Finite Group Databasebeta

← #13106290

Subgroups — #13106290

Order 1600.

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

normal subgroups only

This group has 744 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×D4×D5C211yesmaximal, characteristic, complemented
2C2×#87870C211yesmaximal, characteristic, complemented
3#11717256C211yesmaximal, characteristic, complemented
4C5×#4086C211yesmaximal, characteristic, complemented
5#11717094C214yesmaximal, complemented
6#11717088C214yesmaximal, complemented
7#11717032C211yesmaximal, characteristic, complemented
8#11716979C211yesmaximal, characteristic, complemented
9#11716973C211yesmaximal, characteristic, complemented
10C23×C5×D5C2212yes—
11C2×C52×D4C2211yescharacteristic
12C2×#6045C2211yescharacteristic, complemented
13C2×C5×C22⋊D5C2212yes—
14C2×C5×C22⋊D5C2214yescomplemented
15C5×D4×D5—24no—
16C2×C5×D20C2211yescharacteristic
17C2×C5×D20C2211yescharacteristic, complemented
18C2×C4×C5×D5C2211yescharacteristic
19C2×#6037C2212yes—
20C2×#6037C2214yescomplemented
21C2×D5×Dic5C2212yescomplemented
22#87870—24no—
23C22⋊#1039C2214yescomplemented
24C4×#1039C2211yescharacteristic, complemented
25C4×C5×Dic5C2211yescharacteristic, complemented
26#87780C2212yescomplemented
27#87778C2212yes—
28#87778C2214yescomplemented
29#66139—51nomaximal
30#66049—51nomaximal
31C23×C52C2312yes—
32C22×C5×D5—24no—
33C22×C5×D5—44no—
34C22×C5×D5C2312yes—
35C22×C5×D5C2312yes—
36C22×C5×D5C2311yescharacteristic
37C52×D4—22no—
38C2×C4×C52C2311yescharacteristic
39C2×#1039C2312yes—
40C2×#1039C2312yes—
41#6045—22no—
42C5×C22⋊D5—24no—
43C5×C22⋊D5—44no—
44C2×C5×Dic5C2312yes—
45C2×C5×Dic5C2311yescharacteristic
46C5×D20—22no—
47C5×D20—22no—
48C4×C5×D5D414yescomplemented
49#6037—24no—
50#6037—44no—
51D5×Dic5—24no—
52C22×C5×D4—51no—
53C2×#711—51no—
54C2×D4×D5D511yescharacteristic, complemented
55C2×#709—51no—
56C5×#127—51no—
57#4099—54no—
58#4086—51no—
59#4086—51no—
60#4086D511yescharacteristic, complemented
61#4085—54no—
62#4084—54no—
63#4083—54no—
64#4082—51no—
65#4082—51no—
66#4082—51no—
67#4082—102no—
68#4072—51no—
69#4021—51no—
70#4018—51no—
71C22×C52—22no—
72C22×C52—24no—
73C22×C52C2411yescharacteristic
74C2×C5×D5—22no—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—24no—
78C2×C5×D5—24no—
79C2×C5×D5—48no—
80C2×C5×D5C2×D412yes—
81C4×C52C2×D412yes—
82#1039—22no—
83#1039—22no—
84C5×Dic5—22no—
85C5×Dic5C2×D412yes—
86C24×C5—52no—
87C23×D5C2×D512yes—
88C2×C5×D4—22no—
89C2×C5×D4—52no—
90C2×C5×D4—54no—
91C2×C5×D4—51no—
92C2×C5×D4—51no—
93C2×C5×D4—104no—
94C2×C5×D4C2×D511yescharacteristic
95C2×C5×D4C2×D511yescharacteristic
96C22×C4×C5—51no—
97C2×C22⋊D5—52no—
98C2×C22⋊D5—52no—
99C2×C22⋊D5—54no—
100C2×C22⋊D5—54no—
101C2×C22⋊D5C2×D512yes—
102C2×C22⋊D5C2×D514yescomplemented
103C22×Dic5—52no—
104#711—104no—
105D4×D5—24no—
106#709—104no—
107C2×D20C2×D511yescharacteristic
108C2×D20C2×D511yescharacteristic, complemented
109C2×C4×D5—52no—
110C2×C4×D5C2×D511yescharacteristic
111C2×Dic10—51no—
112C2×Dic10—51no—
113C2×Dic10—102no—
114C42×C5—51no—
115C22⋊Dic5—54no—
116C22⋊Dic5—54no—
117C22⋊Dic5—52no—
118C22⋊Dic5—54no—
119C22⋊Dic5—108no—
120#685—52no—
121#685—54no—
122C4⋊Dic5—52no—
123#683—52no—
124C4×Dic5—51no—
125C4×Dic5—51no—
126C4×Dic5—102no—
127C4×Dic5C2×D511yescharacteristic, complemented
128#535—251noSylow
129C2×C52—42no—
130C2×C52C22×D411yescharacteristic
131C2×C52H(5,𝔽2)12yes—
132C5×D5—22no—
133C5×D5—42no—
134C5×D5—42no—
135C23×C5—24no—
136C23×C5—51no—
137C23×C5—52no—
138C23×C5—52no—
139C23×C5—104no—
140C23×C5—204no—
141C23×C5C22×D512yes—
142C23×C5C22×D512yes—
143C22×D5—24no—
144C22×D5—44no—
145C22×D5C22×D512yes—
146C22×D5C22×D512yes—
147C22×D5C22×D511yescharacteristic
148C5×D4—22no—
149C5×D4—22no—
150C5×D4—44no—
151C5×D4—104no—
152C5×D4—102no—
