Finite Group Databasebeta

← C2×#11717057 #13106454

Subgroups — C2×#11717057 #13106454

Order 1600.

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

normal subgroups only

This group has 664 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×C5×#713C211yesmaximal, characteristic, complemented
2C2×D5×D20C211yesmaximal, characteristic, complemented
3C2×#87811C211yesmaximal, characteristic, complemented
4C2×C5×#688C211yesmaximal, characteristic, complemented
5#11717057C218yesmaximal, direct product factor
6C2×#87771C211yesmaximal, characteristic, complemented
7C2×#87770C211yesmaximal, characteristic, complemented
8C2×#87768C211yesmaximal, characteristic, complemented
9C22×D52—21no—
10C2×C52×Q8C2211yescharacteristic, complemented
11C2×C4⋊(C5⋊D5)C2211yescharacteristic
12C5×#713—22no—
13C5×#713C2214yescomplemented
14C2×C5×D20—21no—
15C2×C5×D20C2211yescharacteristic
16C2×C5×D20C2211yescharacteristic, complemented
17C2×C4×C5×D5—21no—
18C2×C4×C5×D5C2211yescharacteristic
19C2×#6038—21no—
20D5×D20—22no—
21D5×D20C2214yescomplemented
22#87811C2214yescomplemented
23C2×#6029C2211yescharacteristic, complemented
24C5×#688C2214yescomplemented
25C2×C5×#168C2211yescharacteristic, complemented
26#87771C2214yescomplemented
27#87770C2214yescomplemented
28#87768C2214yescomplemented
29C2×#4095—51nomaximal
30C2×#4060—51nomaximal
31C22×C5⋊D5—21no—
32C22×C5×D5—21no—
33C22×C5×D5—21no—
34C22×C5×D5D411yescharacteristic
35C2×D52—24no—
36C2×D52—44no—
37C52×Q8—21no—
38C52×Q8C2312yescomplemented
39C2×C4×C52—21no—
40C2×C4×C52C2311yescharacteristic
41C4⋊(C5⋊D5)—21no—
42C4⋊(C5⋊D5)C2312yes—
43C2×C5×Dic5D411yescharacteristic
44C5×D20—24no—
45C5×D20—21no—
46C5×D20—21no—
47C5×D20C2312yes—
48C5×D20C2312yescomplemented
49C4×C5×D5—24no—
50C4×C5×D5D412yes—
51C4×C5×D5C2312yes—
52#6038—24no—
53#6029C2312yescomplemented
54C5×#168C2312yescomplemented
55C2×C5×#41—51no—
56C2×#713D511yescharacteristic, complemented
57C2×D4×D5—51no—
58C22×D20—51no—
59C2×C5×SD16—51no—
60#4095—58no—
61C2×#688—51no—
62C2×#688—51no—
63C2×#688—102no—
64C2×#688D511yescharacteristic, complemented
65C2×#686—51no—
66C2×#686—51no—
67C2×#686—51no—
68C2×#681—51no—
69#4060—58no—
70C2×D40—51no—
71C2×#676—51no—
72C22×C52C2×D411yescharacteristic
73C2×C5⋊D5—22no—
74C2×C5⋊D5—42no—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—22no—
78C2×C5×D5—42no—
79C2×C5×D5—42no—
80C2×C5×D5C2×D412yes—
81D52—48no—
82C4×C52—22no—
83C4×C52C2×D411yescharacteristic
84C4×C52C2411yescharacteristic
85C5×Dic5C2×D412yes—
86C23×D5—101no—
87C23×D5—101no—
88C5×#41—54no—
89C5×#41—102no—
90C2×C5×Q8—22no—
91C2×C5×Q8C2×D511yescharacteristic, complemented
92C2×C5×Q8C2×D511yescharacteristic, complemented
93C2×C5×D4—51no—
94C2×C5×D4—51no—
95C2×C5×D4—101no—
96C22×C4×C5—51no—
97C22×C4×C5—101no—
98C2×C22⋊D5—101no—
99#713—22no—
100#713C2×D514yescomplemented
101D4×D5—54no—
