Finite Group Databasebeta

← #13107142

Subgroups — #13107142

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×D4×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×#87900C211yesmaximal, characteristic, complemented
3C52×#121C211yesmaximal, characteristic, complemented
4#11717258C212yesmaximal, complemented
5#11717257C212yesmaximal, complemented
6#11717256C211yesmaximal, characteristic, complemented
7C4×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
8#11717217C211yesmaximal, characteristic, complemented
9#11717213C211yesmaximal, characteristic, complemented
10#11717203C212yesmaximal, complemented
11C22⋊C4×C5⋊D5C212yesmaximal, complemented
12C23×C5⋊D5C2212yes—
13C2×C52×D4C2211yescharacteristic
14C2×C52×D4C2212yescomplemented
15C22×C4×C52C2211yescharacteristic, complemented
16C2×C22⋊(C5⋊D5)C2212yes—
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2212yescomplemented
19D4×C5⋊D5—24no—
20#87900—24no—
21C2×C4⋊(C5⋊D5)C2211yescharacteristic
22C2×C4×C5⋊D5C2211yescharacteristic
23C2×C4×C5⋊D5C2212yescomplemented
24C2×C4×C5⋊D5C2212yescomplemented
25C2×#6045C2211yescharacteristic, complemented
26C52×C4⋊C4C2211yescharacteristic, complemented
27C52×C22⋊C4C2212yescomplemented
28C22⋊#1039C2212yescomplemented
29C22⋊#1039C2212yescomplemented
30C22⋊#1039C2211yescharacteristic, complemented
31#87808C2211yescharacteristic
32#87808C2212yes—
33#87808C2212yescomplemented
34C4⋊#1039C2212yescomplemented
35#87806C2211yescharacteristic, complemented
36C4×#1039C2211yescharacteristic, complemented
37#65946—56nomaximal
38C23×C52C2312yes—
39C23×C52C2311yescharacteristic
40C22×C5⋊D5—24no—
41C22×C5⋊D5—24no—
42C22×C5⋊D5—42no—
43C22×C5⋊D5C2312yes—
44C22×C5⋊D5C2311yescharacteristic
45C52×D4—22no—
46C52×D4—42no—
47C2×C4×C52—22no—
48C2×C4×C52C2312yes—
49C2×C4×C52C2311yescharacteristic
50C2×C4×C52C2311yescharacteristic
51C22⋊(C5⋊D5)—24no—
52C22⋊(C5⋊D5)—24no—
53C22⋊(C5⋊D5)—22no—
54C2×#1039C2311yescharacteristic
55C2×#1039C2312yes—
56C2×#1039C2311yescharacteristic
57C2×#1039C2311yescharacteristic
58C4⋊(C5⋊D5)D414yescomplemented
59C4×C5⋊D5—24no—
60C4×C5⋊D5—22no—
61C4×C5⋊D5—42no—
62#6045—22no—
63C2×D4×D5—56no—
64C2×#709—56no—
65C5×#121D516yescomplemented
66#4084—512no—
67#4083—512no—
68#4082—56no—
69C4×C22⋊D5—56no—
70#4043—56no—
71#4039—56no—
72#4029—512no—
73D5×C22⋊C4—512no—
74C22×C52—21no—
75C22×C52—22no—
76C22×C52—21no—
77C22×C52—22no—
78C22×C52—21no—
79C22×C52—22no—
80C22×C52C2411yescharacteristic
81C2×C5⋊D5—21no—
82C2×C5⋊D5—22no—
83C2×C5⋊D5—22no—
84C2×C5⋊D5—21no—
85C2×C5⋊D5—48no—
86C2×C5⋊D5C2×D414yes—
87C2×C5⋊D5#4112yescomplemented
88C4×C52—21no—
89C4×C52—22no—
90C4×C52C2×D412yes—
91#1039—22no—
92#1039—21no—
93#1039—21no—
94#1039—21no—
95C23×D5—512no—
96C2×C5×D4C2×D516yes—
97C2×C5×D4C2×D5112yescomplemented
98C22×C4×C5C2×D516yescomplemented
99C2×C22⋊D5—512no—
100C2×C22⋊D5—512no—
101C2×C22⋊D5—56no—
102D4×D5—1024no—
103#709—1024no—
104C2×D20—56no—
105C2×C4×D5—512no—
106C2×C4×D5—56no—
107C2×C4×D5—512no—
108C2×Dic10—56no—
109C5×C4⋊C4C2×D516yescomplemented
110C5×C22⋊C4C2×D5112yescomplemented
111C22⋊Dic5—512no—
112C22⋊Dic5—512no—
113C22⋊Dic5—56no—
114#685—56no—
115#685—512no—
116#685—512no—
117C4⋊Dic5—512no—
118#683—56no—
119C4×Dic5—56no—
120#546—251noSylow
121C2×C52—42no—
122C2×C52—41no—
123C2×C52C22×D411yescharacteristic
124C2×C52C2×#4111yescharacteristic
125C2×C52H(5,𝔽2)11yescharacteristic
126C5⋊D5—22no—
127C5⋊D5—24no—
128C23×C5C22×D5112yes—
129C23×C5C22×D516yes—
130C22×D5—56no—
131C22×D5—512no—
132C22×D5—1024no—
133C22×D5—1024no—
134C22×D5—2012no—
135C5×D4—212no—
136C5×D4—412no—
137C2×C4×C5—212no—
138C2×C4×C5C22×D516yes—
139C2×C4×C5C22×D5112yes—
140C2×C4×C5C22×D516yes—
141C22⋊D5—1024no—
142C22⋊D5—1024no—
143C22⋊D5—1012no—
144C2×Dic5—56no—
