Finite Group Databasebeta

← #13105811

Subgroups — #13105811

Order 1600.

598 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 128 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#11717227C211yesmaximal, characteristic, complemented
3C5×#4054C212yesmaximal, complemented
4#11716940C211yesmaximal, characteristic, complemented
5C2×#87770C211yesmaximal, characteristic, complemented
6#11716927C212yesmaximal, complemented
7#11716916C214yesmaximal, complemented
8#11716915C212yesmaximal, complemented
9#87900C2211yescharacteristic
10C5×#709C2212yescomplemented
11C2×C5×D20C2212yescomplemented
12D5×C22⋊D5—24no—
13#87877C2214yescomplemented
14D5×D20—22no—
15#87871C2214yescomplemented
16#87870—22no—
17C52×M16C2211yescharacteristic, complemented
18C2×#6029C2211yescharacteristic, complemented
19#87799C2212yescomplemented
20C8×C5⋊D5C2212yescomplemented
21C5×D40C2214yescomplemented
22C5×#677C2214yescomplemented
23#87775C2211yescharacteristic, complemented
24#87772C2212yescomplemented
25#87770C2211yescharacteristic, complemented
26#87770C2212yescomplemented
27#87770C2212yescomplemented
28#66108—52nomaximal
29C22×C5×D5—22no—
30C2×D52—24no—
31C2×C4×C52C2311yescharacteristic
32C22⋊(C5⋊D5)D412yes—
33C4⋊(C5⋊D5)D411yescharacteristic
34C4×C5⋊D5C2312yes—
35#6045D411yescharacteristic
36C5×C22⋊D5—22no—
37C5×D20—22no—
38C5×D20C2314yes—
39C5×D20C2312yes—
40C4×C5×D5—22no—
41C5×Dic10C2312yes—
42#6038—24no—
43#6037—24no—
44D5×Dic5—24no—
45C52×C8C2312yes—
46#6029C2311yescharacteristic
47#6029C2311yescharacteristic
48#4149—52no—
49#4144—52no—
50C5×AGL(1,ℤ/8)—52no—
51#4096—52no—
52C2×#686—52no—
53#4062—54no—
54#4060—54no—
55#4057—54no—
56D5×D8—54no—
57#4054D512yescomplemented
58#4053—52no—
59#4053—102no—
60C22×C52C2×D411yescharacteristic
61C2×C5⋊D5C2×D412yes—
62C2×C5×D5—24no—
63C2×C5×D5—22no—
64C2×C5×D5—82no—
65D52—84no—
66C4×C52C2×D411yescharacteristic
67C4×C52C2411yescharacteristic
68#1039C2×D412yes—
69C5×Dic5—22no—
70C5×#41—52no—
71C2×C5×D4—52no—
72C2×C22⋊D5—104no—
73#713—54no—
74#711—54no—
75#711—102no—
76D4×D5—54no—
77D4×D5—54no—
78D4×D5—102no—
79D4×D5—104no—
80#709—52no—
81#709—102no—
82#709—102no—
83#709C2×D512yescomplemented
84C2×D20—102no—
85C2×D20C2×D512yescomplemented
86C5×SD16—54no—
87C5×D8—54no—
88C5×M16—22no—
89C5×M16C2×D512yescomplemented
90#689—52no—
91#688—52no—
92#687—52no—
93#686—52no—
94#686—54no—
95#686—54no—
96C2×#168—52no—
97C2×#168—102no—
98D40C2×D514yescomplemented
99#677C2×D514yescomplemented
100#676—54no—
101#676—104no—
102C8×D5—54no—
103C8×D5—104no—
104#576—251noSylow
105C2×C52—21no—
106C2×C52C22×D411yescharacteristic
107C5⋊D5—22no—
108C5×D5—42no—
109C5×D5—44no—
110C23×C5—102no—
111C22×D5—22no—
112C22×D5—104no—
113C22×D5—104no—
114C5×Q8—52no—
115C5×D4—54no—
116C5×D4—52no—
117C5×D4—102no—
118C5×D4—102no—
119C2×C4×C5—22no—
120C2×C4×C5—102no—
121C2×C4×C5C22×D512yes—
122C22⋊D5—22no—
123C22⋊D5—54no—
124C22⋊D5—104no—
125C22⋊D5—104no—
126C22⋊D5—104no—
127C22⋊D5—104no—
128C2×Dic5—104no—
129D20—22no—
130D20—52no—
131D20—102no—
