Finite Group Databasebeta

← #11754986

Subgroups — #11754986

Order 960.

504 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 120 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#91824C211yesmaximal, characteristic, complemented
2C5×#940C211yesmaximal, characteristic, complemented
3C3×#4083C211yesmaximal, characteristic, complemented
4#91571C212yesmaximal, complemented
5#91565C212yesmaximal, complemented
6#91563C212yesmaximal, complemented
7#91555C212yesmaximal, complemented
8#91453C212yesmaximal, complemented
9#91363C212yesmaximal, complemented
10#66054—31nomaximal, Hall
11C23×C3×D5C2211yescharacteristic
12C2×C3×C5×D4C2211yescharacteristic
13C22×Dic15C2211yescharacteristic, complemented
14C2×C5×#67C2212yes—
15C2×C5×#67C2212yescomplemented
16C2×C5×D12C2211yescharacteristic, complemented
17C2×C3×C22⋊D5C2212yes—
18C2×#1201C2212yes—
19C2×#1201C2212yes—
20C2×D3×Dic5C2212yescomplemented
21#6719C2212yes—
22#6719C2211yescharacteristic, complemented
23#6712C2212yescomplemented
24C4×C5×Dic3C2211yescharacteristic, complemented
25C3×C22⋊Dic5C2211yescharacteristic
26C3×#685C2212yes—
27#6669C2212yescomplemented
28#6668C2212yescomplemented
29(C2×D3)⋊Dic5C2212yescomplemented
30#6665C2212yes—
31#6665C2212yes—
32#5595—51nomaximal, Hall
33C5×#127—31no—
34#4085—32no—
35#4085—32no—
36#4085—32no—
37#4083—32no—
38#4083D311yescharacteristic, complemented
39#4081—31no—
40#4081—32no—
41#4022—32no—
42C23×C3×C5C2311yescharacteristic
43C23×C3×C5C2311yescharacteristic
44C22×C5×D3C2312yes—
45C22×C3×D5—41no—
46C22×C3×D5—42no—
47C22×C3×D5C2312yes—
48C3×C5×D4—41no—
49C2×C3×C4×C5C2311yescharacteristic
50C2×Dic15—41no—
51C2×Dic15C2312yes—
52C2×Dic15C2312yes—
53C5×#67—42no—
54C5×#67—42no—
55C2×C5×Dic3C2312yes—
56C5×D12—41no—
57C3×C22⋊D5—42no—
58C2×C3×Dic5C2312yes—
59#1201—42no—
60#1201—42no—
61D3×Dic5—42no—
62C3×#120—51no—
63#953—52no—
64#940D511yescharacteristic, complemented
65#939—52no—
66#938—52no—
67#935—51no—
68#884—52no—
69#883—52no—
70#879—52no—
71C23×D5C2×D311yescharacteristic
72C2×C5×D4—32no—
73C2×C5×D4—32no—
74C2×C5×D4—31no—
75C2×C5×D4C2×D311yescharacteristic
76C2×C22⋊D5—32no—
77C2×C22⋊D5—32no—
78C2×C22⋊D5C2×D312yes—
79C22×Dic5—31no—
80C22×Dic5—32no—
81C42×C5—31no—
82C22⋊Dic5—32no—
83C22⋊Dic5—31no—
84C22⋊Dic5—32no—
85C22⋊Dic5C2×D311yescharacteristic
86#685—32no—
87#685—32no—
88#685C2×D312yes—
89#683—32no—
90#683—32no—
91#683—32no—
92#560—151noSylow
93C22×C3×C5—22no—
94C22×C3×C5—22no—
95C22×C3×C5—21no—
96C22×C3×C5—21no—
97C22×C3×C5C2411yescharacteristic
98C2×C5×D3—22no—
99C2×C5×D3—22no—
100C2×C5×D3—22no—
101C2×C3×D5—22no—
102C2×C3×D5—22no—
103C2×C3×D5—22no—
104C2×C3×D5—44no—
105C3×C4×C5—21no—
106Dic15—22no—
107Dic15—22no—
108C3×Dic5—22no—
109C5×Dic3—22no—
110C24×C3—51no—
111C2×C3×D4—52no—
112C2×C3×D4C2×D511yescharacteristic
113C2×#67—52no—
114C2×#67—52no—
115C2×#67C2×D512yes—
116C2×#67C2×D512yescomplemented
117C22×Dic3—51no—
118C2×D12C2×D511yescharacteristic, complemented
119C2×C4×D3—52no—
120C3×C22⋊C4—51no—
121C3×C22⋊C4—52no—
122#217—52no—
123#217—51no—
124#217—52no—
125#217—52no—
126(C2×D3)⋊C4—52no—
127C4⋊Dic3—52no—
128Dic3⋊C4—52no—
129Dic3⋊C4—52no—
130C4×Dic3C2×D511yescharacteristic, complemented
131C23×C5—32no—
132C23×C5C22×D311yescharacteristic
133C23×C5C22×D311yescharacteristic
134C22×D5—42no—
135C22×D5—41no—
136C22×D5C22×D312yes—
137C5×D4—41no—
138C5×D4—121no—
139C5×D4—122no—
140C5×D4—122no—
141C2×C4×C5—32no—
142C2×C4×C5C22×D311yescharacteristic
143C22⋊D5—42no—
144C22⋊D5—122no—
145C22⋊D5—122no—
146C2×Dic5—32no—
147C2×Dic5—32no—
148C2×Dic5—122no—
149C2×Dic5—121no—
150C2×Dic5C22×D312yes—
