Finite Group Databasebeta

← C2×#11716941 #13105914

Subgroups — C2×#11716941 #13105914

Order 1600.

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

normal subgroups only

This group has 664 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×C5×#709C211yesmaximal, characteristic, complemented
2C22×C5×D20C211yesmaximal, characteristic, complemented
3C2×#87804C211yesmaximal, characteristic, complemented
4C2×#87772C212yesmaximal, complemented
5#11716941C218yesmaximal, direct product factor
6C2×#87770C212yesmaximal, complemented
7C23×C5×D5—21no—
8C22×C4×C52C2211yescharacteristic
9C2×C5×C22⋊D5—21no—
10C5×#709—22no—
11C5×#709C2214yescomplemented
12C2×C5×D20—22no—
13C2×C5×D20—21no—
14C2×C5×D20C2212yes—
15C2×C5×D20C2214yescomplemented
16C2×C5×D20C2211yescharacteristic, complemented
17C2×C4×C5×D5—21no—
18C2×C5×Dic10C2211yescharacteristic, complemented
19#87804C2214yescomplemented
20C2×#6029C2212yescomplemented
21#87772C2218yescomplemented
22#87770C2218yescomplemented
23C2×#4095—51nomaximal
24C2×#4078—51nomaximal
25C23×C52D411yescharacteristic
26C22×C5×D5—21no—
27C22×C5×D5—22no—
28C22×C5×D5—24no—
29C22×C5×D5—44no—
30C2×C4×C52D411yescharacteristic
31C2×C4×C52D412yes—
32C2×C4×C52C2311yescharacteristic
33C2×C4×C52C2312yes—
34C5×C22⋊D5—24no—
35C2×C5×Dic5—21no—
36C5×D20—22no—
37C5×D20—24no—
38C5×D20—21no—
39C5×D20C2314yes—
40C5×D20C2312yescomplemented
41C4×C5×D5—24no—
42C5×Dic10—21no—
43C5×Dic10C2312yescomplemented
44#6029C2314yescomplemented
45C2×C5×#41—51no—
46C22×C5×D4—51no—
47C2×#709D511yescharacteristic, complemented
48C22×D20D511yescharacteristic, complemented
49#4095—58no—
50C2×#688—52no—
51C2×#687—52no—
52#4078—58no—
53C2×#686—52no—
54C2×#686—52no—
55C2×#681—51no—
56C2×#681—51no—
57C2×#681—102no—
58C22×C52—22no—
59C22×C52C2×D412yes—
60C22×C52C2×D411yescharacteristic
61C2×C5×D5—22no—
62C2×C5×D5—24no—
63C2×C5×D5—48no—
64C2×C5×D5—42no—
65C2×C5×D5—44no—
66C4×C52C2×D411yescharacteristic
67C4×C52C2×D412yes—
68C4×C52C2411yescharacteristic
69C5×Dic5—22no—
70C24×C5—101no—
71C23×D5—21no—
72C5×#41—54no—
73C5×#41—102no—
74C2×C5×Q8—51no—
75C2×C5×D4—52no—
76C2×C5×D4—54no—
77C2×C5×D4—51no—
78C2×C5×D4—101no—
79C2×C5×D4—101no—
80C2×C5×D4—102no—
81C22×C4×C5—22no—
82C22×C4×C5—101no—
83C22×C4×C5C2×D511yescharacteristic
84C22×C4×C5C2×D511yescharacteristic
85C2×C22⋊D5—21no—
86#709—22no—
87#709C2×D514yescomplemented
88C2×D20—22no—
89C2×D20—21no—
90C2×D20C2×D512yes—
91C2×D20C2×D514yescomplemented
92C2×D20C2×D511yescharacteristic, complemented
93C2×C4×D5—21no—
94C2×Dic10C2×D511yescharacteristic, complemented
95#688—58no—
96#687—58no—
97#686—58no—
98#686—58no—
99#681—54no—
100#681—54no—
101#681—108no—
102C2×#168—52no—
103C2×#168—52no—
104C2×#168—104no—
105C2×AGL(1,ℤ/8)—251noSylow
106C2×C52—22no—
107C2×C52AGL(1,ℤ/8)12yescomplemented
108C2×C52C22×D411yescharacteristic
109C5×D5—42no—
110C5×D5—44no—
111C23×C5—22no—
112C23×C5—101no—
113C23×C5—104no—
114C23×C5—102no—
115C23×C5—204no—
116C23×C5C22⋊D511yescharacteristic
117C23×C5C22⋊D511yescharacteristic
118C22×D5—21no—
119C22×D5—22no—
120C22×D5—24no—
121C22×D5—44no—
122C5×Q8—52no—
123C5×Q8—101no—
124C5×D4—54no—
125C5×D4—52no—
126C5×D4—104no—
127C5×D4—102no—
128C5×D4—104no—
129C5×D4—101no—
130C2×C4×C5—24no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—24no—
134C2×C4×C5—101no—
135C2×C4×C5—104no—
136C2×C4×C5C22⋊D511yescharacteristic
137C2×C4×C5C22⋊D511yescharacteristic
138C2×C4×C5C22⋊D512yes—
139C2×C4×C5C22⋊D512yes—
140C2×C4×C5C22×D511yescharacteristic
141C2×C4×C5C22×D511yescharacteristic
142C2×C4×C5C22×D512yes—
143C2×C4×C5C22×D512yes—