132D20—104no—
133D20C22×D514yes—
134D20C22×D512yes—
135C4×D5—22no—
136C4×D5—54no—
137C4×D5—104no—
138C4×D5—104no—
139Dic10—52no—
140Dic10—102no—
141Dic10C22×D512yes—
142C5×C8—24no—
143C5×C8C22×D514yes—
144#168—52no—
145#168—52no—
146#168—102no—
147#168—102no—
148H(5,𝔽2)—252no—
149AGL(1,ℤ/8)—52no—
150AGL(1,ℤ/8)—254no—
151#135—252no—
152#135—251no—
153C2×D8—252no—
154C2×D8—251no—
155#131—251no—
156C52#57611yescharacteristic, complemented, Sylow
157C22×C5—22no—
158C22×C5—104no—
159C22×C5—102no—
160C22×C5—402no—
161C22×C5D4×D512yes—
162C2×D5—22no—
163C2×D5—24no—
164C2×D5—54no—
165C2×D5—82no—
166C2×D5—104no—
167C2×D5—404no—
168C2×D5—404no—
169C4×C5—22no—
170C4×C5—22no—
171C4×C5—102no—
172C4×C5D4×D512yes—
173C4×C5C23×D512yes—
174Dic5—22no—
175Dic5—54no—
176Dic5—104no—
177#41—52no—
178#41—251no—
179#41—254no—
180#41—502no—
181C2×D4—52no—
182C2×D4—254no—
183C2×D4—502no—
184C2×D4—504no—
185Q16—251no—
186SD16—54no—
187SD16—252no—
188D8—54no—
189D8—251no—
190D8—252no—
191D8—252no—
192M16—252no—
193M16D5211yescharacteristic, complemented
194C2×C8—251no—
195C2×C8—252no—
196C2×C5—22no—
197C2×C5—22no—
198C2×C5—42no—
199C2×C5—202no—
200C2×C5—204no—
201C2×C5C2×D4×D512yes—
202D5—44no—
203D5—42no—
204D5—104no—
205D5—204no—
206C23—102no—
207C23—504no—
208Q8—52no—
209Q8—251no—
210D4—52no—
211D4—54no—
212D4—102no—
213D4—102no—
214D4—252no—
215D4—251no—
216D4—504no—
217D4—504no—
218C2×C4—102no—
219C2×C4—252no—
220C2×C4—504no—
221C2×C4C2×D5211yescharacteristic
222C8—251no—
223C8—251no—
224C8C2×D5212yes—
225C5—22no—
226C5#6610812yescomplemented
227C22—102no—
228C22—104no—
229C22—252no—
230C22—402no—
231C22—2004no—
232C22#8787711yescharacteristic
233C4—102no—
234C4—252no—
235C4#8787711yescharacteristic
236C4C22×D5211yescharacteristic
237C2—21no—
238C2—204no—
239C2—202no—
240C2—502no—
241C2C2×#8787711yescharacteristic
subgroupG/H
centerC2C2×#87877
commutator, cosocleC4×C52C24
FrattiniC4C22×D52
FittingC52×M16C22
socleC2×C52C22×D4
hypercenterM16D52
hypocenterC52#576
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11717472C2semidirect
2. #11717227C2semidirect
3. C5×#4054C2semidirect
4. #11716940C2semidirect
5. C2×#87770C2semidirect
6. #11716927C2semidirect
7. #11716916C2semidirect
8. #11716915C2semidirect
10. C5×#709C22semidirect
11. C2×C5×D20C22semidirect
13. #87877C22semidirect
15. #87871C22semidirect
17. C52×M16C22semidirect
18. C2×#6029C22semidirect
19. #87799C22semidirect
20. C8×C5⋊D5C22semidirect
21. C5×D40C22semidirect
22. C5×#677C22semidirect
23. #87775C22semidirect
24. #87772C22semidirect
25. #87770C22semidirect
26. #87770C22semidirect
27. #87770C22semidirect
57. #4054D5semidirect
83. #709C2×D5semidirect
85. C2×D20C2×D5semidirect
89. C5×M16C2×D5semidirect
98. D40C2×D5semidirect
99. #677C2×D5semidirect
156. C52#576semidirect
193. M16D52semidirect
226. C5#66108semidirect

Maximal subgroups

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

subgroupindex
#117174722
#117172272
C5×#40542
#117169402
C2×#877702
#117169272
#117169162
#117169152
#661085