102D4×D5—102no—
103C2×D20—21no—
104C2×D20—51no—
105C2×D20—51no—
106C2×D20—54no—
107C2×D20—102no—
108C2×D20—102no—
109C2×D20—101no—
110C2×D20C2×D511yescharacteristic
111C2×D20C2×D511yescharacteristic, complemented
112C2×C4×D5—21no—
113C2×C4×D5C2×D511yescharacteristic
114C5×SD16—54no—
115C2×C5×C8—51no—
116#688—54no—
117#688—54no—
118#688—108no—
119#688C2×D514yescomplemented
120#686—54no—
121#686—54no—
122#686—54no—
123#681—54no—
124C2×#168—51no—
125C2×#168—51no—
126C2×#168—102no—
127C2×#168C2×D511yescharacteristic, complemented
128D40—54no—
129#676—54no—
130C2×AGL(1,ℤ/8)—251noSylow
131C2×C52AGL(1,ℤ/8)12yescomplemented
132C2×C52C22×D411yescharacteristic
133C5⋊D5—42no—
134C5×D5—22no—
135C5×D5—42no—
136C5×D5—42no—
137C23×C5—51no—
138C23×C5—101no—
139C23×C5—101no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5—104no—
143C22×D5—101no—
144C22×D5—101no—
145C22×D5—104no—
146C22×D5—202no—
147C22×D5—204no—
148C22×D5—204no—
149C22×D5C22⋊D511yescharacteristic
150C5×Q8—21no—
151C5×Q8—21no—
152C5×Q8—24no—
153C5×Q8—42no—
154C5×Q8C22×D512yescomplemented
155C5×Q8C22×D512yescomplemented
156C5×D4—52no—
157C5×D4—52no—
158C5×D4—104no—
159C5×D4—101no—
160C5×D4—101no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—22no—
164C2×C4×C5—42no—
165C2×C4×C5—51no—
166C2×C4×C5—52no—
167C2×C4×C5—52no—
168C2×C4×C5—104no—
169C2×C4×C5C22×D511yescharacteristic
170C2×C4×C5C22×D511yescharacteristic
171C22⋊D5—104no—
172C2×Dic5C22⋊D511yescharacteristic
173D20—24no—
174D20—21no—
175D20—21no—
176D20—52no—
177D20—52no—
178D20—104no—
179D20—104no—
180D20—101no—
181D20—101no—
182D20—202no—
183D20C22×D512yes—
184D20C22×D512yescomplemented
185C4×D5—24no—
186C4×D5C22⋊D512yes—
187C4×D5C22×D512yes—
188C5×C8—52no—
189#168—52no—
190#168—52no—
191#168—104no—
192#168C22×D512yescomplemented
193C2×#41—51no—
194C22×D4—251no—
195AGL(1,ℤ/8)—258no—
196C2×SD16—51no—
197C2×SD16—251no—
198C2×D8—251no—
199C2×D8—251no—
200C2×M16—251no—
201C52C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
202C22×C5—22no—
203C22×C5—52no—
204C22×C5—102no—
205C22×C5—102no—
206C22×C5—102no—
207C22×C5—202no—
208C22×C5—202no—
209C22×C5D4×D511yescharacteristic
210C22×C5C2×C22⋊D511yescharacteristic
211C2×D5—22no—
212C2×D5—22no—
213C2×D5—22no—
214C2×D5—42no—
215C2×D5—42no—
216C2×D5—102no—
217C2×D5—102no—
218C2×D5—204no—
219C2×D5—208no—
220C2×D5—208no—
221C2×D5—202no—
222C2×D5—202no—
223C2×D5—404no—
224C2×D5C2×C22⋊D512yes—
225C4×C5—22no—
226C4×C5—22no—
227C4×C5—22no—
228C4×C5—22no—
229C4×C5—44no—
230C4×C5—52no—
231C4×C5D4×D511yescharacteristic
232C4×C5C2×C22⋊D511yescharacteristic
233C4×C5C23×D511yescharacteristic
234C4×C5C23×D511yescharacteristic
235Dic5C2×C22⋊D512yes—
236C24—501no—
237#41—54no—
238#41—102no—
239C2×Q8D5211yescharacteristic, complemented
240C2×D4—51no—
241C2×D4—51no—
242C2×D4—101no—
243C2×D4—254no—
244C2×D4—251no—
