Finite Group Databasebeta

← C2×#11716912 #13105764

Subgroups — C2×#11716912 #13105764

Order 1600.

672 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 144 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×D5×D20C211yesmaximal, characteristic, complemented
2C2×#87874C211yesmaximal, characteristic, complemented
3C2×#87801C211yesmaximal, characteristic, complemented
4C2×C5×#677C211yesmaximal, characteristic, complemented
5C2×C5×#676C211yesmaximal, characteristic, complemented
6C2×#87774C211yesmaximal, characteristic, complemented
7C2×#87771C211yesmaximal, characteristic, complemented
8#11716912C218yesmaximal, direct product factor
9C22×D52—21no—
10C2×C4⋊(C5⋊D5)C2211yescharacteristic
11C2×C4⋊(C5⋊D5)C2211yescharacteristic, complemented
12C2×C5×D20C2211yescharacteristic
13C2×C4×C5×D5C2211yescharacteristic
14C2×C5×Dic10C2211yescharacteristic, complemented
15C2×#6038—21no—
16C2×#6038—21no—
17C2×#6036—21no—
18D5×D20—22no—
19D5×D20C2214yescomplemented
20#87874—22no—
21#87874C2214yescomplemented
22C2×C52×C8C2211yescharacteristic, complemented
23#87801C2214yescomplemented
24C2×C5×#168C2211yescharacteristic, complemented
25C5×#677C2214yescomplemented
26C5×#676C2214yescomplemented
27#87774C2214yescomplemented
28#87771C2214yescomplemented
29C2×#4060—51nomaximal
30C2×#4054—51nomaximal
31C22×C5⋊D5—21no—
32C22×C5⋊D5—21no—
33C22×C5×D5—21no—
34C22×C5×D5D411yescharacteristic
35C2×D52—24no—
36C2×D52—44no—
37C2×C4×C52C2311yescharacteristic
38C4⋊(C5⋊D5)—21no—
39C4⋊(C5⋊D5)—21no—
40C4⋊(C5⋊D5)C2312yes—
41C4⋊(C5⋊D5)C2312yescomplemented
42C2×C5×Dic5—21no—
43C2×C5×Dic5D411yescharacteristic
44C5×D20—21no—
45C5×D20C2312yes—
46C4×C5×D5D412yes—
47C4×C5×D5C2312yes—
48C5×Dic10—21no—
49C5×Dic10C2312yescomplemented
50#6038—24no—
51#6038—24no—
52#6036—24no—
53C52×C8C2312yescomplemented
54C5×#168C2312yescomplemented
55C2×#713—51no—
56C2×D4×D5—51no—
57C2×#709—51no—
58C22×D20—51no—
59C2×C5×SD16—51no—
60C2×C5×M16—51no—
61C2×#688—51no—
62C2×#686—51no—
63#4060—58no—
64#4054—58no—
65C2×D40—51no—
66C2×D40—51no—
67C2×D40—51no—
68C2×D40—102no—
69C2×#677—51no—
70C2×#677D511yescharacteristic, complemented
71C2×#676D511yescharacteristic, complemented
72C22×C52C2×D411yescharacteristic
73C2×C5⋊D5—22no—
74C2×C5⋊D5—22no—
75C2×C5⋊D5—42no—
76C2×C5⋊D5—42no—
77C2×C5×D5—22no—
78C2×C5×D5—22no—
79C2×C5×D5—42no—
80C2×C5×D5C2×D412yes—
81D52—48no—
82C4×C52C2×D411yescharacteristic
83C4×C52C2411yescharacteristic
84C5×Dic5—22no—
85C5×Dic5C2×D412yes—
86C23×D5—101no—
87C23×D5—101no—
88C2×C5×Q8—51no—
89C2×C5×D4—51no—
90C22×C4×C5—51no—
91C2×C22⋊D5—101no—
92C2×C22⋊D5—101no—
93#713—54no—
94#713—102no—
95D4×D5—54no—
96D4×D5—102no—
97#709—54no—
98#709—102no—
99C2×D20—51no—
100C2×D20—51no—
101C2×D20—51no—
102C2×D20—51no—
103C2×D20—54no—
104C2×D20—102no—
105C2×D20—102no—
106C2×D20—102no—
107C2×D20—101no—
