Finite Group Databasebeta

← #13105769

Subgroups — #13105769

Order 1600.

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

normal subgroups only

This group has 598 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
1#11717472C212yesmaximal, complemented
2#11717224C211yesmaximal, characteristic, complemented
3C2×C5×D40C211yesmaximal, characteristic, complemented
4C5×#4047C211yesmaximal, characteristic, complemented
5#11716948C212yesmaximal, complemented
6#11716925C212yesmaximal, complemented
7#11716913C214yesmaximal, complemented
8D5×D40C212yesmaximal, complemented
9#87900C2212yescomplemented
10C5×#709C2211yescharacteristic
11C2×C5×D20C2212yescomplemented
12D5×C22⋊D5—24no—
13#87877—22no—
14D5×D20C2214yescomplemented
15#87871—22no—
16#87870C2214yescomplemented
17C2×C52×C8C2211yescharacteristic, complemented
18#87802C2211yescharacteristic, complemented
19#87801C2211yescharacteristic, complemented
20#87800C2212yescomplemented
21C5×#681C2211yescharacteristic, complemented
22C5×D40C2212yescomplemented
23C5×D40C2212yescomplemented
24C5×#676C2212yescomplemented
25C5×C8×D5C2212yescomplemented
26#87773C2214yescomplemented
27#87771C2214yescomplemented
28#66091—51nomaximal
29#66086—51nomaximal
30C22×C5×D5—22no—
31C2×D52—24no—
32C2×C4×C52C2311yescharacteristic
33C22⋊(C5⋊D5)—22no—
34C4⋊(C5⋊D5)C2312yes—
35C4×C5⋊D5—22no—
36#6045C2312yes—
37C5×C22⋊D5D412yes—
38C5×D20—22no—
39C5×D20D411yescharacteristic
40C5×D20C2314yes—
41C4×C5×D5C2312yes—
42C5×Dic10D411yescharacteristic
43#6038—24no—
44#6037—24no—
45D5×Dic5—24no—
46C52×C8C2311yescharacteristic
47C52×C8C2311yescharacteristic
48C5×#168C2312yes—
49#4149—52no—
50#4144—52no—
51C2×C5×D8—51no—
52C5×#131—51no—
53#4078—52no—
54#4058—52no—
55#4057—54no—
56D5×D8—52no—
57#4054—52no—
58#4054—54no—
59#4050—51no—
60#4050—51no—
61#4050—52no—
62#4050—102no—
63C2×D40—52no—
64C2×D40D511yescharacteristic, complemented
65#4047D511yescharacteristic, complemented
66C22×C52C2×D411yescharacteristic
67C2×C5⋊D5—22no—
68C2×C5×D5—24no—
69C2×C5×D5—82no—
70C2×C5×D5C2×D412yes—
71D52—84no—
72C4×C52C2×D411yescharacteristic
73C4×C52C2411yescharacteristic
74#1039—22no—
75C5×Dic5C2×D412yes—
76C5×#41—51no—
77C2×C5×D4—52no—
78C2×C22⋊D5—104no—
79#713—102no—
80#711—54no—
81#711—102no—
82D4×D5—54no—
83D4×D5—102no—
84D4×D5—104no—
85D4×D5—102no—
86#709—52no—
87#709—52no—
88#709—54no—
89#709—104no—
90#709C2×D511yescharacteristic
91C2×D20—54no—
92C2×D20C2×D512yescomplemented
93C5×D8—52no—
94C5×D8—52no—
95C5×M16—51no—
96C5×M16—52no—
97C2×C5×C8—22no—
98C2×C5×C8—52no—
99C2×C5×C8C2×D511yescharacteristic, complemented
100C2×C5×C8C2×D511yescharacteristic, complemented
101#687—54no—
102#686—54no—
103#681C2×D511yescharacteristic, complemented
104Dic20—51no—
105Dic20—51no—
106Dic20—102no—
107D40—54no—
108D40—51no—
109D40—51no—
110D40—102no—
111D40C2×D512yescomplemented
112D40C2×D512yescomplemented
113#677—54no—
114#677—52no—
115#677—52no—
116#677—104no—
117#676C2×D512yescomplemented
118C8×D5C2×D512yescomplemented
119#576—251noSylow
120C2×C52—21no—
121C2×C52C22×D411yescharacteristic
122C5⋊D5—42no—
123C5×D5—22no—
124C5×D5—44no—
125C23×C5—102no—
126C22×D5—22no—
127C22×D5—104no—
128C22×D5—104no—
129C5×Q8—51no—
130C5×D4—52no—
131C5×D4—51no—
132C5×D4—54no—
133C5×D4—102no—
134C2×C4×C5—22no—
135C2×C4×C5—52no—
136C2×C4×C5C22×D511yescharacteristic
137C2×C4×C5C22×D511yescharacteristic
138C22⋊D5—104no—
139C22⋊D5—102no—
140C22⋊D5—102no—
141C22⋊D5—104no—
142C22⋊D5—104no—
143C22⋊D5—204no—
144C22⋊D5D2012yes—
145C2×Dic5—104no—
146D20—22no—
147D20—52no—
148D20—52no—
149D20—104no—
150D20—104no—
151D20D2011yescharacteristic
152D20C22×D514yes—
