Finite Group Databasebeta

← #11753806

Subgroups — #11753806

Order 960.

476 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4⋊C4×D15C211yesmaximal, characteristic, complemented
2C5×#893C211yesmaximal, characteristic, complemented
3C3×#4038C211yesmaximal, characteristic, complemented
4#91466C211yesmaximal, characteristic, complemented
5#91461C212yesmaximal, complemented
6#91452C212yesmaximal, complemented
7#91449C211yesmaximal, characteristic, complemented
8C4×#1203C212yesmaximal, complemented
9C4×#1202C211yesmaximal, characteristic, complemented
10#91381C211yesmaximal, characteristic, complemented
11#91372C212yesmaximal, complemented
12#65892—31nomaximal, Hall
13C2×C4×D15C2211yescharacteristic
14C2×C4×D15C2212yes—
15C2×C5×D12C2211yescharacteristic
16C2×C4×C5×D3C2212yes—
17C2×C3×C4×D5C2211yescharacteristic
18C2×#1203C2212yes—
19C2×#1202C2211yescharacteristic
20C2×#1201C2212yes—
21C3×C5×C4⋊C4C2211yescharacteristic
22C4⋊Dic15C2211yescharacteristic
23#6712C2212yes—
24C5×(C2×D3)⋊C4C2212yes—
25C4×C5×Dic3C2211yescharacteristic
26C3×#685C2212yes—
27C3×C4⋊Dic5C2211yescharacteristic
28C3×C4×Dic5C2212yes—
29Dic3⋊Dic5C2211yescharacteristic
30#6669C2212yes—
31#6668C2212yes—
32#6667C2211yescharacteristic
33(C2×D3)⋊Dic5C2212yes—
34(C2×D3)⋊Dic5C2212yes—
35#6665C2211yescharacteristic
36#5530—51nomaximal, Hall
37C4×C5×D4—31no—
38#4084—31no—
39#4084—31no—
40#4075—32no—
41C4×C22⋊D5—32no—
42#4038D311yescharacteristic, complemented
43D5×C4⋊C4—31no—
44#4030—32no—
45#4025—32no—
46C4×D20—31no—
47#4016—31no—
48C22×D15C2311yescharacteristic
49C22×C5×D3C2312yes—
50C22×C3×D5C2311yescharacteristic
51C2×C3×C4×C5C2312yes—
52C2×C3×C4×C5C2311yescharacteristic
53C2×Dic15C2311yescharacteristic
54C2×Dic15C2312yes—
55C4×D15—22no—
56C4×D15—42no—
57C2×C5×Dic3C2311yescharacteristic
58C2×C5×Dic3C2311yescharacteristic
59C5×D12—22no—
60C4×C5×D3—42no—
61C2×C3×Dic5C2311yescharacteristic
62C2×C3×Dic5C2312yes—
63C3×C4×D5—22no—
64#1203—24no—
65#1202—22no—
66#1201—42no—
67C3×#117—51no—
68#930—51no—
69C4×#67—51no—
70#895—51no—
71#893D511yescharacteristic, complemented
72D3×C4⋊C4—51no—
73#890—51no—
74#884—52no—
75#883—52no—
76#876—52no—
77C4×D12—52no—
78C2×C5×D4—31no—
79C22×C4×C5—32no—
80C2×C22⋊D5—32no—
81C2×C22⋊D5—32no—
82C2×D20—31no—
83C2×C4×D5—31no—
84C2×C4×D5—32no—
85C2×C4×D5C2×D311yescharacteristic
86C5×C4⋊C4C2×D311yescharacteristic
87C5×C22⋊C4—32no—
88C42×C5—31no—
89C22⋊Dic5—32no—
90C22⋊Dic5—32no—
91#685—31no—
92#685—31no—
93#685C2×D312yes—
94C4⋊Dic5—31no—
95C4⋊Dic5—31no—
96C4⋊Dic5C2×D311yescharacteristic
97#683—32no—
98#683—32no—
99#683—32no—
100C4×Dic5C2×D312yes—
101#553—151noSylow
102C22×C3×C5C2411yescharacteristic
103C2×D15—21no—
104C2×D15—21no—
105C2×D15#4112yes—
106C2×C5×D3—22no—
107C2×C5×D3—22no—
108C2×C5×D3—22no—
109C2×C3×D5—21no—
110C2×C3×D5—21no—
111C2×C3×D5—21no—
112C3×C4×C5—22no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—22no—
116C3×Dic5—22no—
117C3×Dic5—21no—
118C5×Dic3—21no—
119C5×Dic3—21no—
120C22×C3×C4—51no—
121C2×#67—52no—
122C2×#67—51no—
123C2×D12—52no—
124C2×D12C2×D511yescharacteristic
125C2×C4×D3—51no—
126C2×C4×D3—52no—
127C2×C4×D3C2×D512yes—
128C3×C4⋊C4—51no—
129C3×C4⋊C4C2×D511yescharacteristic
130C3×C22⋊C4—52no—
131C3×C42—52no—
132#217—51no—
133(C2×D3)⋊C4—52no—
134(C2×D3)⋊C4—52no—
135(C2×D3)⋊C4—51no—
136(C2×D3)⋊C4C2×D512yes—
137C4⋊Dic3—51no—
138C4⋊Dic3—52no—
139Dic3⋊C4—52no—
140Dic3⋊C4—52no—
141Dic3⋊C4—51no—
142C4×Dic3C2×D511yescharacteristic
143C23×C5—32no—
144C22×D5—31no—
145C22×D5C22×D311yescharacteristic
146C5×D4—62no—
147C2×C4×C5—31no—
148C2×C4×C5—31no—
149C2×C4×C5—122no—
150C2×C4×C5C22×D312yes—
151C2×C4×C5C22×D311yescharacteristic
