Finite Group Databasebeta

← #13106517

Subgroups — #13106517

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×#713C211yesmaximal, characteristic, complemented
2C2×D5×D20C211yesmaximal, characteristic, complemented
3#11717267C211yesmaximal, characteristic, complemented
4C5×#4093C211yesmaximal, characteristic, complemented
5#11717040C214yesmaximal, complemented
6#11717032C211yesmaximal, characteristic, complemented
7#11717029C211yesmaximal, characteristic, complemented
8#11717024C214yesmaximal, complemented
9#11716973C211yesmaximal, characteristic, complemented
10C22×D52C2212yes—
11C2×C52×Q8C2211yescharacteristic, complemented
12C2×C4⋊(C5⋊D5)C2211yescharacteristic
13C5×#713—24no—
14C2×C5×D20C2211yescharacteristic
15C2×C5×D20C2212yes—
16C2×C5×D20C2211yescharacteristic, complemented
17C2×C4×C5×D5C2211yescharacteristic
18C2×C4×C5×D5C2212yescomplemented
19C2×#6038C2212yes—
20C2×#6038C2214yescomplemented
21C2×#6037C2214yescomplemented
22D5×D20—24no—
23#87808C2214yescomplemented
24C4×#1039C2211yescharacteristic, complemented
25C5×#685C2214yescomplemented
26C4×C5×Dic5C2211yescharacteristic, complemented
27#87780C2212yescomplemented
28#87778C2212yes—
29#66164—51nomaximal
30#66008—51nomaximal
31C22×C5⋊D5C2312yes—
32C22×C5×D5C2312yes—
33C22×C5×D5C2312yes—
34C22×C5×D5C2311yescharacteristic
35C2×D52—24no—
36C2×D52—44no—
37C52×Q8—22no—
38C2×C4×C52C2312yes—
39C2×C4×C52C2311yescharacteristic
40C2×#1039C2312yes—
41C4⋊(C5⋊D5)—22no—
42C2×C5×Dic5C2312yes—
43C2×C5×Dic5C2311yescharacteristic
44C5×D20—24no—
45C5×D20—22no—
46C5×D20—22no—
47C4×C5×D5—24no—
48C4×C5×D5D414yescomplemented
49#6038—24no—
50#6038—44no—
51#6037—44no—
52C2×C5×#41—51no—
53C2×#713D511yescharacteristic, complemented
54C2×D4×D5—51no—
55C22×D20—51no—
56C5×#124—51no—
57#4093—51no—
58#4093—51no—
59#4093—102no—
60#4093D511yescharacteristic, complemented
61#4086—51no—
62#4086—51no—
63#4086—51no—
64#4083—54no—
65#4076—54no—
66#4072—51no—
67#4030—54no—
68#4028—54no—
69#4020—51no—
70#4018—51no—
71C22×C52C2411yescharacteristic
72C2×C5⋊D5—22no—
73C2×C5⋊D5—24no—
74C2×C5×D5—22no—
75C2×C5×D5—22no—
76C2×C5×D5—22no—
77C2×C5×D5—24no—
78C2×C5×D5—24no—
79C2×C5×D5C2×D412yes—
80D52—48no—
81C4×C52—22no—
82C4×C52C2×D412yes—
83#1039—22no—
84C5×Dic5—22no—
85C5×Dic5C2×D412yes—
86C23×D5—52no—
87C23×D5—52no—
88C5×#41—104no—
89C2×C5×Q8—22no—
90C2×C5×Q8C2×D511yescharacteristic, complemented
91C2×C5×Q8C2×D511yescharacteristic, complemented
92C2×C5×D4—51no—
93C2×C5×D4—52no—
94C2×C5×D4—51no—
95C22×C4×C5—51no—
96C22×C4×C5—52no—
97C2×C22⋊D5—54no—
98C2×C22⋊D5—54no—
99C2×C22⋊D5—52no—
100C2×C22⋊D5—54no—
101#713—24no—
102D4×D5—104no—
103C2×D20—52no—
104C2×D20—51no—
105C2×D20—51no—
106C2×D20—54no—
107C2×D20—102no—
108C2×D20—104no—
109C2×D20C2×D511yescharacteristic
110C2×D20C2×D512yes—
111C2×D20C2×D511yescharacteristic, complemented
112C2×C4×D5C2×D511yescharacteristic
113C2×C4×D5C2×D512yescomplemented
114C5×C22⋊C4—54no—
115C42×C5—51no—
116C22⋊Dic5—52no—
117#685—52no—
118#685—54no—
119#685—54no—
120#685—108no—
121#685C2×D514yescomplemented
122C4⋊Dic5—52no—
123#683—52no—
124C4×Dic5—51no—
125C4×Dic5—51no—
126C4×Dic5—102no—
127C4×Dic5C2×D511yescharacteristic, complemented
128#535—251noSylow
129C2×C52C22×D411yescharacteristic
130C2×C52H(5,𝔽2)12yes—
131C5⋊D5—42no—
132C5×D5—22no—
133C5×D5—42no—
134C5×D5—42no—
135C23×C5—51no—
136C23×C5—52no—
137C23×C5—52no—
138C22×D5—52no—
139C22×D5—52no—
140C22×D5—104no—
141C22×D5—104no—
142C22×D5—104no—
143C22×D5—204no—
144C22×D5—204no—
145C22×D5C22×D512yes—
146C22×D5C22×D512yes—
147C22×D5C22×D511yescharacteristic
148C5×Q8—22no—
149C5×Q8—22no—
150C5×Q8—44no—
151C5×D4—104no—
152C5×D4—102no—
153C5×D4—102no—
154C2×C4×C5—22no—
155C2×C4×C5—24no—
156C2×C4×C5—51no—
157C2×C4×C5—54no—
158C2×C4×C5—52no—
159C2×C4×C5—104no—
160C2×C4×C5C22×D512yes—
