5semidirect
110. C5×C22⋊C4C2×D5semidirect
111. C42×C5C2×D5semidirect
155. #124C5⋊D5semidirect
164. C52#546semidirect
188. C2×Q8C2×C5⋊D5semidirect
208. C22⋊C4C2×C5⋊D5semidirect
209. C22⋊C4C2×C5⋊D5semidirect
210. C42C2×C5⋊D5semidirect
243. C5#66010semidirect

Maximal subgroups

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

subgroupindex
C2×#879042
C2×D4×C5⋊D52
C52×#1242
#117172662
#117172582
#117172042
#117172032
#117172022
C22⋊C4×C5⋊D52
C4×C4⋊(C5⋊D5)2
#660105