Finite Group Databasebeta

← #13105095

Subgroups — #13105095

Order 1600.

950 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 160 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×#87877C212yesmaximal, complemented
2C2×D5×D20C212yesmaximal, complemented
3C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
4C5×#4020C211yesmaximal, characteristic, complemented
5C4×C5×D20C211yesmaximal, characteristic, complemented
6#11717040C214yesmaximal, complemented
7#11717034C212yesmaximal, complemented
8#11717026C212yesmaximal, complemented
9C22×D52C2214yescomplemented
10C2×C4⋊(C5⋊D5)C2211yescharacteristic
11C2×C4×C5⋊D5C2212yescomplemented
12C2×C5×D20C2211yescharacteristic
13C2×C5×D20C2212yescomplemented
14C2×C5×D20C2212yescomplemented
15C2×C5×D20C2212yescomplemented
16C2×C4×C5×D5C2212yescomplemented
17C2×#6038C2214yescomplemented
18C2×#6037C2214yescomplemented
19#87877—28no—
20D5×D20—28no—
21C42×C52C2211yescharacteristic, complemented
22#87808C2212yescomplemented
23C4⋊#1039C2211yescharacteristic, complemented
24C5×#685C2212yescomplemented
25C5×C4⋊Dic5C2211yescharacteristic, complemented
26#87778C2214yescomplemented
27#66051—51nomaximal
28D4×D20—51nomaximal
29C22×C5⋊D5C2312yes—
30C22×C5×D5C2314yes—
31C22×C5×D5C2312yes—
32C2×D52—28no—
33C2×D52—28no—
34C2×D52—44no—
35C2×C4×C52C2311yescharacteristic
36C2×C4×C52C2311yescharacteristic
37C2×C4×C52C2311yescharacteristic
38C2×#1039C2312yes—
39C4⋊(C5⋊D5)D414yescomplemented
40C4×C5⋊D5—24no—
41C2×C5×Dic5C2312yes—
42C5×D20—24no—
43C5×D20—24no—
44C5×D20—42no—
45C5×D20D414yescomplemented
46C4×C5×D5—24no—
47#6038—28no—
48#6037—28no—
49C2×D4×D5—52no—
50C2×D4×D5—52no—
51C2×D4×D5—52no—
52C22×D20—52no—
53C5×#127—51no—
54C4×C5×D4—51no—
55#4084—52no—
56#4084—52no—
57#4083—54no—
58#4076—52no—
59#4041—52no—
60#4028—54no—
61#4020D511yescharacteristic, complemented
62C4×D20—51no—
63C4×D20—51no—
64C4×D20—102no—
65C4×D20D511yescharacteristic, complemented
66C22×C52C2411yescharacteristic
67C2×C5⋊D5—22no—
68C2×C5⋊D5—22no—
69C2×C5⋊D5C2×D414yescomplemented
70C2×C5×D5—22no—
71C2×C5×D5—24no—
72C2×C5×D5—24no—
73C2×C5×D5—22no—
74C2×C5×D5—24no—
75C2×C5×D5C2×D414yescomplemented
76D52—416no—
77C4×C52—21no—
78C4×C52C2×D412yescomplemented
79C4×C52C2×D412yescomplemented
80#1039—22no—
81C5×Dic5—22no—
82C23×D5—54no—
83C23×D5—54no—
84C2×C5×D4—51no—
85C2×C5×D4—52no—
86C2×C5×D4—52no—
87C2×C5×D4—52no—
88C22×C4×C5—52no—
89C2×C22⋊D5—54no—
90C2×C22⋊D5—54no—
91C2×C22⋊D5—54no—
92D4×D5—108no—
93D4×D5—108no—
94D4×D5—108no—
95C2×D20—51no—
96C2×D20—51no—
97C2×D20—54no—
98C2×D20—102no—
99C2×D20—108no—
100C2×D20C2×D511yescharacteristic
101C2×D20C2×D512yescomplemented
102C2×D20C2×D512yescomplemented
103C2×D20C2×D512yescomplemented
104C2×C4×D5—52no—
105C2×C4×D5—52no—
106C2×C4×D5—104no—
