Finite Group Databasebeta

← #13107356

Subgroups — #13107356

Order 1600.

1336 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 293 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×C52×#41C211yesmaximal, characteristic, complemented
2C2×D4×C5⋊D5C211yesmaximal, characteristic, complemented
3#11717267C211yesmaximal, characteristic, complemented
4#11717266C211yesmaximal, characteristic, complemented
5#11717259C212yesmaximal, complemented
6#11717257C212yesmaximal, complemented
7D4×#1039C211yesmaximal, characteristic, complemented
8#11717250C211yesmaximal, characteristic, complemented
9#11717249C212yesmaximal, complemented
10C4×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
11C2×#87808C212yesmaximal, complemented
12C23×C5⋊D5C2212yes—
13C52×#41—24no—
14C2×C52×Q8C2211yescharacteristic, complemented
15C2×C52×D4C2211yescharacteristic
16C2×C52×D4C2212yescomplemented
17C22×C4×C52C2212yescomplemented
18C22×C4×C52C2211yescharacteristic, complemented
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C2×C22⋊(C5⋊D5)C2212yes—
21C2×C22⋊(C5⋊D5)C2212yescomplemented
22C22×#1039C2212yescomplemented
23D4×C5⋊D5—24no—
24C2×C4⋊(C5⋊D5)C2211yescharacteristic
25C2×C4×C5⋊D5C2211yescharacteristic
26C22⋊#1039C2211yescharacteristic, complemented
27C22⋊#1039C2212yescomplemented
28#87808C2211yescharacteristic
29#87808C2212yes—
30#87808C2212yescomplemented
31#87808C2212yescomplemented
32#87808C2212yescomplemented
33C4⋊#1039C2211yescharacteristic, complemented
34#87806C2211yescharacteristic, complemented
35#87806C2212yescomplemented
36C4×#1039C2211yescharacteristic, complemented
37#66163—56nomaximal
38C23×C52C2311yescharacteristic
39C23×C52C2312yes—
40C22×C5⋊D5—24no—
41C22×C5⋊D5—24no—
42C22×C5⋊D5—42no—
43C22×C5⋊D5C2312yes—
44C22×C5⋊D5C2311yescharacteristic
45C52×Q8—22no—
46C52×D4—24no—
47C52×D4D414yescomplemented
48C2×C4×C52—24no—
49C2×C4×C52—22no—
50C2×C4×C52C2311yescharacteristic
51C2×C4×C52C2312yes—
52C2×C4×C52C2311yescharacteristic
53C22⋊(C5⋊D5)—24no—
54C22⋊(C5⋊D5)—22no—
55C22⋊(C5⋊D5)—42no—
56C2×#1039—42no—
57C2×#1039C2311yescharacteristic
58C2×#1039C2312yes—
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C4⋊(C5⋊D5)—22no—
62C4×C5⋊D5—22no—
63C2×C5×#41D516yescomplemented
64C2×D4×D5—56no—
65#4093—56no—
66#4092—56no—
67#4085—512no—
68#4083—512no—
69D4×Dic5—56no—
70#4076—56no—
71#4075—512no—
72C4×C22⋊D5—56no—
73C2×#685—512no—
74C22×C52—22no—
75C22×C52—22no—
76C22×C52—21no—
77C22×C52—21no—
78C22×C52—21no—
79C22×C52C2×D414yes—
80C22×C52C2411yescharacteristic
81C2×C5⋊D5—21no—
82C2×C5⋊D5—21no—
83C2×C5⋊D5—22no—
84C2×C5⋊D5—22no—
85C2×C5⋊D5—22no—
86C2×C5⋊D5—48no—
87C2×C5⋊D5#4112yescomplemented
88C4×C52—21no—
89C4×C52—22no—
90C4×C52C2×D412yes—
91#1039—21no—
92#1039—21no—
93#1039—21no—
94#1039—22no—
95C23×D5—512no—
96C5×#41—224no—
97C2×C5×Q8C2×D516yescomplemented
98C2×C5×D4C2×D516yes—
99C2×C5×D4C2×D5112yescomplemented
100C22×C4×C5C2×D5112yescomplemented
101C22×C4×C5C2×D516yescomplemented
102C2×C22⋊D5—512no—
103C2×C22⋊D5—56no—
104C2×C22⋊D5—512no—
105C22×Dic5—512no—
106D4×D5—1024no—
107C2×D20—56no—
108C2×C4×D5—56no—
109C22⋊Dic5—56no—
110C22⋊Dic5—512no—
111#685—56no—
112#685—512no—
113#685—512no—
114#685—512no—
115#685—512no—
116C4⋊Dic5—56no—
117#683—56no—
118#683—512no—
119C4×Dic5—56no—
120#546—251noSylow
121C2×C52—24no—
122C2×C52—41no—
123C2×C52C22×D411yescharacteristic
124C2×C52C2×#4111yescharacteristic
125C2×C52H(5,𝔽2)11yescharacteristic
126C5⋊D5—22no—
127C5⋊D5—42no—
128C23×C5C22×D516yes—
129C23×C5C22×D5112yes—
130C22×D5—56no—
131C22×D5—512no—
132C22×D5—1024no—
133C22×D5—1024no—
134C22×D5—2012no—
135C5×Q8—212no—
136C5×D4—224no—
137C5×D4C22⋊D5124yescomplemented
138C2×C4×C5—224no—
139C2×C4×C5—212no—
140C2×C4×C5C22×D516yes—
141C2×C4×C5C22×D5112yes—
