Finite Group Databasebeta

← #11755272

Subgroups — #11755272

Order 960.

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

normal subgroups only

This group has 668 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×C3×#713C211yesmaximal, characteristic, complemented
2C2×D5×D12C211yesmaximal, characteristic, complemented
3#91836C211yesmaximal, characteristic, complemented
4C5×#947C211yesmaximal, characteristic, complemented
5#91474C214yesmaximal, complemented
6#91463C211yesmaximal, characteristic, complemented
7#91458C211yesmaximal, characteristic, complemented
8#91448C214yesmaximal, complemented
9#91357C211yesmaximal, characteristic, complemented
10#66008—31nomaximal, Hall
11C22×D3×D5C2212yes—
12C2×C3×C5×Q8C2211yescharacteristic, complemented
13C2×D60C2211yescharacteristic
14C2×C5×D12C2211yescharacteristic
15C3×#713—24no—
16C2×C3×D20C2212yes—
17C2×C3×D20C2211yescharacteristic, complemented
18C2×C3×C4×D5C2211yescharacteristic
19C2×C3×C4×D5C2212yescomplemented
20C2×#1203C2212yes—
21C2×#1202C2214yescomplemented
22C2×#1201C2214yescomplemented
23D5×D12—24no—
24#6714C2214yescomplemented
25C4×Dic15C2211yescharacteristic, complemented
26C5×(C2×D3)⋊C4C2214yescomplemented
27C4×C5×Dic3C2211yescharacteristic, complemented
28#6668C2212yescomplemented
29#6665C2212yes—
30#5830—51nomaximal, Hall
31C2×#713D311yescharacteristic, complemented
32C2×D4×D5—31no—
33C5×#124—31no—
34#4093—31no—
35#4086—31no—
36#4030—34no—
37#4028—34no—
38#4020—31no—
39#4018—31no—
40C22×D15C2312yes—
41C22×C5×D3C2312yes—
42C22×C3×D5C2312yes—
43C22×C3×D5C2311yescharacteristic
44C2×D3×D5—24no—
45C2×D3×D5—44no—
46C3×C5×Q8—22no—
47C2×C3×C4×C5C2312yes—
48C2×C3×C4×C5C2311yescharacteristic
49C2×Dic15C2312yes—
50D60—22no—
51C2×C5×Dic3C2312yes—
52C5×D12—22no—
53C2×C3×Dic5C2311yescharacteristic
54C3×D20—24no—
55C3×D20—22no—
56C3×C4×D5—24no—
57C3×C4×D5D414yescomplemented
58#1203—24no—
59#1202—44no—
60#1201—44no—
61C2×C3×#41—51no—
62C22×D12—51no—
63#947—51no—
64#947D511yescharacteristic, complemented
65#940—51no—
66#940—51no—
67#937—54no—
68#930—54no—
69#926—51no—
70C23×D5—32no—
71C2×C5×Q8C2×D311yescharacteristic, complemented
72C2×C5×D4—31no—
73C2×C22⋊D5—34no—
74C2×C22⋊D5—32no—
75#713—24no—
76D4×D5—64no—
77C2×D20—31no—
78C2×D20—34no—
79C2×D20C2×D312yes—
80C2×D20C2×D311yescharacteristic, complemented
81C2×C4×D5C2×D311yescharacteristic
82C2×C4×D5C2×D312yescomplemented
83C5×C22⋊C4—34no—
84C42×C5—31no—
85#685—32no—
86#685—34no—
87#683—32no—
88C4×Dic5—31no—
89#535—151noSylow
90C22×C3×C5C2411yescharacteristic
91C2×D15—22no—
92C2×D15—24no—
93C2×C5×D3—22no—
94C2×C5×D3—24no—
95C2×C3×D5—22no—
96C2×C3×D5—22no—
97C2×C3×D5—24no—
98C2×C3×D5C2×D412yes—
99D3×D5—48no—
100C3×C4×C5—22no—
101C3×C4×C5C2×D412yes—
102Dic15—22no—
103C3×Dic5C2×D412yes—
104C5×Dic3—22no—
105C23×D3—52no—
106C3×#41—104no—
107C2×C3×Q8C2×D511yescharacteristic, complemented
108C2×C3×D4—52no—
109C2×C3×D4—51no—
110C22×C3×C4—51no—
111C22×C3×C4—52no—
112C2×#67—54no—
113C2×#67—54no—
114C2×D12—52no—
115C2×D12—51no—
116C2×D12—104no—
117C2×D12C2×D511yescharacteristic
118#217—52no—
119(C2×D3)⋊C4—54no—
120(C2×D3)⋊C4C2×D514yescomplemented
121C4⋊Dic3—52no—
122C4×Dic3—51no—
123C4×Dic3C2×D511yescharacteristic, complemented
124C23×C5—32no—
125C22×D5—32no—
126C22×D5—64no—
127C22×D5—124no—
128C22×D5C22×D312yes—
129C22×D5C22×D311yescharacteristic
130C5×Q8—22no—
131C5×D4—62no—
132C2×C4×C5—32no—
133C2×C4×C5C22×D312yes—
134C2×C4×C5C22×D311yescharacteristic
135C22⋊D5—64no—
136C22⋊D5—124no—
137C2×Dic5—32no—
138C2×Dic5C22×D311yescharacteristic
139D20—24no—
140D20—22no—
141D20—62no—
142D20—124no—
143C4×D5—24no—
144C4×D5#6714yescomplemented