161C2×C4×C5C22×D512yes—
162C2×C4×C5C22×D511yescharacteristic
163C2×C4×C5C22×D511yescharacteristic
164C22⋊D5—104no—
165C22⋊D5—204no—
166C22⋊D5—204no—
167C22⋊D5—204no—
168C2×Dic5—52no—
169C2×Dic5—52no—
170C2×Dic5—104no—
171C2×Dic5C22×D512yes—
172C2×Dic5C22×D511yescharacteristic
173D20—24no—
174D20—22no—
175D20—22no—
176D20—104no—
177D20—102no—
178D20—102no—
179D20—204no—
180D20—204no—
181C4×D5—24no—
182C4×D5C22⋊D514yescomplemented
183C2×#41—51no—
184C22×D4—251no—
185#127—251no—
186#127—251no—
187#124—51no—
188#124—251no—
189#121—254no—
190#120—254no—
191#117—251no—
192C52#53511yescharacteristic, complemented, Sylow
193C22×C5—22no—
194C22×C5—52no—
195C22×C5—102no—
196C22×C5—102no—
197C22×C5—104no—
198C22×C5—104no—
199C22×C5—102no—
200C22×C5C23×D511yescharacteristic
201C22×C5C23×D511yescharacteristic
202C2×D5—22no—
203C2×D5—22no—
204C2×D5—24no—
205C2×D5—24no—
206C2×D5—22no—
207C2×D5—102no—
208C2×D5—104no—
209C2×D5—104no—
210C2×D5—102no—
211C2×D5—204no—
212C2×D5—208no—
213C2×D5—208no—
214C2×D5—208no—
215C2×D5C2×C22⋊D512yes—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—24no—
219C4×C5—44no—
220C4×C5—52no—
221C4×C5—102no—
222C4×C5D4×D512yes—
223C4×C5C2×C22⋊D512yes—
224Dic5—22no—
225Dic5—102no—
226Dic5—102no—
227Dic5—204no—
228Dic5C2×C22⋊D512yes—
229C24—252no—
230#41—104no—
231C2×Q8D5211yescharacteristic, complemented
232C2×D4—52no—
233C2×D4—51no—
234C2×D4—51no—
235C2×D4—252no—
236C2×D4—254no—
237C2×D4—254no—
238C2×D4—251no—
239C2×D4—504no—
240C22×C4—51no—
241C22×C4—52no—
242C4⋊C4—252no—
243C22⋊C4—54no—
244C22⋊C4—252no—
245C22⋊C4—254no—
246C42—51no—
247C42—251no—
248C2×C5—24no—
249C2×C5—22no—
250C2×C5—102no—
251C2×C5—202no—
252C2×C5—202no—
253C2×C5C2×D4×D511yescharacteristic
254C2×C5#414912yes—
255C2×C5#414912yes—
256C2×C5C22×C22⋊D511yescharacteristic
257D5—22no—
258D5—42no—
259D5—42no—
260D5—202no—
261D5—202no—
262D5—404no—
263C23—51no—
264C23—52no—
265C23—52no—
266C23—252no—
267C23—504no—
268C23—1004no—
269Q8—22no—
270D4—104no—
271D4—102no—
272D4—102no—
273D4—504no—
274D4—502no—
275D4—1004no—
276D4—1004no—
277C2×C4—51no—
278C2×C4—54no—
279C2×C4—52no—
280C2×C4—104no—
281C2×C4—252no—
282C2×C4C2×D5212yes—
283C2×C4C2×D5211yescharacteristic
284C5—22no—
285C5#6600811yescharacteristic, complemented
286C5#6616411yescharacteristic, complemented
287C22—52no—
288C22—102no—
289C22—104no—
290C22—104no—
291C22—102no—
292C22—102no—
293C22—504no—
294C22—502no—
295C22—1008no—
296C22C22×D5211yescharacteristic
297C4—22no—
298C4—52no—
299C4—102no—
300C4—502no—
301C4D5×C22⋊D512yes—
302C2—102no—
303C2—202no—
304C2—202no—
305C2—1002no—
306C2#1171748512yes—
307C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×Q8C22
hypercenterC2×Q8D52
hypocenterC52#535
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×#713C2semidirect
2. C2×D5×D20C2semidirect
3. #11717267C2semidirect
4. C5×#4093C2semidirect
5. #11717040C2semidirect
6. #11717032C2semidirect
7. #11717029C2semidirect
8. #11717024C2semidirect
9. #11716973C2semidirect
11. C2×C52×Q8C22semidirect
16. C2×C5×D20C22semidirect
18. C2×C4×C5×D5C22semidirect
20. C2×#6038C22semidirect
21. C2×#6037C22semidirect
23. #87808C22semidirect
24. C4×#1039C22semidirect
25. C5×#685C22semidirect
26. C4×C5×Dic5C22semidirect
27. #87780C22semidirect
48. C4×C5×D5D4semidirect
53. C2×#713D5semidirect
60. #4093D5semidirect
90. C2×C5×Q8C2×D5semidirect
91. C2×C5×Q8C2×D5semidirect
111. C2×D20C2×D5semidirect
113. C2×C4×D5C2×D5semidirect
121. #685C2×D5semidirect
127. C4×Dic5C2×D5semidirect
182. C4×D5C22⋊D5semidirect
192. C52#535semidirect
231. C2×Q8D52semidirect
285. C5#66008semidirect
286. C5#66164semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7132
C2×D5×D202
#117172672
C5×#40932
#117170402
#117170322
#117170292
#117170242
#117169732
#661645
#660085