Finite Group Databasebeta

← #13106607

Subgroups — #13106607

Order 1600.

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

normal subgroups only

This group has 928 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
1C22×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C22×C5×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×C5×C22⋊Dic5C211yesmaximal, characteristic, complemented
4#11717099C212yesmaximal, complemented
5#11717096C214yesmaximal, complemented
6#11717095C212yesmaximal, complemented
7#11717084C212yesmaximal, complemented
8#11717080C212yesmaximal, complemented
9C24×C52C2211yescharacteristic
10C23×C5⋊D5C2211yescharacteristic, complemented
11C23×C5×D5C2211yescharacteristic, complemented
12C2×C22⋊(C5⋊D5)—24no—
13C2×C22⋊(C5⋊D5)C2212yes—
14C2×C22⋊(C5⋊D5)C2212yescomplemented
15C22×#1039C2211yescharacteristic, complemented
16C2×C5×C22⋊D5—24no—
17C2×C5×C22⋊D5C2212yes—
18C2×C5×C22⋊D5C2212yescomplemented
19C22×C5×Dic5C2211yescharacteristic, complemented
20C22×C5×Dic5C2212yescomplemented
21C2×#6038C2214yescomplemented
22C5×C22⋊Dic5C2212yescomplemented
23C5×C22⋊Dic5C2212yescomplemented
24#87780C2214yescomplemented
25#87779C2214yescomplemented
26#87778C2214yescomplemented
27#66138—51nomaximal
28#65827—51nomaximal
29C23×C52—42no—
30C23×C52D414yescomplemented
31C23×C52C2311yescharacteristic
32C23×C52C2311yescharacteristic
33C23×C52C2311yescharacteristic
34C22×C5⋊D5—22no—
35C22×C5⋊D5—42no—
36C22×C5⋊D5C2312yes—
37C22×C5×D5—22no—
38C22×C5×D5—42no—
39C22×C5×D5C2312yes—
40C22⋊(C5⋊D5)—24no—
41C22⋊(C5⋊D5)—24no—
42C2×#1039—22no—
43C2×#1039C2312yes—
44C5×C22⋊D5—24no—
45C5×C22⋊D5—24no—
46C2×C5×Dic5—22no—
47C2×C5×Dic5—42no—
48C2×C5×Dic5C2314yes—
49C2×C5×Dic5C2312yes—
50#6038—44no—
51C22×C5×D4—51no—
52C22×C22⋊D5—51no—
53C22×C22⋊D5—51no—
54C22×C22⋊D5—102no—
55C22×C22⋊D5D511yescharacteristic, complemented
56C2×C5×C22⋊C4—51no—
57#4099—52no—
58C2×C22⋊Dic5D511yescharacteristic, complemented
59#4085—54no—
60#4083—52no—
61#4081—52no—
62#4076—54no—
63#4075—52no—
64#4032—52no—
65#4032—52no—
66#4028—52no—
67#4028—52no—
68C22×C52—22no—
69C22×C52—22no—
70C22×C52—22no—
71C22×C52—44no—
72C22×C52C2×D412yes—
73C22×C52C2×D412yes—
74C22×C52C2×D412yes—
75C22×C52C2411yescharacteristic
76C2×C5⋊D5—22no—
77C2×C5⋊D5—24no—
78C2×C5⋊D5—44no—
79C2×C5×D5—22no—
80C2×C5×D5—24no—
81C2×C5×D5—44no—
82#1039—22no—
83C5×Dic5—22no—
84C5×Dic5—24no—
85C24×C5—22no—
86C24×C5—51no—
87C24×C5C2×D511yescharacteristic
88C24×C5C2×D511yescharacteristic
89C23×D5—51no—
90C23×D5—51no—
91C23×D5—102no—
92C23×D5C2×D511yescharacteristic, complemented
93C2×C5×D4—52no—
94C2×C5×D4—52no—
95C2×C5×D4—104no—
96C22×C4×C5—51no—
97C22×C4×C5—52no—
98C2×C22⋊D5—24no—
99C2×C22⋊D5—52no—
100C2×C22⋊D5—52no—
101C2×C22⋊D5—52no—
102C2×C22⋊D5—52no—
103C2×C22⋊D5—54no—
104C2×C22⋊D5—104no—
