Finite Group Databasebeta

← #13107334

Subgroups — #13107334

Order 1600.

1336 conjugacy classes of subgroups; only the 307 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 307 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
2C2×#87902C211yesmaximal, characteristic, complemented
3#11717273C212yesmaximal, complemented
4C2×C22⋊#1039C212yesmaximal, complemented
5#11717259C212yesmaximal, complemented
6#11717258C211yesmaximal, characteristic, complemented
7#11717256C211yesmaximal, characteristic, complemented
8#11717255C212yesmaximal, complemented
9D4×#1039C211yesmaximal, characteristic, complemented
10C4×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
11#11717244C211yesmaximal, characteristic, complemented
12C24×C52C2212yes—
13C2×C52×D4—24no—
14C2×C52×D4C2212yes—
15C2×C52×D4C2211yescharacteristic
16C2×C52×D4C2211yescharacteristic
17C22×C4×C52C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C2×C22⋊(C5⋊D5)C2212yescomplemented
20C2×C22⋊(C5⋊D5)C2212yescomplemented
21C22×#1039C2212yescomplemented
22C22×#1039C2212yescomplemented
23#87902—24no—
24C2×C4×C5⋊D5C2211yescharacteristic, complemented
25C2×#6045C2211yescharacteristic, complemented
26C22⋊#1039C2212yes—
27C22⋊#1039C2211yescharacteristic
28C22⋊#1039C2212yescomplemented
29C22⋊#1039C2212yescomplemented
30C22⋊#1039C2212yescomplemented
31C22⋊#1039C2212yescomplemented
32#87808C2211yescharacteristic, complemented
33C4⋊#1039C2211yescharacteristic, complemented
34#87806C2212yescomplemented
35#87806C2211yescharacteristic, complemented
36C4×#1039C2211yescharacteristic, complemented
37#66140—56nomaximal
38C23×C52—24no—
39C23×C52—24no—
40C23×C52—42no—
41C23×C52C2312yes—
42C23×C52C2312yes—
43C23×C52C2311yescharacteristic
44C22×C5⋊D5C2311yescharacteristic
45C52×D4—24no—
46C52×D4—22no—
47C52×D4D414yescomplemented
48C2×C4×C52—22no—
49C2×C4×C52C2311yescharacteristic
50C2×C4×C52C2311yescharacteristic
51C22⋊(C5⋊D5)—24no—
52C22⋊(C5⋊D5)—22no—
53C22⋊(C5⋊D5)—42no—
54C2×#1039—24no—
55C2×#1039—42no—
56C2×#1039C2311yescharacteristic
57C2×#1039C2312yes—
58C2×#1039C2311yescharacteristic
59C2×#1039C2312yes—
60C2×#1039C2311yescharacteristic
61C4×C5⋊D5—22no—
62#6045—22no—
63C22×C5×D4D516yescomplemented
64C2×#711—56no—
65#4099—512no—
66C2×C22⋊Dic5—512no—
67#4085—512no—
68#4084—56no—
69#4082—56no—
70#4081—512no—
71D4×Dic5—56no—
72C4×C22⋊D5—56no—
73#4070—56no—
74C22×C52—21no—
75C22×C52—22no—
76C22×C52—22no—
77C22×C52—21no—
78C22×C52—22no—
79C22×C52—22no—
80C22×C52—22no—
81C22×C52—48no—
82C22×C52C2×D414yes—
83C22×C52#4112yescomplemented
84C22×C52C2411yescharacteristic
85C2×C5⋊D5—21no—
86C2×C5⋊D5—21no—
87C2×C5⋊D5—21no—
88C4×C52—21no—
89C4×C52C2×D412yes—
90#1039—22no—
91#1039—21no—
92#1039—21no—
93#1039—21no—
94#1039—22no—
95C24×C5C2×D5112yes—
96C2×C5×D4—224no—
97C2×C5×D4C2×D5112yes—
98C2×C5×D4C2×D516yes—
99C2×C5×D4C2×D516yes—
100C22×C4×C5C2×D516yes—
101C2×C22⋊D5—56no—
102C2×C22⋊D5—512no—
103C2×C22⋊D5—512no—
104C22×Dic5—512no—
105C22×Dic5—512no—
106#711—1024no—
107C2×C4×D5—56no—
108C2×Dic10—56no—
109C22⋊Dic5—56no—
110C22⋊Dic5—512no—
111C22⋊Dic5—512no—
112C22⋊Dic5—512no—
113C22⋊Dic5—512no—
114C22⋊Dic5—512no—
115#685—56no—
116C4⋊Dic5—56no—
117#683—512no—
118#683—56no—
119C4×Dic5—56no—
120#546—251noSylow
121C2×C52—22no—
122C2×C52—24no—
123C2×C52—42no—
124C2×C52C22×D411yescharacteristic
125C2×C52C2×#4111yescharacteristic
126C2×C52H(5,𝔽2)11yescharacteristic
127C5⋊D5—41no—
128C23×C5—224no—
129C23×C5—224no—
130C23×C5—412no—
131C23×C5C22×D5112yes—
132C23×C5C22×D516yes—
133C23×C5C22×D5112yes—
134C22×D5—56no—
135C5×D4—224no—
136C5×D4—212no—
137C5×D4C22⋊D5124yescomplemented
138C2×C4×C5—212no—
139C2×C4×C5C22×D516yes—
