Finite Group Databasebeta

← #13105771

Subgroups — #13105771

Order 1600.

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

normal subgroups only

This group has 612 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#11717473C212yesmaximal, complemented
2C2×#87801C211yesmaximal, characteristic, complemented
3C5×#4050C211yesmaximal, characteristic, complemented
4C5×#4047C211yesmaximal, characteristic, complemented
5#11716943C212yesmaximal, complemented
6#11716926C212yesmaximal, complemented
7#11716912C214yesmaximal, complemented
8D5×D40C212yesmaximal, complemented
9C2×C4⋊(C5⋊D5)C2212yescomplemented
10C5×#709C2211yescharacteristic
11C5×#709C2212yescomplemented
12#87886—24no—
13D5×D20—22no—
14D5×D20C2214yescomplemented
15#87874—22no—
16#87874C2214yescomplemented
17C2×C52×C8C2211yescharacteristic, complemented
18#87801C2212yescomplemented
19#87801C2212yescomplemented
20C5×#681C2211yescharacteristic, complemented
21C5×Dic20C2211yescharacteristic, complemented
22C5×D40C2211yescharacteristic, complemented
23C5×#677C2212yescomplemented
24C5×#676C2212yescomplemented
25C5×C8×D5C2212yescomplemented
26#87774C2214yescomplemented
27#87771C2214yescomplemented
28#66103—51nomaximal
29#66086—51nomaximal
30C22×C5⋊D5—22no—
31C2×D52—24no—
32C2×C4×C52C2311yescharacteristic
33C4⋊(C5⋊D5)—22no—
34C4⋊(C5⋊D5)C2314yes—
35C5×C22⋊D5—22no—
36C5×C22⋊D5D412yes—
37C5×D20D411yescharacteristic
38C5×D20C2312yes—
39C4×C5×D5—22no—
40C4×C5×D5C2312yes—
41C5×Dic10D411yescharacteristic
42C5×Dic10C2312yes—
43#6038—24no—
44#6038—24no—
45#6036—24no—
46C52×C8C2311yescharacteristic
47C52×C8C2311yescharacteristic
48C5×#168C2312yes—
49#4149—52no—
50#4149—52no—
51C5×#135—51no—
52C5×#131—51no—
53#4095—52no—
54#4065—52no—
55#4060—54no—
56D5×D8—52no—
57#4054—52no—
58#4054—54no—
59#4050—52no—
60#4050D511yescharacteristic, complemented
61C2×D40—51no—
62C2×D40—51no—
63C2×D40—52no—
64C2×D40—102no—
65#4047D511yescharacteristic, complemented
66C22×C52C2×D411yescharacteristic
67C2×C5⋊D5—24no—
68C2×C5⋊D5—82no—
69C2×C5×D5—22no—
70C2×C5×D5C2×D412yes—
71D52—84no—
72C4×C52C2×D411yescharacteristic
73C4×C52C2411yescharacteristic
74C5×Dic5—22no—
75C5×Dic5C2×D412yes—
76C5×#41—51no—
77C5×#41—52no—
78#713—54no—
79#713—102no—
80D4×D5—54no—
81D4×D5—102no—
82D4×D5—104no—
83D4×D5—104no—
84#709—54no—
85#709—102no—
86#709C2×D511yescharacteristic
87#709C2×D512yescomplemented
88C2×D20—52no—
89C2×D20—52no—
90C2×D20—54no—
91C2×D20—104no—
92C2×D20—102no—
93C5×Q16—51no—
94C5×SD16—52no—
95C5×D8—51no—
96C5×M16—51no—
97C5×M16—52no—
98C2×C5×C8—22no—
99C2×C5×C8—52no—
100C2×C5×C8C2×D511yescharacteristic, complemented
101C2×C5×C8C2×D511yescharacteristic, complemented
102#688—54no—
103#686—54no—
104#681C2×D511yescharacteristic, complemented
105Dic20C2×D511yescharacteristic, complemented
106D40—54no—
107D40—52no—
108D40—52no—
109D40—52no—
110D40—52no—
111D40—104no—
112D40—104no—
113D40C2×D511yescharacteristic, complemented
114#677—54no—
115#677C2×D512yescomplemented
116#676C2×D512yescomplemented
117C8×D5C2×D512yescomplemented
118#576—251noSylow
119C2×C52—21no—
120C2×C52C22×D411yescharacteristic
121C5⋊D5—44no—
122C5×D5—22no—
123C5×D5—42no—
124C22×D5—104no—
125C22×D5—102no—
126C22×D5—102no—
127C22×D5—104no—
128C22×D5—204no—
129C5×Q8—51no—
130C5×Q8—52no—
131C5×D4—52no—
132C5×D4—51no—
133C5×D4—52no—
134C5×D4—102no—
135C2×C4×C5—22no—
136C2×C4×C5—52no—
137C2×C4×C5—102no—
138C2×C4×C5C22×D511yescharacteristic
139C2×C4×C5C22×D511yescharacteristic
140C22⋊D5—22no—
141C22⋊D5—104no—
142C22⋊D5—104no—
143C22⋊D5D2012yes—
144D20—54no—
145D20—54no—