145C2×#41—51no—
146C22×D4—151no—
147#127—151no—
148#127—151no—
149#124—31no—
150#124—151no—
151#121—154no—
152#120—154no—
153#117—151no—
154C2×C3×C5C22×D411yescharacteristic
155C2×C3×C5H(5,𝔽2)12yes—
156D15—42no—
157C3×D5—22no—
158C3×D5—42no—
159C5×D3—42no—
160C23×C3—51no—
161C23×C3—52no—
162C22×D3—52no—
163C22×D3—104no—
164C22×D3—204no—
165C22×D3C22×D512yes—
166C3×Q8—22no—
167C3×D4—104no—
168C3×D4—102no—
169C2×C3×C4—51no—
170C2×C3×C4—54no—
171C2×C3×C4—104no—
172C2×C3×C4C22×D512yes—
173C2×C3×C4C22×D511yescharacteristic
174#67—204no—
175#67—204no—
176C2×Dic3—52no—
177C2×Dic3C22×D512yes—
178D12—22no—
179D12—104no—
180D12—102no—
181C22×C5—64no—
182C22×C5—62no—
183C22×C5C23×D311yescharacteristic
184C2×D5—22no—
185C2×D5—24no—
186C2×D5—22no—
187C2×D5—64no—
188C2×D5—62no—
189C2×D5—128no—
190C2×D5C2×#6712yes—
191C4×C5—22no—
192C4×C5—62no—
193C4×C5C2×#6712yes—
194Dic5—62no—
195Dic5C2×#6712yes—
196C24—152no—
197#41—104no—
198C2×Q8D3×D511yescharacteristic, complemented
199C2×D4—31no—
200C2×D4—52no—
201C2×D4—51no—
202C2×D4—154no—
203C2×D4—152no—
204C2×D4—154no—
205C2×D4—151no—
206C2×D4—304no—
207C22×C4—51no—
208C22×C4—52no—
209C4⋊C4—152no—
210C22⋊C4—34no—
211C22⋊C4—152no—
212C22⋊C4—154no—
213C42—31no—
214C42—151no—
215C3×C5#53511yescharacteristic, complemented, Hall
216C22×C3—52no—
217C22×C3—102no—
218C22×C3—102no—
219C22×C3—104no—
220C22×C3C23×D511yescharacteristic
221C2×D3—22no—
222C2×D3—24no—
223C2×D3—102no—
224C2×D3—104no—
225C2×D3—208no—
226C3×C4—22no—
227C3×C4—52no—
228C3×C4D4×D512yes—
229Dic3—22no—
230Dic3—102no—
231C2×C5—122no—
232C2×C5#100912yes—
233C2×C5C22×#6711yescharacteristic
234D5—22no—
235D5—42no—
236D5—122no—
237C23—32no—
238C23—51no—
239C23—52no—
240C23—152no—
241C23—304no—
242C23—604no—
243Q8—22no—
244D4—62no—
245D4—104no—
246D4—102no—
247D4—304no—
248D4—302no—
249D4—604no—
250D4—604no—
251C2×C4—32no—
252C2×C4—51no—
253C2×C4—54no—
254C2×C4—104no—
255C2×C4—152no—
256C2×C4C2×D3×D512yes—
257C2×C4C2×D3×D511yescharacteristic
258C2×C3—102no—
259C2×C3—202no—
260C2×C3C2×D4×D511yescharacteristic
261C2×C3#414912yes—
262D3—42no—
263D3—202no—
264C5#583011yescharacteristic, complemented, Sylow
265C22—52no—
266C22—64no—
267C22—62no—
268C22—102no—
269C22—104no—
270C22—102no—
271C22—304no—
272C22—302no—
273C22—608no—
274C22C22×D3×D511yescharacteristic
275C4—22no—
276C4—52no—
277C4—62no—
278C4—302no—
279C4D5×#6712yes—
280C3#6600811yescharacteristic, complemented, Sylow
281C2—102no—
282C2—122no—
283C2—202no—
284C2—602no—
285C2#9203512yes—
286C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
hypercenterC2×Q8D3×D5
hypocenterC3×C5#535
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#713C2semidirect
2. C2×D5×D12C2semidirect
3. #91836C2semidirect
4. C5×#947C2semidirect
5. #91474C2semidirect
6. #91463C2semidirect
7. #91458C2semidirect
8. #91448C2semidirect
9. #91357C2semidirect
12. C2×C3×C5×Q8C22semidirect
17. C2×C3×D20C22semidirect
19. C2×C3×C4×D5C22semidirect
21. C2×#1202C22semidirect
22. C2×#1201C22semidirect
24. #6714C22semidirect
25. C4×Dic15C22semidirect
26. C5×(C2×D3)⋊C4C22semidirect
27. C4×C5×Dic3C22semidirect
28. #6668C22semidirect
31. C2×#713D3semidirect
57. C3×C4×D5D4semidirect
64. #947D5semidirect
71. C2×C5×Q8C2×D3semidirect
80. C2×D20C2×D3semidirect
82. C2×C4×D5C2×D3semidirect
107. C2×C3×Q8C2×D5semidirect
120. (C2×D3)⋊C4C2×D5semidirect
123. C4×Dic3C2×D5semidirect
144. C4×D5#67semidirect
198. C2×Q8D3×D5semidirect
215. C3×C5#535semidirect
264. C5#5830semidirect
280. C3#66008semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7132
C2×D5×D122
#918362
C5×#9472
#914742
#914632
#914582
#914482
#913572
#660083
#58305