Finite Group Databasebeta

← C2×#11716916 #13105773

Subgroups — C2×#11716916 #13105773

Order 1600.

664 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 136 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×#87877C211yesmaximal, characteristic, complemented
2C2×#87871C211yesmaximal, characteristic, complemented
3C2×#87799C211yesmaximal, characteristic, complemented
4C2×C5×D40C211yesmaximal, characteristic, complemented
5C2×C5×#677C211yesmaximal, characteristic, complemented
6C2×#87772C211yesmaximal, characteristic, complemented
7C2×#87770C211yesmaximal, characteristic, complemented
8#11716916C218yesmaximal, direct product factor
9C22×D52—21no—
10C2×C4×C5⋊D5C2211yescharacteristic
11C2×C5×D20C2211yescharacteristic
12C2×C5×D20C2211yescharacteristic
13C2×C5×D20C2211yescharacteristic, complemented
14C2×C5×Dic10C2211yescharacteristic, complemented
15C2×#6038—21no—
16C2×#6037—21no—
17C2×D5×Dic5—21no—
18#87877—22no—
19#87877C2214yescomplemented
20#87871—22no—
21#87871C2214yescomplemented
22C2×C52×C8C2211yescharacteristic, complemented
23C2×#6029C2211yescharacteristic, complemented
24#87799C2214yescomplemented
25C5×D40C2214yescomplemented
26C5×#677C2214yescomplemented
27#87772C2214yescomplemented
28#87770C2214yescomplemented
29C2×#4060—51nomaximal
30C2×#4057—51nomaximal
31C22×C5⋊D5D411yescharacteristic
32C22×C5×D5—21no—
33C22×C5×D5—21no—
34C22×C5×D5—21no—
35C2×D52—24no—
36C2×D52—44no—
37C2×C4×C52C2311yescharacteristic
38C2×#1039D411yescharacteristic
39C4×C5⋊D5D412yes—
40C4×C5⋊D5C2312yes—
41C2×C5×Dic5—21no—
42C5×D20—21no—
43C5×D20—21no—
44C5×D20—21no—
45C5×D20C2312yes—
46C5×D20C2312yes—
47C5×D20C2312yescomplemented
48C5×Dic10—21no—
49C5×Dic10C2312yescomplemented
50#6038—24no—
51#6037—24no—
52D5×Dic5—24no—
53C52×C8C2312yescomplemented
54#6029C2312yescomplemented
55C2×#713—51no—
56C2×#711—51no—
57C2×D4×D5—51no—
58C2×D4×D5—51no—
59C2×C5×SD16—51no—
60C2×C5×D8—51no—
61C2×#688—51no—
62C2×#687—51no—
63C2×#686—51no—
64C2×#686—51no—
65#4060—58no—
66#4057—58no—
67C2×D40D511yescharacteristic, complemented
68C2×#677D511yescharacteristic, complemented
69C2×#676—51no—
70C2×#676—51no—
71C2×#676—102no—
72C22×C52C2×D411yescharacteristic
73C2×C5⋊D5—22no—
74C2×C5⋊D5C2×D412yes—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—42no—
79C2×C5×D5—42no—
80C2×C5×D5—42no—
81D52—48no—
82C4×C52C2×D411yescharacteristic
83C4×C52C2411yescharacteristic
84#1039C2×D412yes—
85C5×Dic5—22no—
86C23×D5—101no—
87C23×D5—101no—
88C2×C5×Q8—51no—
89C2×C5×D4—51no—
90C2×C5×D4—51no—
91C2×C5×D4—51no—
92C2×C22⋊D5—101no—
93C2×C22⋊D5—101no—
94C2×C22⋊D5—101no—
95C22×Dic5—101no—
96#713—54no—
97#713—102no—
98#711—54no—
99#711—102no—
100D4×D5—54no—
101D4×D5—54no—
102D4×D5—102no—
103D4×D5—102no—
104C2×D20—101no—