107C2×C4×D5C2×D512yescomplemented
108C5×C4⋊C4—51no—
109C5×C22⋊C4—52no—
110C42×C5—22no—
111C42×C5C2×D511yescharacteristic, complemented
112C42×C5C2×D511yescharacteristic, complemented
113C22⋊Dic5—54no—
114#685—54no—
115#685—52no—
116#685—52no—
117#685—104no—
118#685C2×D512yescomplemented
119C4⋊Dic5—51no—
120C4⋊Dic5—51no—
121C4⋊Dic5—102no—
122C4⋊Dic5C2×D511yescharacteristic, complemented
123D42—251noSylow
124C2×C52C22×D411yescharacteristic
125C2×C52C22×D411yescharacteristic
126C2×C52H(5,𝔽2)11yescharacteristic
127C5⋊D5—24no—
128C5×D5—24no—
129C5×D5—44no—
130C23×C5—52no—
131C23×C5—54no—
132C22×D5—52no—
133C22×D5—52no—
134C22×D5—104no—
135C22×D5—108no—
136C22×D5—108no—
137C22×D5—108no—
138C22×D5—108no—
139C22×D5—204no—
140C22×D5—204no—
141C22×D5C22×D514yes—
142C22×D5C22×D512yes—
143C5×D4—54no—
144C5×D4—104no—
145C5×D4—104no—
146C5×D4—202no—
147C2×C4×C5—22no—
148C2×C4×C5—22no—
149C2×C4×C5—22no—
150C2×C4×C5—52no—
151C2×C4×C5—104no—
152C2×C4×C5C22×D511yescharacteristic
153C2×C4×C5C22×D511yescharacteristic
154C2×C4×C5C22×D511yescharacteristic
155C2×C4×C5C22×D511yescharacteristic
156C2×C4×C5C22×D511yescharacteristic
157C2×C4×C5C22×D511yescharacteristic
158C22⋊D5—108no—
159C22⋊D5—108no—
160C22⋊D5—108no—
161C2×Dic5—52no—
162C2×Dic5—52no—
163C2×Dic5—104no—
164C2×Dic5C22×D512yes—
165D20—24no—
166D20—24no—
167D20—42no—
168D20—54no—
169D20—54no—
170D20—108no—
171D20—108no—
172D20D2014yescomplemented
173C4×D5—24no—
174C4×D5—104no—
175C4×D5—104no—
176C4×D5—208no—
177C22×D4—252no—
178C22×D4—252no—
179#127—51no—
180#121—252no—
181#121—252no—
182#120—254no—
183C4×D4—51no—
184C4×D4—251no—
185C52D4211yescharacteristic, complemented, Sylow
186C22×C5—22no—
187C22×C5—54no—
188C22×C5—102no—
189C22×C5—102no—
190C22×C5—104no—
191C22×C5—104no—
192C22×C5—104no—
193C22×C5C23×D511yescharacteristic
194C22×C5C23×D511yescharacteristic
195C2×D5—22no—
196C2×D5—22no—
197C2×D5—24no—
198C2×D5—24no—
199C2×D5—24no—
200C2×D5—54no—
201C2×D5—54no—
202C2×D5—108no—
203C2×D5—102no—
204C2×D5—102no—
205C2×D5—102no—
206C2×D5—102no—
207C2×D5—204no—
208C2×D5—204no—
209C2×D5—2016no—
210C2×D5—2016no—
211C2×D5C2×D2014yescomplemented
212C4×C5—21no—
213C4×C5—21no—
214C4×C5—24no—
215C4×C5—24no—
216C4×C5—42no—
217C4×C5—102no—
218C4×C5C2×D2012yescomplemented
219C4×C5D4×D512yescomplemented
220C4×C5D4×D512yescomplemented
221C4×C5D4×D512yescomplemented
222Dic5—22no—
223Dic5—102no—
224Dic5—102no—
225Dic5—204no—
226C24—254no—
227C2×D4—51no—
228C2×D4—52no—
229C2×D4—52no—
230C2×D4—52no—
231C2×D4—254no—
232C2×D4—254no—
233C2×D4—251no—
234C2×D4—508no—
235C2×D4—508no—
236C22×C4—52no—
237C22×C4—252no—
238C4⋊C4—51no—
239C4⋊C4—251no—
240C22⋊C4—52no—
241C22⋊C4—254no—
242C22⋊C4—252no—
243C42D5211yescharacteristic, complemented
244C2×C5—22no—
245C2×C5—22no—
246C2×C5—22no—
247C2×C5—104no—