144C22⋊D5—24no—
145C2×Dic5—21no—
146D20—22no—
147D20—24no—
148D20—21no—
149D20C22×D514yes—
150D20C22×D512yescomplemented
151C4×D5—24no—
152Dic10—21no—
153Dic10C22×D512yescomplemented
154#168—54no—
155#168—54no—
156#168—108no—
157C2×#41—51no—
158C22×D4—51no—
159AGL(1,ℤ/8)—258no—
160C2×SD16—252no—
161C2×D8—252no—
162C2×M16—251no—
163C52C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
164C22×C5—22no—
165C22×C5—22no—
166C22×C5—24no—
167C22×C5—22no—
168C22×C5—44no—
169C22×C5—102no—
170C22×C5—104no—
171C22×C5—208no—
172C22×C5—202no—
173C22×C5—204no—
174C22×C5C2×C22⋊D512yes—
175C22×C5C2×C22⋊D512yes—
176C22×C5C2×C22⋊D511yescharacteristic
177C22×C5C2×C22⋊D511yescharacteristic
178C2×D5—22no—
179C2×D5—24no—
180C2×D5—48no—
181C2×D5—44no—
182C2×D5—42no—
183C4×C5—24no—
184C4×C5—22no—
185C4×C5—22no—
186C4×C5—102no—
187C4×C5C2×C22⋊D511yescharacteristic
188C4×C5C2×C22⋊D512yes—
189C4×C5C2×C22⋊D512yes—
190C4×C5C2×C22⋊D511yescharacteristic
191C4×C5C23×D511yescharacteristic
192C4×C5C23×D511yescharacteristic
193Dic5—22no—
194C24—101no—
195#41—54no—
196#41—102no—
197C2×Q8—51no—
198C2×D4—51no—
199C2×D4—52no—
200C2×D4—54no—
201C2×D4—101no—
202C2×D4—102no—
203C2×D4—101no—
204C22×C4—101no—
205C22×C4D5211yescharacteristic
206SD16—258no—
207D8—258no—
208M16—254no—
209C2×C8—252no—
210C2×C5—22no—
211C2×C5—22no—
212C2×C5—24no—
213C2×C5—22no—
214C2×C5—44no—
215C2×C5—202no—
216C2×C5—204no—
217C2×C5#407812yescomplemented
218C2×C5#409512yescomplemented
219C2×C5C22×C22⋊D511yescharacteristic
220C2×C5C22×C22⋊D511yescharacteristic
221D5—44no—
222D5—42no—
223C23—104no—
224C23—101no—
225C23—102no—
226C23—204no—
227C23#603711yescharacteristic
228Q8—52no—
229Q8—101no—
230D4—52no—
231D4—54no—
232D4—104no—
233D4—104no—
234D4—102no—
235D4—101no—
236C2×C4—101no—
237C2×C4—104no—
238C2×C4#603712yes—
239C2×C4#603711yescharacteristic
240C2×C4C2×D5212yes—
241C2×C4C2×D5211yescharacteristic
242C8—254no—
243C5—22no—
244C5C2×#407811yescharacteristic, complemented
245C5C2×#409511yescharacteristic, complemented
246C22—22no—
247C22—102no—
248C22—104no—
249C22—208no—
250C22—204no—
251C22—202no—
252C22C2×#603712yes—
253C22C2×#603711yescharacteristic
254C4—102no—
255C4C2×#603712yes—
256C4C2×#603711yescharacteristic
257C4C22×D5211yescharacteristic
258C2—22no—
259C2—204no—
260C2—202no—
261C2#1171694112yesdirect product factor
262C2C22×#603711yescharacteristic
subgroupG/H
centerC22C2×#6037
commutator, cosocleC4×C52C24
FrattiniC4C22×D52
FittingC22×C4×C52C22
socleC22×C52C2×D4
hypercenterC22×C4D52
hypocenterC52C2×AGL(1,ℤ/8)
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#709C2semidirect
2. C22×C5×D20C2semidirect
3. C2×#87804C2semidirect
4. C2×#87772C2semidirect
5. #11716941C2direct (N × H)
6. C2×#87770C2semidirect
11. C5×#709C22semidirect
15. C2×C5×D20C22semidirect
16. C2×C5×D20C22semidirect
18. C2×C5×Dic10C22semidirect
19. #87804C22semidirect
20. C2×#6029C22semidirect
21. #87772C22semidirect
22. #87770C22semidirect
40. C5×D20C23semidirect
43. C5×Dic10C23semidirect
44. #6029C23semidirect
47. C2×#709D5semidirect
48. C22×D20D5semidirect
87. #709C2×D5semidirect
91. C2×D20C2×D5semidirect
92. C2×D20C2×D5semidirect
94. C2×Dic10C2×D5semidirect
107. C2×C52AGL(1,ℤ/8)semidirect
150. D20C22×D5semidirect
153. Dic10C22×D5semidirect
163. C52C2×AGL(1,ℤ/8)semidirect
217. C2×C5#4078semidirect
218. C2×C5#4095semidirect
244. C5C2×#4078semidirect
245. C5C2×#4095semidirect
261. C2#11716941direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#7092
C22×C5×D202
C2×#878042
C2×#877722
#117169412
C2×#877702
C2×#40955
C2×#40785