140C2×C4×C5C22×D516yes—
141C22⋊D5—1012no—
142C22⋊D5—1024no—
143C22⋊D5—2012no—
144C2×Dic5—56no—
145C2×Dic5—56no—
146C2×Dic5—512no—
147C2×Dic5—512no—
148C2×Dic5—56no—
149C2×Dic5—1024no—
150C2×Dic5—2012no—
151C4×D5—1012no—
152Dic10—1012no—
153C2×#41—251no—
154C22×D4C5⋊D511yescharacteristic, complemented
155#124—251no—
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—26no—
168C22×C5—26no—
169C22×C5—212no—
170C22×C5—212no—
171C22×C5—212no—
172C22×C5—448no—
173C22×C5#711112yescomplemented
174C22×C5C2×C22⋊D5124yes—
175C22×C5C23×D516yes—
176C2×D5—106no—
177C2×D5—106no—
178C2×D5—106no—
179C4×C5—26no—
180C4×C5C2×C22⋊D5112yes—
181Dic5—106no—
182Dic5—106no—
183Dic5—1012no—
184Dic5—1012no—
185Dic5—106no—
186C24C2×C5⋊D512yes—
187#41—504no—
188C2×Q8—251no—
189C2×D4—24no—
190C2×D4—251no—
191C2×D4—252no—
192C2×D4—252no—
193C2×D4C2×C5⋊D512yes—
194C2×D4C2×C5⋊D511yescharacteristic
195C2×D4C2×C5⋊D511yescharacteristic
196C22×C4—252no—
197C22×C4—252no—
198C22×C4—251no—
199C22×C4C2×C5⋊D511yescharacteristic
200C4⋊C4—251no—
201C4⋊C4—251no—
202C4⋊C4—252no—
203C22⋊C4—251no—
204C22⋊C4—252no—
205C22⋊C4—252no—
206C22⋊C4—252no—
207C22⋊C4—252no—
208C22⋊C4—252no—
209C22⋊C4—251no—
210C42—251no—
211C2×C5—212no—
212C2×C5—224no—
213C2×C5—412no—
214C2×C5C2×#71116yes—
215C2×C5#414416yes—
216C2×C5C22×C22⋊D516yes—
217D5—206no—
218C23—24no—
219C23—24no—
220C23—42no—
221C23—251no—
222C23C22×C5⋊D512yes—
223C23C22×C5⋊D511yescharacteristic
224C23C22×C5⋊D512yes—
225Q8—502no—
226D4—24no—
227D4—22no—
228D4—502no—
229D4—504no—
230D4—1002no—
231D4C22⋊(C5⋊D5)14yescomplemented
232C2×C4—22no—
233C2×C4—251no—
234C2×C4—251no—
235C2×C4—252no—
236C2×C4—251no—
237C2×C4—252no—
238C2×C4—504no—
239C2×C4—502no—
240C2×C4—1002no—
241C2×C4C22×C5⋊D511yescharacteristic
242C2×C4C22×C5⋊D511yescharacteristic
243C5#6614016yescomplemented
244C22—22no—
245C22—22no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—48no—
252C22—501no—
253C22—501no—
254C22—501no—
255C22#8790212yescomplemented
256C22C2×C22⋊(C5⋊D5)14yes—
257C22C23×C5⋊D511yescharacteristic
258C4—21no—
259C4—501no—
260C4—501no—
261C4—502no—
262C4—501no—
263C4—502no—
264C4C2×C22⋊(C5⋊D5)12yes—
265C2—22no—
266C2—24no—
267C2—42no—
268C2—1001no—
269C2C2×#8790211yescharacteristic
270C2#1171751711yescharacteristic
271C2C22×C22⋊(C5⋊D5)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×C5⋊D5
commutator, socle, cosocleC22×C52C24
FittingC22×C52×D4C2
hypercenterC22×D4C5⋊D5
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C52×D4C2semidirect
2. C2×#87902C2semidirect
3. #11717273C2semidirect
4. C2×C22⋊#1039C2semidirect
5. #11717259C2semidirect
6. #11717258C2semidirect
7. #11717256C2semidirect
8. #11717255C2semidirect
9. D4×#1039C2semidirect
10. C4×C22⋊(C5⋊D5)C2semidirect
11. #11717244C2semidirect
19. C2×C22⋊(C5⋊D5)C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
21. C22×#1039C22semidirect
22. C22×#1039C22semidirect
24. C2×C4×C5⋊D5C22semidirect
25. C2×#6045C22semidirect
28. C22⋊#1039C22semidirect
29. C22⋊#1039C22semidirect
30. C22⋊#1039C22semidirect
31. C22⋊#1039C22semidirect
32. #87808C22semidirect
33. C4⋊#1039C22semidirect
34. #87806C22semidirect
35. #87806C22semidirect
36. C4×#1039C22semidirect
47. C52×D4D4semidirect
63. C22×C5×D4D5semidirect
83. C22×C52#41semidirect
137. C5×D4C22⋊D5semidirect
154. C22×D4C5⋊D5semidirect
164. C52#546semidirect
173. C22×C5#711semidirect
231. D4C22⋊(C5⋊D5)semidirect
243. C5#66140semidirect
255. C22#87902semidirect

Maximal subgroups

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

subgroupindex
C22×C52×D42
C2×#879022
#117172732
C2×C22⋊#10392
#117172592
#117172582
#117172562
#117172552
D4×#10392
C4×C22⋊(C5⋊D5)2
#117172442
#661405