142C2×C4×C5C22×D516yes—
143C22⋊D5—1012no—
144C22⋊D5—1024no—
145C22⋊D5—2012no—
146C2×Dic5—56no—
147C2×Dic5—56no—
148C2×Dic5—56no—
149C2×Dic5—512no—
150C2×Dic5—2012no—
151D20—1012no—
152C4×D5—1012no—
153C2×#41C5⋊D511yescharacteristic, complemented
154C22×D4—251no—
155#124—251no—
156#123—252no—
157C22⋊Q8—251no—
158#121—252no—
159#121—251no—
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—26no—
168C22×C5—26no—
169C22×C5—26no—
170C22×C5C2×C22⋊D5124yes—
171C22×C5C23×D516yes—
172C2×D5—512no—
173C2×D5—106no—
174C2×D5—106no—
175C2×D5—1012no—
176C2×D5—1012no—
177C2×D5—1012no—
178C2×D5—2048no—
179C4×C5—26no—
180C4×C5—212no—
181C4×C5C2×C22⋊D5112yes—
182Dic5—106no—
183Dic5—106no—
184Dic5—1012no—
185Dic5—106no—
186C24—252no—
187#41—24no—
188C2×Q8C2×C5⋊D511yescharacteristic, complemented
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×C4C2×C5⋊D512yescomplemented
199C22×C4C2×C5⋊D511yescharacteristic, complemented
200C4⋊C4—251no—
201C4⋊C4—251no—
202C4⋊C4—252no—
203C22⋊C4—252no—
204C22⋊C4—251no—
205C22⋊C4—251no—
206C22⋊C4—252no—
207C22⋊C4—252no—
208C22⋊C4—252no—
209C22⋊C4—252no—
210C42—251no—
211C2×C5—224no—
212C2×C5—46no—
213C2×C5D5×#4116yes—
214C2×C5#414916yes—
215C2×C5C22×C22⋊D516yes—
216D5—1012no—
217D5—2012no—
218C23—251no—
219C23—252no—
220C23—504no—
221C23—504no—
222C23—1002no—
223C23C22×C5⋊D511yescharacteristic
224C23C22×C5⋊D512yes—
225Q8—22no—
226D4—24no—
227D4—502no—
228D4—504no—
229D4—502no—
230D4—1002no—
231D4C22⋊(C5⋊D5)14yescomplemented
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—251no—
235C2×C4—251no—
236C2×C4—252no—
237C2×C4—251no—
238C2×C4—502no—
239C2×C4—1002no—
240C2×C4C22×C5⋊D511yescharacteristic
241C2×C4C22×C5⋊D512yes—
242C2×C4C22×C5⋊D511yescharacteristic
243C5#6616316yescomplemented
244C22—22no—
245C22—22no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—252no—
250C22—501no—
251C22—501no—
252C22—502no—
253C22—502no—
254C22—502no—
255C22—1008no—
256C22C2×C22⋊(C5⋊D5)14yes—
257C22C23×C5⋊D511yescharacteristic
258C4—21no—
259C4—22no—
260C4—501no—
261C4—501no—
262C4—502no—
263C4—501no—
264C4C2×C22⋊(C5⋊D5)12yes—
265C2—24no—
266C2—41no—
267C2—502no—
268C2—1002no—
269C2#41×C5⋊D511yescharacteristic
270C2#1171752211yescharacteristic
271C2C22×C22⋊(C5⋊D5)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC2×C52×#41C2
hypercenterC2×#41C5⋊D5
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C52×#41C2semidirect
2. C2×D4×C5⋊D5C2semidirect
3. #11717267C2semidirect
4. #11717266C2semidirect
5. #11717259C2semidirect
6. #11717257C2semidirect
7. D4×#1039C2semidirect
8. #11717250C2semidirect
9. #11717249C2semidirect
10. C4×C22⋊(C5⋊D5)C2semidirect
11. C2×#87808C2semidirect
14. C2×C52×Q8C22semidirect
16. C2×C52×D4C22semidirect
17. C22×C4×C52C22semidirect
18. C22×C4×C52C22semidirect
21. C2×C22⋊(C5⋊D5)C22semidirect
22. C22×#1039C22semidirect
26. C22⋊#1039C22semidirect
27. C22⋊#1039C22semidirect
30. #87808C22semidirect
31. #87808C22semidirect
32. #87808C22semidirect
33. C4⋊#1039C22semidirect
34. #87806C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
47. C52×D4D4semidirect
63. C2×C5×#41D5semidirect
87. C2×C5⋊D5#41semidirect
97. C2×C5×Q8C2×D5semidirect
99. C2×C5×D4C2×D5semidirect
100. C22×C4×C5C2×D5semidirect
101. C22×C4×C5C2×D5semidirect
137. C5×D4C22⋊D5semidirect
153. C2×#41C5⋊D5semidirect
164. C52#546semidirect
188. C2×Q8C2×C5⋊D5semidirect
195. C2×D4C2×C5⋊D5semidirect
198. C22×C4C2×C5⋊D5semidirect
199. C22×C4C2×C5⋊D5semidirect
231. D4C22⋊(C5⋊D5)semidirect
243. C5#66163semidirect

Maximal subgroups

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

subgroupindex
C2×C52×#412
C2×D4×C5⋊D52
#117172672
#117172662
#117172592
#117172572
D4×#10392
#117172502
#117172492
C4×C22⋊(C5⋊D5)2
C2×#878082
#661635