105C2×D20C2×D511yescharacteristic
106C2×D20C2×D511yescharacteristic
107C2×D20C2×D511yescharacteristic, complemented
108C2×C4×D5—51no—
109C2×C4×D5—51no—
110C2×C4×D5—102no—
111C2×C4×D5—101no—
112C2×Dic10C2×D511yescharacteristic, complemented
113C5×SD16—54no—
114C5×D8—54no—
115C2×C5×C8—22no—
116C2×C5×C8C2×D511yescharacteristic, complemented
117C2×C5×C8C2×D511yescharacteristic, complemented
118#688—54no—
119#687—54no—
120#686—54no—
121#686—54no—
122C2×#168—51no—
123C2×#168—51no—
124C2×#168—102no—
125D40C2×D514yescomplemented
126#677C2×D514yescomplemented
127#676—54no—
128#676—54no—
129#676—108no—
130C2×AGL(1,ℤ/8)—251noSylow
131C2×C52AGL(1,ℤ/8)12yescomplemented
132C2×C52C22×D411yescharacteristic
133C5⋊D5—22no—
134C5×D5—42no—
135C5×D5—42no—
136C5×D5—42no—
137C23×C5—101no—
138C23×C5—101no—
139C23×C5—101no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—51no—
144C22×D5—51no—
145C22×D5—102no—
146C22×D5—104no—
147C22×D5—104no—
148C22×D5—204no—
149C22×D5—204no—
150C5×Q8—52no—
151C5×Q8—101no—
152C5×D4—52no—
153C5×D4—52no—
154C5×D4—52no—
155C5×D4—101no—
156C5×D4—101no—
157C5×D4—101no—
158C2×C4×C5—22no—
159C2×C4×C5—101no—
160C2×C4×C5C22×D511yescharacteristic
161C2×C4×C5C22×D511yescharacteristic
162C22⋊D5—104no—
163C22⋊D5—104no—
164C22⋊D5—104no—
165C2×Dic5—21no—
166C2×Dic5—51no—
167C2×Dic5—51no—
168C2×Dic5—102no—
169C2×Dic5—104no—
170D20—21no—
171D20—21no—
172D20—21no—
173D20—104no—
174D20C22×D512yes—
175D20C22×D512yes—
176D20C22×D512yescomplemented
177C4×D5—52no—
178C4×D5—52no—
179C4×D5—52no—
180C4×D5—52no—
181C4×D5—104no—
182C4×D5—104no—
183C4×D5—104no—
184Dic10—21no—
185Dic10C22×D512yescomplemented
186C5×C8—24no—
187C5×C8C22×D512yescomplemented
188C5×C8C22×D512yescomplemented
189#168—52no—
190#168—52no—
191#168—104no—
192C2×#41—251no—
193C22×D4—251no—
194AGL(1,ℤ/8)—258no—
195C2×SD16—51no—
196C2×SD16—251no—
197C2×D8—51no—
198C2×D8—251no—
199C2×M16—251no—
200C52C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
201C22×C5—22no—
202C22×C5—102no—
203C22×C5—102no—
204C22×C5—102no—
205C22×C5—202no—
206C22×C5—202no—
207C22×C5—202no—
208C22×C5D4×D511yescharacteristic
209C22×C5D4×D511yescharacteristic
210C2×D5—22no—
211C2×D5—22no—
212C2×D5—22no—
213C2×D5—42no—
214C2×D5—42no—
215C2×D5—42no—
216C2×D5—52no—
217C2×D5—52no—
218C2×D5—104no—
219C2×D5—102no—
220C2×D5—102no—
221C2×D5—204no—
222C2×D5—208no—
223C2×D5—208no—
224C4×C5—22no—
225C4×C5—22no—
226C4×C5—102no—
227C4×C5D4×D511yescharacteristic
228C4×C5D4×D511yescharacteristic
229C4×C5C23×D511yescharacteristic
230C4×C5C23×D511yescharacteristic
231Dic5—22no—
232Dic5—52no—
233Dic5—52no—
234Dic5—104no—
235C24—501no—
236#41—254no—
237#41—502no—
238C2×Q8—51no—
239C2×D4—51no—
240C2×D4—51no—
241C2×D4—51no—
242C2×D4—254no—