105C2×C22⋊D5—104no—
106C2×C22⋊D5—1016no—
107C2×C22⋊D5—104no—
108C2×C22⋊D5—104no—
109C2×C22⋊D5C2×D512yes—
110C2×C22⋊D5C2×D512yescomplemented
111C22×Dic5—51no—
112C22×Dic5—51no—
113C22×Dic5—102no—
114C22×Dic5C2×D511yescharacteristic, complemented
115C22×Dic5C2×D512yescomplemented
116C2×D20—54no—
117C5×C22⋊C4—52no—
118C5×C22⋊C4—52no—
119C22⋊Dic5—54no—
120C22⋊Dic5C2×D512yescomplemented
121C22⋊Dic5C2×D512yescomplemented
122#685—54no—
123#685—54no—
124#685—54no—
125C4⋊Dic5—54no—
126#683—54no—
127#534—251noSylow
128C2×C52—22no—
129C2×C52—22no—
130C2×C52—22no—
131C2×C52C22×D411yescharacteristic
132C2×C52H(5,𝔽2)12yes—
133C5⋊D5—42no—
134C5×D5—42no—
135C23×C5—28no—
136C23×C5—22no—
137C23×C5—22no—
138C23×C5—22no—
139C23×C5—42no—
140C23×C5—42no—
141C23×C5—48no—
142C23×C5—52no—
143C23×C5—102no—
144C23×C5—202no—
145C23×C5D2014yescomplemented
146C23×C5C22⋊D514yescomplemented
147C23×C5C22×D511yescharacteristic
148C23×C5C22×D511yescharacteristic
149C23×C5C22×D511yescharacteristic
150C23×C5C22×D511yescharacteristic
151C23×C5C22×D511yescharacteristic
152C23×C5C22×D511yescharacteristic
153C22×D5—22no—
154C22×D5—42no—
155C22×D5—52no—
156C22×D5—52no—
157C22×D5—104no—
158C22×D5—108no—
159C22×D5—102no—
160C22×D5—102no—
161C22×D5—202no—
162C22×D5—202no—
163C22×D5—208no—
164C22×D5C22×D512yes—
165C5×D4—104no—
166C5×D4—104no—
167C2×C4×C5—52no—
168C2×C4×C5—54no—
169C2×C4×C5—102no—
170C2×C4×C5—202no—
171C22⋊D5—24no—
172C22⋊D5—24no—
173C22⋊D5—1016no—
174C22⋊D5—1016no—
175C22⋊D5—104no—
176C22⋊D5—104no—
177C22⋊D5—104no—
178C22⋊D5—104no—
179C22⋊D5—204no—
180C2×Dic5—22no—
181C2×Dic5—42no—
182C2×Dic5—52no—
183C2×Dic5—52no—
184C2×Dic5—104no—
185C2×Dic5—108no—
186C2×Dic5—102no—
187C2×Dic5—102no—
188C2×Dic5C22×D514yes—
189C2×Dic5C22×D512yes—
190D20—204no—
191C22×D4—51no—
192C22×D4—251no—
193#123—252no—
194#123—252no—
195#121—254no—
196#120—252no—
197#120—252no—
198C2×C22⋊C4—51no—
199C52#53411yescharacteristic, complemented, Sylow
200C22×C5—28no—
201C22×C5—22no—
202C22×C5—22no—
203C22×C5—22no—
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—22no—
207C22×C5—24no—
208C22×C5—24no—
209C22×C5—24no—
210C22×C5—22no—
211C22×C5—44no—
212C22×C5—44no—
213C22×C5—44no—
214C22×C5—44no—
215C22×C5—416no—
216C22×C5—102no—
217C22×C5—104no—
218C22×C5—204no—
219C22×C5C2×D2012yes—
220C22×C5C2×D2012yes—
221C22×C5C2×D2012yes—
222C22×C5C2×C22⋊D512yes—
223C22×C5C2×C22⋊D512yes—
224C22×C5C2×C22⋊D512yes—
225C22×C5C23×D511yescharacteristic
226C22×C5C23×D511yescharacteristic
227C2×D5—24no—
228C2×D5—22no—
229C2×D5—44no—
230C2×D5—108no—
231C2×D5—102no—
232C2×D5—104no—
233C2×D5—104no—
234C2×D5—102no—
235C2×D5—208no—
236C2×D5—2016no—
237C2×D5—204no—
238C2×D5—204no—
239C4×C5—102no—
240C4×C5—104no—
241Dic5—24no—
242Dic5—22no—
243Dic5—108no—
244Dic5—102no—
245Dic5—102no—
246C24—51no—