151#127—31no—
152#126—152no—
153#123—152no—
154#123—151no—
155#121—152no—
156#121—152no—
157#121—152no—
158#120—51no—
159#120—152no—
160C2×C3×C5—41no—
161C2×C3×C5—41no—
162C2×C3×C5H(5,𝔽2)12yes—
163C2×C3×C5H(5,𝔽2)11yescharacteristic
164C3×D5—42no—
165C5×D3—42no—
166C23×C3—52no—
167C23×C3—202no—
168C23×C3—201no—
169C23×C3C22×D511yescharacteristic
170C23×C3C22×D511yescharacteristic
171C22×D3C22×D512yes—
172C3×D4—41no—
173C3×D4—202no—
174C2×C3×C4—52no—
175C2×C3×C4C22×D511yescharacteristic
176#67—42no—
177#67—42no—
178#67—202no—
179#67—202no—
180C2×Dic3—52no—
181C2×Dic3—52no—
182C2×Dic3—201no—
183C2×Dic3C22×D512yes—
184D12—41no—
185C4×D3—202no—
186C22×C5—21no—
187C22×C5—22no—
188C22×C5—21no—
189C22×C5—22no—
190C22×C5—62no—
191C22×C5—62no—
192C22×C5—62no—
193C22×C5C23×D311yescharacteristic
194C2×D5—22no—
195C2×D5—22no—
196C2×D5—22no—
197C2×D5—44no—
198C4×C5—21no—
199C4×C5—62no—
200Dic5—22no—
201Dic5—62no—
202Dic5—62no—
203C24—51no—
204C2×D4—32no—
205C2×D4—32no—
206C2×D4—31no—
207C2×D4—52no—
208C2×D4—152no—
209C2×D4—152no—
210C2×D4D3×D511yescharacteristic
211C22×C4—152no—
212C22×C4—151no—
213C4⋊C4—152no—
214C4⋊C4—152no—
215C4⋊C4—152no—
216C22⋊C4—51no—
217C22⋊C4—52no—
218C22⋊C4—152no—
219C22⋊C4—152no—
220C22⋊C4—152no—
221C22⋊C4—152no—
222C22⋊C4—151no—
223C42—31no—
224C3×C5#56011yescharacteristic, complemented, Hall
225C22×C3—21no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—102no—
230C22×C3—102no—
231C22×C3—102no—
232C22×C3—204no—
233C22×C3C23×D511yescharacteristic
234C2×D3—22no—
235C2×D3—22no—
236C2×D3—22no—
237C3×C4—21no—
238C3×C4—102no—
239Dic3—22no—
240Dic3—102no—
241Dic3—102no—
242C2×C5—41no—
243C2×C5—41no—
244C2×C5—122no—
245C2×C5#100412yes—
246C2×C5#100411yescharacteristic
247D5—42no—
248C23—32no—
249C23—52no—
250C23—202no—
251C23—201no—
252C23C2×D3×D511yescharacteristic
253C23C2×D3×D511yescharacteristic
254D4—41no—
255D4—121no—
256D4—122no—
257D4—122no—
258D4—202no—
259D4—602no—
260D4—602no—
261C2×C4—32no—
262C2×C4—52no—
263C2×C4—152no—
264C2×C4—152no—
265C2×C4—602no—
266C2×C4—601no—
267C2×C4C2×D3×D511yescharacteristic
268C2×C3—41no—
269C2×C3—41no—
270C2×C3—202no—
271C2×C3#414412yes—
272C2×C3#414411yescharacteristic
273D3—42no—
274C5#559511yescharacteristic, complemented, Sylow
275C22—22no—
276C22—22no—
277C22—21no—
278C22—21no—
279C22—62no—
280C22—62no—
281C22—62no—
282C22—102no—
283C22—102no—
284C22—102no—
285C22—204no—
286C22C22×D3×D511yescharacteristic
287C4—21no—
288C4—62no—
289C4—102no—
290C4—302no—
291C4—302no—
292C3#6605411yescharacteristic, complemented, Sylow
293C2—41no—
294C2—41no—
295C2—122no—
296C2—202no—
297C2#9202512yes—
298C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#560
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #91824C2semidirect
2. C5×#940C2semidirect
3. C3×#4083C2semidirect
4. #91571C2semidirect
5. #91565C2semidirect
6. #91563C2semidirect
7. #91555C2semidirect
8. #91453C2semidirect
9. #91363C2semidirect
13. C22×Dic15C22semidirect
15. C2×C5×#67C22semidirect
16. C2×C5×D12C22semidirect
20. C2×D3×Dic5C22semidirect
22. #6719C22semidirect
23. #6712C22semidirect
24. C4×C5×Dic3C22semidirect
27. #6669C22semidirect
28. #6668C22semidirect
29. (C2×D3)⋊Dic5C22semidirect
38. #4083D3semidirect
64. #940D5semidirect
116. C2×#67C2×D5semidirect
118. C2×D12C2×D5semidirect
130. C4×Dic3C2×D5semidirect
224. C3×C5#560semidirect
274. C5#5595semidirect
292. C3#66054semidirect

Maximal subgroups

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

subgroupindex
#918242
C5×#9402
C3×#40832
#915712
#915652
#915632
#915552
#914532
#913632
#660543
#55955