243C2×D4—501no—
244C2×D4—502no—
245C2×D4—501no—
246C22×C4—251no—
247C22×C4—501no—
248SD16—54no—
249SD16—254no—
250D8—54no—
251D8—254no—
252M16—254no—
253C2×C8—251no—
254C2×C8D5211yescharacteristic, complemented
255C2×C5—24no—
256C2×C5—22no—
257C2×C5—202no—
258C2×C5—202no—
259C2×C5—202no—
260C2×C5#405712yescomplemented
261C2×C5#406012yescomplemented
262C2×C5C2×D4×D511yescharacteristic
263C2×C5C2×D4×D511yescharacteristic
264D5—42no—
265D5—42no—
266D5—42no—
267D5—102no—
268D5—102no—
269D5—204no—
270C23—101no—
271C23—101no—
272C23—101no—
273C23—251no—
274C23—504no—
275C23—1004no—
276Q8—52no—
277Q8—101no—
278D4—52no—
279D4—52no—
280D4—52no—
281D4—101no—
282D4—101no—
283D4—101no—
284D4—504no—
285D4—504no—
286C2×C4—101no—
287C2×C4—251no—
288C2×C4—252no—
289C2×C4—252no—
290C2×C4—504no—
291C2×C4C2×D5211yescharacteristic
292C8—252no—
293C8C2×D5212yescomplemented
294C5—22no—
295C5C2×#405711yescharacteristic, complemented
296C5C2×#406011yescharacteristic, complemented
297C22—102no—
298C22—102no—
299C22—102no—
300C22—202no—
301C22—202no—
302C22—202no—
303C22—252no—
304C22—502no—
305C22—1008no—
306C22#8787711yescharacteristic
307C4—102no—
308C4—252no—
309C4#8787711yescharacteristic
310C4C22×D5211yescharacteristic
311C2—202no—
312C2—202no—
313C2—202no—
314C2—502no—
315C2#1171691612yesdirect product factor
316C2C2×#8787711yescharacteristic
subgroupG/H
centerC22#87877
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×#87877C2semidirect
2. C2×#87871C2semidirect
3. C2×#87799C2semidirect
4. C2×C5×D40C2semidirect
5. C2×C5×#677C2semidirect
6. C2×#87772C2semidirect
7. C2×#87770C2semidirect
8. #11716916C2direct (N × H)
13. C2×C5×D20C22semidirect
14. C2×C5×Dic10C22semidirect
19. #87877C22semidirect
21. #87871C22semidirect
22. C2×C52×C8C22semidirect
23. C2×#6029C22semidirect
24. #87799C22semidirect
25. C5×D40C22semidirect
26. C5×#677C22semidirect
27. #87772C22semidirect
28. #87770C22semidirect
47. C5×D20C23semidirect
49. C5×Dic10C23semidirect
53. C52×C8C23semidirect
54. #6029C23semidirect
67. C2×D40D5semidirect
68. C2×#677D5semidirect
107. C2×D20C2×D5semidirect
112. C2×Dic10C2×D5semidirect
116. C2×C5×C8C2×D5semidirect
117. C2×C5×C8C2×D5semidirect
125. D40C2×D5semidirect
126. #677C2×D5semidirect
131. C2×C52AGL(1,ℤ/8)semidirect
176. D20C22×D5semidirect
185. Dic10C22×D5semidirect
187. C5×C8C22×D5semidirect
188. C5×C8C22×D5semidirect
200. C52C2×AGL(1,ℤ/8)semidirect
254. C2×C8D52semidirect
260. C2×C5#4057semidirect
261. C2×C5#4060semidirect
293. C8C2×D52semidirect
295. C5C2×#4057semidirect
296. C5C2×#4060semidirect
315. C2#11716916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#878772
C2×#878712
C2×#877992
C2×C5×D402
C2×C5×#6772
C2×#877722
C2×#877702
#117169162
C2×#40605
C2×#40575