108C2×D20—101no—
109C2×D20C2×D511yescharacteristic
110C2×C4×D5—101no—
111C2×C4×D5—101no—
112C2×C4×D5C2×D511yescharacteristic
113C2×Dic10C2×D511yescharacteristic, complemented
114C5×SD16—54no—
115C5×M16—54no—
116C2×C5×C8—22no—
117C2×C5×C8—51no—
118C2×C5×C8C2×D511yescharacteristic, complemented
119C2×C5×C8C2×D511yescharacteristic, complemented
120#688—54no—
121#686—54no—
122C2×#168C2×D511yescharacteristic, complemented
123D40—54no—
124D40—54no—
125D40—54no—
126D40—108no—
127#677—54no—
128#677C2×D514yescomplemented
129#676C2×D514yescomplemented
130C2×AGL(1,ℤ/8)—251noSylow
131C2×C52AGL(1,ℤ/8)12yescomplemented
132C2×C52C22×D411yescharacteristic
133C5⋊D5—42no—
134C5⋊D5—42no—
135C5×D5—22no—
136C5×D5—42no—
137C23×C5—51no—
138C23×C5—101no—
139C22×D5—21no—
140C22×D5—104no—
141C22×D5—101no—
142C22×D5—101no—
143C22×D5—101no—
144C22×D5—101no—
145C22×D5—104no—
146C22×D5—202no—
147C22×D5—202no—
148C22×D5—204no—
149C22×D5—204no—
150C22×D5D2011yescharacteristic
151C5×Q8—52no—
152C5×Q8—101no—
153C5×D4—52no—
154C5×D4—101no—
155C2×C4×C5—22no—
156C2×C4×C5—51no—
157C2×C4×C5—52no—
158C2×C4×C5—52no—
159C2×C4×C5—101no—
160C2×C4×C5C22×D511yescharacteristic
161C2×C4×C5C22×D511yescharacteristic
162C22⋊D5—104no—
163C22⋊D5—104no—
164C2×Dic5—21no—
165C2×Dic5D2011yescharacteristic
166D20—21no—
167D20—52no—
168D20—52no—
169D20—52no—
170D20—52no—
171D20—104no—
172D20—104no—
173D20—104no—
174D20—104no—
175D20—101no—
176D20—101no—
177D20—101no—
178D20—101no—
179D20—202no—
180D20—202no—
181D20C22×D512yes—
182C4×D5—104no—
183C4×D5—104no—
184C4×D5D2012yes—
185C4×D5C22×D512yes—
186Dic10—21no—
187Dic10C22×D512yescomplemented
188C5×C8—24no—
189C5×C8—52no—
190C5×C8C22×D512yescomplemented
191C5×C8C22×D512yescomplemented
192#168C22×D512yescomplemented
193C2×#41—251no—
194C22×D4—251no—
195AGL(1,ℤ/8)—258no—
196C2×SD16—51no—
197C2×SD16—251no—
198C2×D8—251no—
199C2×D8—251no—
200C2×M16—51no—
201C52C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
202C22×C5—22no—
203C22×C5—52no—
204C22×C5—102no—
205C22×C5—102no—
206C22×C5—202no—
207C22×C5C2×D2011yescharacteristic
208C22×C5D4×D511yescharacteristic
209C2×D5—22no—
210C2×D5—22no—
211C2×D5—42no—
212C2×D5—102no—
213C2×D5—102no—
214C2×D5—102no—
215C2×D5—102no—
216C2×D5—204no—
217C2×D5—204no—
218C2×D5—208no—
219C2×D5—208no—
220C2×D5—202no—
221C2×D5—202no—
222C2×D5—202no—
223C2×D5—202no—
224C2×D5—404no—
225C2×D5—404no—
226C2×D5C2×D2012yes—
227C4×C5—22no—
228C4×C5—22no—
229C4×C5—52no—
230C4×C5—102no—
231C4×C5C2×D2011yescharacteristic
232C4×C5D4×D511yescharacteristic
233C4×C5C23×D511yescharacteristic
234C4×C5C23×D511yescharacteristic
235Dic5—22no—
236Dic5C2×D2012yes—
237C24—501no—
238#41—254no—
239#41—502no—
240C2×Q8—51no—
241C2×D4—51no—
242C2×D4—254no—
243C2×D4—251no—
244C2×D4—251no—
245C2×D4—502no—
246C2×D4—501no—
247C2×D4—501no—
248C22×C4—51no—