247C24—251no—
248C24D5211yescharacteristic
249C2×D4—52no—
250C2×D4—52no—
251C2×D4—104no—
252C2×D4—254no—
253C2×D4—252no—
254C2×D4—252no—
255C2×D4—504no—
256C22×C4—51no—
257C22×C4—52no—
258C22×C4—251no—
259C4⋊C4—254no—
260C22⋊C4—52no—
261C22⋊C4—52no—
262C22⋊C4—254no—
263C22⋊C4—254no—
264C2×C5—28no—
265C2×C5—22no—
266C2×C5—22no—
267C2×C5—22no—
268C2×C5—22no—
269C2×C5—22no—
270C2×C5—22no—
271C2×C5—22no—
272C2×C5—24no—
273C2×C5—44no—
274C2×C5—44no—
275C2×C5—202no—
276C2×C5C22×D2011yescharacteristic
277C2×C5#414412yes—
278C2×C5#414412yes—
279C2×C5C22×C22⋊D511yescharacteristic
280D5—42no—
281D5—208no—
282D5—202no—
283D5—202no—
284C23—42no—
285C23—52no—
286C23—102no—
287C23—202no—
288C23—252no—
289C23—502no—
290C23—1002no—
291C23#603814yescomplemented
292C23C2×D5211yescharacteristic
293C23C2×D5211yescharacteristic
294C23C2×D5211yescharacteristic
295D4—104no—
296D4—104no—
297D4—504no—
298D4—504no—
299D4—1004no—
300C2×C4—54no—
301C2×C4—52no—
302C2×C4—102no—
303C2×C4—202no—
304C2×C4—252no—
305C2×C4—502no—
306C5—22no—
307C5#6582711yescharacteristic, complemented
308C5#6613811yescharacteristic, complemented
309C22—22no—
310C22—22no—
311C22—22no—
312C22—44no—
313C22—102no—
314C22—104no—
315C22—204no—
316C22—502no—
317C22—504no—
318C22—1004no—
319C22C2×#603812yes—
320C22C2×#603812yes—
321C22C2×#603812yes—
322C22C22×D5211yescharacteristic
323C4—102no—
324C4—104no—
325C4—502no—
326C2—22no—
327C2—22no—
328C2—22no—
329C2—202no—
330C2—1002no—
331C2C22×#603811yescharacteristic
332C2#1171749512yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC24×C52C22
hypercenterC24D52
hypocenterC52#534
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C5⋊D5)C2semidirect
2. C22×C5×C22⋊D5C2semidirect
3. C2×C5×C22⋊Dic5C2semidirect
4. #11717099C2semidirect
5. #11717096C2semidirect
6. #11717095C2semidirect
7. #11717084C2semidirect
8. #11717080C2semidirect
10. C23×C5⋊D5C22semidirect
11. C23×C5×D5C22semidirect
14. C2×C22⋊(C5⋊D5)C22semidirect
15. C22×#1039C22semidirect
18. C2×C5×C22⋊D5C22semidirect
19. C22×C5×Dic5C22semidirect
20. C22×C5×Dic5C22semidirect
21. C2×#6038C22semidirect
22. C5×C22⋊Dic5C22semidirect
23. C5×C22⋊Dic5C22semidirect
24. #87780C22semidirect
25. #87779C22semidirect
26. #87778C22semidirect
30. C23×C52D4semidirect
55. C22×C22⋊D5D5semidirect
58. C2×C22⋊Dic5D5semidirect
92. C23×D5C2×D5semidirect
110. C2×C22⋊D5C2×D5semidirect
114. C22×Dic5C2×D5semidirect
115. C22×Dic5C2×D5semidirect
120. C22⋊Dic5C2×D5semidirect
121. C22⋊Dic5C2×D5semidirect
145. C23×C5D20semidirect
146. C23×C5C22⋊D5semidirect
199. C52#534semidirect
291. C23#6038semidirect
307. C5#65827semidirect
308. C5#66138semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C5⋊D5)2
C22×C5×C22⋊D52
C2×C5×C22⋊Dic52
#117170992
#117170962
#117170952
#117170842
#117170802
#661385
#658275