153C5×D4—102no—
154C5×D4—204no—
155C2×C4×C5—22no—
156C2×C4×C5—51no—
157C2×C4×C5—54no—
158C2×C4×C5—52no—
159C2×C4×C5C22×D511yescharacteristic
160C2×C4×C5C22×D511yescharacteristic
161C22⋊D5—24no—
162C22⋊D5—44no—
163C22⋊D5—104no—
164C22⋊D5—104no—
165C22⋊D5—204no—
166C22⋊D5—204no—
167C2×Dic5—52no—
168C2×Dic5—52no—
169C2×Dic5—52no—
170C2×Dic5—52no—
171C2×Dic5—104no—
172C2×Dic5—104no—
173C2×Dic5—104no—
174C2×Dic5C22×D512yes—
175C2×Dic5C22×D511yescharacteristic
176D20—22no—
177D20—22no—
178C4×D5—104no—
179C4×D5C22⋊D514yescomplemented
180Dic10—102no—
181Dic10—102no—
182Dic10—204no—
183C2×#41—251no—
184C22×D4—51no—
185#127—51no—
186#127—251no—
187#124—251no—
188#124—251no—
189#121—254no—
190#120—254no—
191#117—251no—
192C52#53511yescharacteristic, complemented, Sylow
193C22×C5—22no—
194C22×C5—24no—
195C22×C5—24no—
196C22×C5—22no—
197C22×C5—22no—
198C22×C5—44no—
199C22×C5—48no—
200C22×C5—52no—
201C22×C5—102no—
202C22×C5—102no—
203C22×C5—104no—
204C22×C5—104no—
205C22×C5—102no—
206C22×C5—208no—
207C22×C5C23×D511yescharacteristic
208C22×C5C23×D511yescharacteristic
209C2×D5—22no—
210C2×D5—22no—
211C2×D5—24no—
212C2×D5—24no—
213C2×D5—22no—
214C2×D5—48no—
215C2×D5C2×C22⋊D512yes—
216C4×C5—24no—
217C4×C5—52no—
218C4×C5—102no—
219C4×C5D4×D512yes—
220C4×C5C2×C22⋊D512yes—
221Dic5—22no—
222Dic5—102no—
223Dic5—102no—
224Dic5—102no—
225Dic5—102no—
226Dic5—204no—
227Dic5—204no—
228Dic5C2×C22⋊D512yes—
229C24—52no—
230#41—504no—
231C2×Q8—251no—
232C2×D4—52no—
233C2×D4—54no—
234C2×D4—51no—
235C2×D4—51no—
236C2×D4—104no—
237C2×D4—252no—
238C2×D4—254no—
239C2×D4D5211yescharacteristic
240C22×C4—51no—
241C22×C4—252no—
242C4⋊C4—252no—
243C22⋊C4—252no—
244C22⋊C4—254no—
245C22⋊C4—254no—
246C42—51no—
247C42—251no—
248C2×C5—24no—
249C2×C5—22no—
250C2×C5—42no—
251C2×C5—42no—
252C2×C5—84no—
253C2×C5—102no—
254C2×C5—202no—
255C2×C5—202no—
256C2×C5C2×D4×D511yescharacteristic
257C2×C5#414412yes—
258C2×C5#414412yes—
259C2×C5C22×C22⋊D511yescharacteristic
260D5—22no—
261D5—42no—
262D5—42no—
263C23—51no—
264C23—52no—
265C23—52no—
266C23—104no—
267C23—204no—
268C23C2×D5212yes—
269Q8—502no—
270D4—22no—
271D4—104no—
272D4—102no—
273D4—102no—
274D4—204no—
275D4—504no—
276D4—1004no—
277C2×C4—51no—
278C2×C4—54no—
279C2×C4—52no—
280C2×C4—252no—
281C2×C4—252no—
282C2×C4—504no—
283C2×C4C2×D5211yescharacteristic
284C5—22no—
285C5#6604911yescharacteristic, complemented
286C5#6613911yescharacteristic, complemented
287C22—24no—
288C22—22no—
289C22—52no—
290C22—102no—
291C22—104no—
292C22—104no—
293C22—102no—
294C22—102no—
295C22—208no—
296C22C22×D5211yescharacteristic
297C4—52no—
298C4—102no—
299C4—502no—
300C4—502no—
301C4D5×C22⋊D512yes—
302C2—42no—
303C2—102no—
304C2—202no—
305C2—202no—
306C2#1171747812yes—
307C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#535
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×D4×D5C2semidirect
2. C2×#87870C2semidirect
3. #11717256C2semidirect
4. C5×#4086C2semidirect
5. #11717094C2semidirect
6. #11717088C2semidirect
7. #11717032C2semidirect
8. #11716979C2semidirect
9. #11716973C2semidirect
12. C2×#6045C22semidirect
14. C2×C5×C22⋊D5C22semidirect
17. C2×C5×D20C22semidirect
20. C2×#6037C22semidirect
21. C2×D5×Dic5C22semidirect
23. C22⋊#1039C22semidirect
24. C4×#1039C22semidirect
25. C4×C5×Dic5C22semidirect
26. #87780C22semidirect
28. #87778C22semidirect
48. C4×C5×D5D4semidirect
54. C2×D4×D5D5semidirect
60. #4086D5semidirect
102. C2×C22⋊D5C2×D5semidirect
108. C2×D20C2×D5semidirect
127. C4×Dic5C2×D5semidirect
179. C4×D5C22⋊D5semidirect
192. C52#535semidirect
285. C5#66049semidirect
286. C5#66139semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×D52
C2×#878702
#117172562
C5×#40862
#117170942
#117170882
#117170322
#117169792
#117169732
#661395
#660495