245C2×D4—502no—
246C2×D4—501no—
247C22×C4—51no—
248C22×C4—101no—
249SD16—54no—
250SD16—254no—
251D8—254no—
252D8—254no—
253M16—254no—
254C2×C8—51no—
255C2×C8—251no—
256C2×C5—22no—
257C2×C5—24no—
258C2×C5—102no—
259C2×C5—202no—
260C2×C5—202no—
261C2×C5#406012yescomplemented
262C2×C5#409512yescomplemented
263C2×C5C2×D4×D511yescharacteristic
264C2×C5C22×C22⋊D511yescharacteristic
265D5—22no—
266D5—42no—
267D5—42no—
268D5—202no—
269D5—202no—
270D5—404no—
271C23—51no—
272C23—101no—
273C23—101no—
274C23—504no—
275C23—501no—
276C23—1004no—
277Q8—21no—
278Q8C2×D5212yescomplemented
279D4—52no—
280D4—52no—
281D4—104no—
282D4—101no—
283D4—101no—
284D4—252no—
285D4—504no—
286D4—501no—
287C2×C4—21no—
288C2×C4—51no—
289C2×C4—52no—
290C2×C4—52no—
291C2×C4—104no—
292C2×C4C2×D5211yescharacteristic
293C8—52no—
294C8—252no—
295C5—22no—
296C5C2×#406011yescharacteristic, complemented
297C5C2×#409511yescharacteristic, complemented
298C22—52no—
299C22—102no—
300C22—102no—
301C22—102no—
302C22—202no—
303C22—202no—
304C22—502no—
305C22—1008no—
306C22—1002no—
307C22D5×C22⋊D511yescharacteristic
308C4—22no—
309C4—52no—
310C4D5×C22⋊D511yescharacteristic
311C4C22×D5211yescharacteristic
312C2—102no—
313C2—202no—
314C2—202no—
315C2—1002no—
316C2#1171705712yesdirect product factor
317C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
centerC22D5×C22⋊D5
commutator, cosocleC4×C52C24
FrattiniC4C22×D52
FittingC2×C52×Q8C22
socleC22×C52C2×D4
hypercenterC2×Q8D52
hypocenterC52C2×AGL(1,ℤ/8)
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×#713C2semidirect
2. C2×D5×D20C2semidirect
3. C2×#87811C2semidirect
4. C2×C5×#688C2semidirect
5. #11717057C2direct (N × H)
6. C2×#87771C2semidirect
7. C2×#87770C2semidirect
8. C2×#87768C2semidirect
10. C2×C52×Q8C22semidirect
13. C5×#713C22semidirect
16. C2×C5×D20C22semidirect
21. D5×D20C22semidirect
22. #87811C22semidirect
23. C2×#6029C22semidirect
24. C5×#688C22semidirect
25. C2×C5×#168C22semidirect
26. #87771C22semidirect
27. #87770C22semidirect
28. #87768C22semidirect
38. C52×Q8C23semidirect
48. C5×D20C23semidirect
53. #6029C23semidirect
54. C5×#168C23semidirect
56. C2×#713D5semidirect
64. C2×#688D5semidirect
91. C2×C5×Q8C2×D5semidirect
92. C2×C5×Q8C2×D5semidirect
100. #713C2×D5semidirect
111. C2×D20C2×D5semidirect
119. #688C2×D5semidirect
127. C2×#168C2×D5semidirect
131. C2×C52AGL(1,ℤ/8)semidirect
154. C5×Q8C22×D5semidirect
155. C5×Q8C22×D5semidirect
184. D20C22×D5semidirect
192. #168C22×D5semidirect
201. C52C2×AGL(1,ℤ/8)semidirect
239. C2×Q8D52semidirect
261. C2×C5#4060semidirect
262. C2×C5#4095semidirect
278. Q8C2×D52semidirect
296. C5C2×#4060semidirect
297. C5C2×#4095semidirect
316. C2#11717057direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#7132
C2×D5×D202
C2×#878112
C2×C5×#6882
#117170572
C2×#877712
C2×#877702
C2×#877682
C2×#40955
C2×#40605