146D20—108no—
147D20—104no—
148D20—104no—
149D20—102no—
150D20—102no—
151D20—204no—
152D20D2011yescharacteristic
153D20C22×D512yes—
154C4×D5—22no—
155C4×D5—104no—
156C4×D5—104no—
157C4×D5C22×D512yes—
158Dic10D2011yescharacteristic
159Dic10C22×D512yes—
160C5×C8—22no—
161C5×C8—22no—
162C5×C8—52no—
163C5×C8C22×D511yescharacteristic
164C5×C8C22×D511yescharacteristic
165C5×C8C22×D511yescharacteristic
166C5×C8C22×D511yescharacteristic
167#168C22×D512yes—
168H(5,𝔽2)—252no—
169AGL(1,ℤ/8)—254no—
170AGL(1,ℤ/8)—252no—
171#135—51no—
172#135—252no—
173C2×D8—252no—
174C2×D8—251no—
175#131—51no—
176C52#57611yescharacteristic, complemented, Sylow
177C22×C5—22no—
178C22×C5—52no—
179C22×C5—102no—
180C22×C5C2×D2011yescharacteristic
181C22×C5D4×D511yescharacteristic
182C2×D5—22no—
183C2×D5—104no—
184C2×D5—104no—
185C2×D5—208no—
186C2×D5—404no—
187C2×D5—404no—
188C2×D5—402no—
189C2×D5—402no—
190C2×D5—408no—
191C2×D5C2×D2012yes—
192C4×C5—22no—
193C4×C5—22no—
194C4×C5—52no—
195C4×C5—102no—
196C4×C5C2×D2011yescharacteristic
197C4×C5D4×D511yescharacteristic
198C4×C5C23×D511yescharacteristic
199C4×C5C23×D511yescharacteristic
200Dic5—22no—
201Dic5C2×D2012yes—
202#41—51no—
203#41—52no—
204#41—254no—
205#41—502no—
206C2×D4—254no—
207C2×D4—252no—
208C2×D4—502no—
209C2×D4—504no—
210Q16—51no—
211SD16—52no—
212SD16—254no—
213D8—51no—
214D8—254no—
215D8—252no—
216D8—252no—
217M16—51no—
218M16—52no—
219C2×C8—52no—
220C2×C8D5211yescharacteristic, complemented
221C2×C5—24no—
222C2×C5—21no—
223C2×C5—21no—
224C2×C5—22no—
225C2×C5—102no—
226C2×C5—202no—
227C2×C5C22×D2011yescharacteristic
228C2×C5C2×D4×D511yescharacteristic
229D5—22no—
230D5—42no—
231D5—204no—
232D5—204no—
233D5—408no—
234C23—504no—
235C23—502no—
236Q8—51no—
237Q8—52no—
238D4—52no—
239D4—51no—
240D4—52no—
241D4—102no—
242D4—254no—
243D4—504no—
244D4—504no—
245D4—502no—
246C2×C4—52no—
247C2×C4—102no—
248C2×C4—504no—
249C2×C4C2×D5211yescharacteristic
250C8—52no—
251C8C2×D5211yescharacteristic
252C8C2×D5211yescharacteristic
253C5—22no—
254C5#6608611yescharacteristic, complemented
255C5#6610311yescharacteristic, complemented
256C22—52no—
257C22—102no—
258C22—504no—
259C22—2002no—
260C22—2004no—
261C22D5×D2011yescharacteristic
262C4—52no—
263C4—102no—
264C4D5×D2011yescharacteristic
265C4C22×D5211yescharacteristic
266C2—21no—
267C2—102no—
268C2—202no—
269C2—1004no—
270C2C2×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. #11717473C2semidirect
2. C2×#87801C2semidirect
3. C5×#4050C2semidirect
4. C5×#4047C2semidirect
5. #11716943C2semidirect
6. #11716926C2semidirect
7. #11716912C2semidirect
8. D5×D40C2semidirect
9. C2×C4⋊(C5⋊D5)C22semidirect
11. C5×#709C22semidirect
14. D5×D20C22semidirect
16. #87874C22semidirect
17. C2×C52×C8C22semidirect
18. #87801C22semidirect
19. #87801C22semidirect
20. C5×#681C22semidirect
21. C5×Dic20C22semidirect
22. C5×D40C22semidirect
23. C5×#677C22semidirect
24. C5×#676C22semidirect
25. C5×C8×D5C22semidirect
26. #87774C22semidirect
27. #87771C22semidirect
60. #4050D5semidirect
65. #4047D5semidirect
87. #709C2×D5semidirect
100. C2×C5×C8C2×D5semidirect
101. C2×C5×C8C2×D5semidirect
104. #681C2×D5semidirect
105. Dic20C2×D5semidirect
113. D40C2×D5semidirect
115. #677C2×D5semidirect
116. #676C2×D5semidirect
117. C8×D5C2×D5semidirect
176. C52#576semidirect
220. C2×C8D52semidirect
254. C5#66086semidirect
255. C5#66103semidirect

Maximal subgroups

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

subgroupindex
#117174732
C2×#878012
C5×#40502
C5×#40472
#117169432
#117169262
#117169122
D5×D402
#661035
#660865