152C22⋊D5—64no—
153C22⋊D5—122no—
154C2×Dic5—32no—
155C2×Dic5—31no—
156C2×Dic5C22×D311yescharacteristic
157C2×Dic5C22×D312yes—
158D20—62no—
159C4×D5—22no—
160C4×D5—62no—
161C4×D5—122no—
162#126—152no—
163#125—151no—
164#123—152no—
165#121—152no—
166#121—151no—
167#121—151no—
168C4×D4—31no—
169C4×D4—151no—
170C4×D4—152no—
171#117—51no—
172C2×C4⋊C4—151no—
173C2×C3×C5C2×#4111yescharacteristic
174C2×C3×C5C2×#4111yescharacteristic
175C2×C3×C5H(5,𝔽2)11yescharacteristic
176D15—22no—
177C3×D5—41no—
178C5×D3—42no—
179C23×C3—51no—
180C22×D3—51no—
181C22×D3C22×D512yes—
182C2×C3×C4—52no—
183C2×C3×C4—51no—
184C2×C3×C4—102no—
185C2×C3×C4C22×D512yes—
186C2×C3×C4C22×D511yescharacteristic
187#67—102no—
188#67—202no—
189C2×Dic3—51no—
190C2×Dic3—52no—
191C2×Dic3C22×D511yescharacteristic
192C2×Dic3C22×D511yescharacteristic
193D12—22no—
194D12—104no—
195C4×D3—42no—
196C4×D3—102no—
197C4×D3—202no—
198C22×C5—62no—
199C22×C5—62no—
200C22×C5—62no—
201C22×C5C23×D311yescharacteristic
202C2×D5—21no—
203C2×D5—21no—
204C2×D5—21no—
205C2×D5—32no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—22no—
209C4×C5—61no—
210C4×C5—61no—
211C4×C5#23512yes—
212Dic5—21no—
213Dic5—22no—
214Dic5—61no—
215Dic5—62no—
216C2×D4—31no—
217C2×D4—151no—
218C2×D4—152no—
219C2×D4—152no—
220C22×C4—32no—
221C22×C4—51no—
222C22×C4—151no—
223C22×C4—152no—
224C4⋊C4—51no—
225C4⋊C4—152no—
226C4⋊C4—151no—
227C4⋊C4—152no—
228C4⋊C4—151no—
229C4⋊C4—152no—
230C4⋊C4D3×D511yescharacteristic
231C22⋊C4—32no—
232C22⋊C4—52no—
233C22⋊C4—151no—
234C22⋊C4—151no—
235C22⋊C4—152no—
236C22⋊C4—152no—
237C42—31no—
238C42—52no—
239C3×C5#55311yescharacteristic, complemented, Hall
240C22×C3—101no—
241C22×C3—101no—
242C22×C3—101no—
243C22×C3C23×D511yescharacteristic
244C2×D3—22no—
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—52no—
248C2×D3—101no—
249C2×D3—101no—
250C3×C4—22no—
251C3×C4—101no—
252C3×C4—102no—
253C3×C4#70912yes—
254Dic3—21no—
255Dic3—21no—
256Dic3—101no—
257Dic3—102no—
258C2×C5—122no—
259C2×C5C2×#23511yescharacteristic
260C2×C5D3×#4111yescharacteristic
261C2×C5#100911yescharacteristic
262D5—41no—
263D5—62no—
264C23—32no—
265C23—51no—
266C23—151no—
267D4—62no—
268D4—302no—
269D4—304no—
270D4—602no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—51no—
274C2×C4—52no—
275C2×C4—102no—
276C2×C4—122no—
277C2×C4—151no—
278C2×C4—152no—
279C2×C4—302no—
280C2×C4—602no—
281C2×C4C2×D3×D512yes—
282C2×C4C2×D3×D511yescharacteristic
283C2×C3—201no—
284C2×C3C2×#70911yescharacteristic
285C2×C3#414411yescharacteristic
286C2×C3D5×#4111yescharacteristic
287D3—42no—
288D3—102no—
289C5#553011yescharacteristic, complemented, Sylow
290C22—62no—
291C22—62no—
292C22—62no—
293C22—101no—
294C22—101no—
295C22—101no—
296C22—152no—
297C22—301no—
298C22—301no—
299C22C22×D3×D511yescharacteristic
300C4—22no—
301C4—61no—
302C4—61no—
303C4—101no—
304C4—102no—
305C4—301no—
306C4—302no—
307C4#677112yes—
308C3#6589211yescharacteristic, complemented, Sylow
309C2—122no—
310C2—201no—
311C2—302no—
312C2C2×#677111yescharacteristic
313C2#9202711yescharacteristic
314C2#9203411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×D15C2semidirect
2. C5×#893C2semidirect
3. C3×#4038C2semidirect
4. #91466C2semidirect
5. #91461C2semidirect
6. #91452C2semidirect
7. #91449C2semidirect
8. C4×#1203C2semidirect
9. C4×#1202C2semidirect
10. #91381C2semidirect
11. #91372C2semidirect
42. #4038D3semidirect
71. #893D5semidirect
239. C3×C5#553semidirect
289. C5#5530semidirect
308. C3#65892semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D152
C5×#8932
C3×#40382
#914662
#914612
#914522
#914492
C4×#12032
C4×#12022
#913812
#913722
#658923
#55305