153C4×D5—104no—
154C4×D5—102no—
155C4×D5—102no—
156C4×D5—204no—
157C4×D5C22×D512yes—
158Dic10—52no—
159Dic10—52no—
160Dic10—104no—
161Dic10D2011yescharacteristic
162C5×C8—22no—
163C5×C8—22no—
164C5×C8—52no—
165C5×C8C22×D511yescharacteristic
166C5×C8C22×D511yescharacteristic
167C5×C8C22×D511yescharacteristic
168C5×C8C22×D511yescharacteristic
169#168C22×D512yes—
170H(5,𝔽2)—252no—
171AGL(1,ℤ/8)—254no—
172AGL(1,ℤ/8)—252no—
173#135—252no—
174#135—251no—
175C2×D8—51no—
176C2×D8—252no—
177#131—51no—
178C52#57611yescharacteristic, complemented, Sylow
179C22×C5—22no—
180C22×C5—52no—
181C22×C5—104no—
182C22×C5—402no—
183C22×C5C2×D2011yescharacteristic
184C22×C5D4×D511yescharacteristic
185C2×D5—24no—
186C2×D5—82no—
187C2×D5—102no—
188C2×D5—102no—
189C2×D5—204no—
190C2×D5—404no—
191C2×D5—404no—
192C2×D5C2×D2012yes—
193C4×C5—22no—
194C4×C5—22no—
195C4×C5—52no—
196C4×C5C2×D2011yescharacteristic
197C4×C5D4×D511yescharacteristic
198C4×C5C23×D511yescharacteristic
199C4×C5C23×D511yescharacteristic
200Dic5—102no—
201Dic5—102no—
202Dic5—204no—
203Dic5C2×D2012yes—
204#41—51no—
205#41—254no—
206#41—252no—
207#41—502no—
208C2×D4—52no—
209C2×D4—254no—
210C2×D4—502no—
211C2×D4—504no—
212Q16—251no—
213SD16—254no—
214SD16—252no—
215D8—52no—
216D8—52no—
217D8—254no—
218D8—251no—
219M16—51no—
220M16—52no—
221C2×C8—52no—
222C2×C8D5211yescharacteristic, complemented
223C2×C5—21no—
224C2×C5—21no—
225C2×C5—22no—
226C2×C5—42no—
227C2×C5—102no—
228C2×C5—204no—
229C2×C5C22×D2011yescharacteristic
230C2×C5C2×D4×D511yescharacteristic
231D5—22no—
232D5—44no—
233D5—202no—
234D5—202no—
235D5—404no—
236C23—102no—
237C23—504no—
238Q8—51no—
239Q8—252no—
240D4—52no—
241D4—51no—
242D4—54no—
243D4—102no—
244D4—252no—
245D4—504no—
246D4—504no—
247D4—502no—
248C2×C4—52no—
249C2×C4—504no—
250C2×C4—502no—
251C2×C4C2×D5211yescharacteristic
252C8—52no—
253C8C2×D5211yescharacteristic
254C8C2×D5211yescharacteristic
255C5—22no—
256C5#6608611yescharacteristic, complemented
257C5#6609111yescharacteristic, complemented
258C22—52no—
259C22—104no—
260C22—402no—
261C22—502no—
262C22—2004no—
263C22D5×D2011yescharacteristic
264C4—52no—
265C4—502no—
266C4D5×D2011yescharacteristic
267C4C22×D5211yescharacteristic
268C2—21no—
269C2—102no—
270C2—204no—
271C2—1002no—
272C2C2×D5×D2011yescharacteristic
subgroupG/H
centerC2C2×D5×D20
commutator, cosocleC4×C52C24
FrattiniC4C22×D52
FittingC2×C52×C8C22
socleC2×C52C22×D4
hypercenterC2×C8D52
hypocenterC52#576
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11717472C2semidirect
2. #11717224C2semidirect
3. C2×C5×D40C2semidirect
4. C5×#4047C2semidirect
5. #11716948C2semidirect
6. #11716925C2semidirect
7. #11716913C2semidirect
8. D5×D40C2semidirect
9. #87900C22semidirect
11. C2×C5×D20C22semidirect
14. D5×D20C22semidirect
16. #87870C22semidirect
17. C2×C52×C8C22semidirect
18. #87802C22semidirect
19. #87801C22semidirect
20. #87800C22semidirect
21. C5×#681C22semidirect
22. C5×D40C22semidirect
23. C5×D40C22semidirect
24. C5×#676C22semidirect
25. C5×C8×D5C22semidirect
26. #87773C22semidirect
27. #87771C22semidirect
64. C2×D40D5semidirect
65. #4047D5semidirect
92. C2×D20C2×D5semidirect
99. C2×C5×C8C2×D5semidirect
100. C2×C5×C8C2×D5semidirect
103. #681C2×D5semidirect
111. D40C2×D5semidirect
112. D40C2×D5semidirect
117. #676C2×D5semidirect
118. C8×D5C2×D5semidirect
178. C52#576semidirect
222. C2×C8D52semidirect
256. C5#66086semidirect
257. C5#66091semidirect

Maximal subgroups

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

subgroupindex
#117174722
#117172242
C2×C5×D402
C5×#40472
#117169482
#117169252
#117169132
D5×D402
#660915
#660865