248C2×C5—204no—
249C2×C5C22×D2011yescharacteristic
250C2×C5C2×D4×D511yescharacteristic
251C2×C5C2×D4×D511yescharacteristic
252C2×C5C2×D4×D511yescharacteristic
253C2×C5#414411yescharacteristic
254C2×C5#414911yescharacteristic
255D5—24no—
256D5—44no—
257D5—104no—
258D5—104no—
259D5—208no—
260C23—52no—
261C23—54no—
262C23—252no—
263C23—508no—
264C23—508no—
265C23—1004no—
266D4—54no—
267D4—104no—
268D4—104no—
269D4—202no—
270D4—254no—
271D4—508no—
272D4—508no—
273C2×C4—52no—
274C2×C4—104no—
275C2×C4—252no—
276C2×C4—504no—
277C2×C4C2×D5211yescharacteristic
278C2×C4C2×D5211yescharacteristic
279C2×C4C2×D5211yescharacteristic
280C5—22no—
281C5D4×D2011yescharacteristic, complemented
282C5#6605111yescharacteristic, complemented
283C22—54no—
284C22—102no—
285C22—102no—
286C22—104no—
287C22—104no—
288C22—104no—
289C22—254no—
290C22—502no—
291C22—502no—
292C22—10016no—
293C22C22×D5211yescharacteristic
294C4—21no—
295C4—102no—
296C4—502no—
297C4D5×D2012yescomplemented
298C4#8787712yescomplemented
299C2—104no—
300C2—204no—
301C2—504no—
302C2C2×D5×D2011yescharacteristic
303C2C2×#8787711yescharacteristic
304C2#1171747211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
hypercenterC42D52
hypocenterC52D42
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#87877C2semidirect
2. C2×D5×D20C2semidirect
3. C4×C4⋊(C5⋊D5)C2semidirect
4. C5×#4020C2semidirect
5. C4×C5×D20C2semidirect
6. #11717040C2semidirect
7. #11717034C2semidirect
8. #11717026C2semidirect
9. C22×D52C22semidirect
11. C2×C4×C5⋊D5C22semidirect
13. C2×C5×D20C22semidirect
14. C2×C5×D20C22semidirect
15. C2×C5×D20C22semidirect
16. C2×C4×C5×D5C22semidirect
17. C2×#6038C22semidirect
18. C2×#6037C22semidirect
21. C42×C52C22semidirect
22. #87808C22semidirect
23. C4⋊#1039C22semidirect
24. C5×#685C22semidirect
25. C5×C4⋊Dic5C22semidirect
26. #87778C22semidirect
39. C4⋊(C5⋊D5)D4semidirect
45. C5×D20D4semidirect
61. #4020D5semidirect
65. C4×D20D5semidirect
69. C2×C5⋊D5C2×D4semidirect
75. C2×C5×D5C2×D4semidirect
78. C4×C52C2×D4semidirect
79. C4×C52C2×D4semidirect
101. C2×D20C2×D5semidirect
102. C2×D20C2×D5semidirect
103. C2×D20C2×D5semidirect
107. C2×C4×D5C2×D5semidirect
111. C42×C5C2×D5semidirect
112. C42×C5C2×D5semidirect
118. #685C2×D5semidirect
122. C4⋊Dic5C2×D5semidirect
172. D20D20semidirect
185. C52D42semidirect
211. C2×D5C2×D20semidirect
218. C4×C5C2×D20semidirect
219. C4×C5D4×D5semidirect
220. C4×C5D4×D5semidirect
221. C4×C5D4×D5semidirect
243. C42D52semidirect
281. C5D4×D20semidirect
282. C5#66051semidirect
297. C4D5×D20semidirect
298. C4#87877semidirect

Maximal subgroups

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

subgroupindex
C2×#878772
C2×D5×D202
C4×C4⋊(C5⋊D5)2
C5×#40202
C4×C5×D202
#117170402
#117170342
#117170262
#660515
D4×D205