145C2×Dic5—56no—
146C2×Dic5—56no—
147C2×Dic5—512no—
148D20—524no—
149C4×D5—1024no—
150C4×D5—1012no—
151C4×D5—2012no—
152Dic10—1012no—
153C2×#41—251no—
154C22×D4—251no—
155#124—251no—
156#123—252no—
157C22⋊Q8—251no—
158#121—252no—
159#121C5⋊D511yescharacteristic, complemented
160#120—252no—
161C4×D4—251no—
162C4×D4—251no—
163C2×C22⋊C4—252no—
164C52#54611yescharacteristic, complemented, Sylow
165C22×C5—26no—
166C22×C5—212no—
167C22×C5—26no—
168C22×C5—212no—
169C22×C5—26no—
170C22×C5—212no—
171C22×C5C23×D516yes—
172C2×D5—524no—
173C2×D5—512no—
174C2×D5—106no—
175C2×D5—1012no—
176C2×D5—106no—
177C2×D5—1012no—
178C2×D5—2048no—
179C4×C5—26no—
180C4×C5—212no—
181C4×C5D4×D5112yes—
182Dic5—1012no—
183Dic5—106no—
184Dic5—106no—
185Dic5—106no—
186C24—252no—
187#41—504no—
188C2×Q8—251no—
189C2×D4—251no—
190C2×D4—252no—
191C2×D4—252no—
192C2×D4—251no—
193C2×D4—504no—
194C2×D4C2×C5⋊D511yescharacteristic
195C2×D4C2×C5⋊D512yescomplemented
196C22×C4—252no—
197C22×C4—251no—
198C22×C4—252no—
199C22×C4C2×C5⋊D511yescharacteristic, complemented
200C4⋊C4—251no—
201C4⋊C4—252no—
202C4⋊C4C2×C5⋊D511yescharacteristic, complemented
203C22⋊C4—252no—
204C22⋊C4—251no—
205C22⋊C4—252no—
206C22⋊C4—251no—
207C22⋊C4—252no—
208C22⋊C4—252no—
209C22⋊C4C2×C5⋊D512yescomplemented
210C42—251no—
211C2×C5—412no—
212C2×C5—46no—
213C2×C5C2×D4×D516yes—
214C2×C5#414416yes—
215C2×C5D5×#4116yes—
216D5—1024no—
217D5—1012no—
218C23—251no—
219C23—252no—
220C23—504no—
221C23—504no—
222C23—1002no—
223C23C22×C5⋊D512yes—
224C23C22×C5⋊D511yescharacteristic
225Q8—502no—
226D4—22no—
227D4—42no—
228D4—254no—
229D4—504no—
230D4—504no—
231D4—502no—
232C2×C4—22no—
233C2×C4—251no—
234C2×C4—251no—
235C2×C4—252no—
236C2×C4—251no—
237C2×C4—504no—
238C2×C4—502no—
239C2×C4—1002no—
240C2×C4C22×C5⋊D512yes—
241C2×C4C22×C5⋊D511yescharacteristic
242C2×C4C22×C5⋊D511yescharacteristic
243C5#6594616yescomplemented
244C22—21no—
245C22—22no—
246C22—21no—
247C22—22no—
248C22—21no—
249C22—22no—
250C22—254no—
251C22—252no—
252C22—501no—
253C22—502no—
254C22—501no—
255C22—502no—
256C22—1008no—
257C22C23×C5⋊D511yescharacteristic
258C4—21no—
259C4—22no—
260C4—501no—
261C4—502no—
262C4—501no—
263C4—501no—
264C4D4×C5⋊D512yes—
265C2—42no—
266C2—41no—
267C2—502no—
268C2—504no—
269C2C2×D4×C5⋊D511yescharacteristic
270C2#1171751711yescharacteristic
271C2#41×C5⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC52×#121C2
hypercenter#121C5⋊D5
hypocenterC52#546
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. C2×#87900C2semidirect
3. C52×#121C2semidirect
4. #11717258C2semidirect
5. #11717257C2semidirect
6. #11717256C2semidirect
7. C4×C22⋊(C5⋊D5)C2semidirect
8. #11717217C2semidirect
9. #11717213C2semidirect
10. #11717203C2semidirect
11. C22⋊C4×C5⋊D5C2semidirect
14. C2×C52×D4C22semidirect
15. C22×C4×C52C22semidirect
18. C2×C22⋊(C5⋊D5)C22semidirect
23. C2×C4×C5⋊D5C22semidirect
24. C2×C4×C5⋊D5C22semidirect
25. C2×#6045C22semidirect
26. C52×C4⋊C4C22semidirect
27. C52×C22⋊C4C22semidirect
28. C22⋊#1039C22semidirect
29. C22⋊#1039C22semidirect
30. C22⋊#1039C22semidirect
33. #87808C22semidirect
34. C4⋊#1039C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
58. C4⋊(C5⋊D5)D4semidirect
65. C5×#121D5semidirect
87. C2×C5⋊D5#41semidirect
97. C2×C5×D4C2×D5semidirect
98. C22×C4×C5C2×D5semidirect
109. C5×C4⋊C4C2×D5semidirect
110. C5×C22⋊C4C2×D5semidirect
159. #121C5⋊D5semidirect
164. C52#546semidirect
195. C2×D4C2×C5⋊D5semidirect
199. C22×C4C2×C5⋊D5semidirect
202. C4⋊C4C2×C5⋊D5semidirect
209. C22⋊C4C2×C5⋊D5semidirect
243. C5#65946semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C5⋊D52
C2×#879002
C52×#1212
#117172582
#117172572
#117172562
C4×C22⋊(C5⋊D5)2
#117172172
#117172132
#117172032
C22⋊C4×C5⋊D52
#659465