249C22×C4—501no—
250SD16—54no—
251SD16—254no—
252D8—254no—
253D8—254no—
254M16—54no—
255C2×C8—51no—
256C2×C8D5211yescharacteristic, complemented
257C2×C5—24no—
258C2×C5—22no—
259C2×C5—102no—
260C2×C5—202no—
261C2×C5#405412yescomplemented
262C2×C5#406012yescomplemented
263C2×C5C22×D2011yescharacteristic
264C2×C5C2×D4×D511yescharacteristic
265D5—22no—
266D5—42no—
267D5—202no—
268D5—202no—
269D5—202no—
270D5—202no—
271D5—404no—
272D5—404no—
273C23—51no—
274C23—101no—
275C23—504no—
276C23—501no—
277C23—501no—
278C23—1004no—
279Q8—52no—
280Q8—101no—
281D4—52no—
282D4—101no—
283D4—252no—
284D4—252no—
285D4—504no—
286D4—504no—
287D4—501no—
288D4—501no—
289C2×C4—51no—
290C2×C4—52no—
291C2×C4—52no—
292C2×C4—101no—
293C2×C4—504no—
294C2×C4C2×D5211yescharacteristic
295C8—52no—
296C8C2×D5212yescomplemented
297C5—22no—
298C5C2×#405411yescharacteristic, complemented
299C5C2×#406011yescharacteristic, complemented
300C22—52no—
301C22—102no—
302C22—102no—
303C22—202no—
304C22—502no—
305C22—502no—
306C22—1008no—
307C22—1002no—
308C22—1002no—
309C22D5×D2011yescharacteristic
310C4—52no—
311C4—102no—
312C4D5×D2011yescharacteristic
313C4C22×D5211yescharacteristic
314C2—102no—
315C2—202no—
316C2—1002no—
317C2—1002no—
318C2#1171691212yesdirect product factor
319C2C2×D5×D2011yescharacteristic
subgroupG/H
centerC22D5×D20
commutator, cosocleC4×C52C24
FrattiniC4C22×D52
FittingC2×C52×C8C22
socleC22×C52C2×D4
hypercenterC2×C8D52
hypocenterC52C2×AGL(1,ℤ/8)
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×D5×D20C2semidirect
2. C2×#87874C2semidirect
3. C2×#87801C2semidirect
4. C2×C5×#677C2semidirect
5. C2×C5×#676C2semidirect
6. C2×#87774C2semidirect
7. C2×#87771C2semidirect
8. #11716912C2direct (N × H)
11. C2×C4⋊(C5⋊D5)C22semidirect
14. C2×C5×Dic10C22semidirect
19. D5×D20C22semidirect
21. #87874C22semidirect
22. C2×C52×C8C22semidirect
23. #87801C22semidirect
24. C2×C5×#168C22semidirect
25. C5×#677C22semidirect
26. C5×#676C22semidirect
27. #87774C22semidirect
28. #87771C22semidirect
41. C4⋊(C5⋊D5)C23semidirect
49. C5×Dic10C23semidirect
53. C52×C8C23semidirect
54. C5×#168C23semidirect
70. C2×#677D5semidirect
71. C2×#676D5semidirect
113. C2×Dic10C2×D5semidirect
118. C2×C5×C8C2×D5semidirect
119. C2×C5×C8C2×D5semidirect
122. C2×#168C2×D5semidirect
128. #677C2×D5semidirect
129. #676C2×D5semidirect
131. C2×C52AGL(1,ℤ/8)semidirect
187. Dic10C22×D5semidirect
190. C5×C8C22×D5semidirect
191. C5×C8C22×D5semidirect
192. #168C22×D5semidirect
201. C52C2×AGL(1,ℤ/8)semidirect
256. C2×C8D52semidirect
261. C2×C5#4054semidirect
262. C2×C5#4060semidirect
296. C8C2×D52semidirect
298. C5C2×#4054semidirect
299. C5C2×#4060semidirect
318. C2#11716912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×D202
C2×#878742
C2×#878012
C2×C5×#6772
C2×C5×#6762
C2×#877742
C2×#877712
#117169122
C2×#40605
C2×#40545