Finite Group Databasebeta

← #11755279

Subgroups — #11755279

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×C5×#239C211yesmaximal, characteristic, complemented
2C2×D3×D20C211yesmaximal, characteristic, complemented
3#91836C211yesmaximal, characteristic, complemented
4C3×#4093C211yesmaximal, characteristic, complemented
5#91475C214yesmaximal, complemented
6#91463C211yesmaximal, characteristic, complemented
7#91454C214yesmaximal, complemented
8#91451C211yesmaximal, characteristic, complemented
9#91371C211yesmaximal, characteristic, complemented
10#66164—31nomaximal, Hall
11C22×D3×D5C2212yes—
12C2×C3×C5×Q8C2211yescharacteristic, complemented
13C2×D60C2211yescharacteristic
14C5×#239—24no—
15C2×C5×D12C2212yes—
16C2×C5×D12C2211yescharacteristic, complemented
17C2×C4×C5×D3C2211yescharacteristic
18C2×C4×C5×D3C2212yescomplemented
19C2×C3×D20C2211yescharacteristic
20C2×#1203C2214yescomplemented
21C2×#1202C2212yes—
22C2×#1201C2214yescomplemented
23D3×D20—24no—
24#6714C2214yescomplemented
25C4×Dic15C2211yescharacteristic, complemented
26C3×#685C2214yescomplemented
27C3×C4×Dic5C2211yescharacteristic, complemented
28Dic3⋊Dic5C2212yescomplemented
29(C2×D3)⋊Dic5C2212yes—
30#5674—51nomaximal, Hall
31C2×C5×#41—31no—
32C22×D20—31no—
33#4093—31no—
34#4093D311yescharacteristic, complemented
35#4086—31no—
36#4086—31no—
37#4083—34no—
38#4076—34no—
39#4072—31no—
40C22×D15C2312yes—
41C22×C5×D3C2312yes—
42C22×C5×D3C2311yescharacteristic
43C22×C3×D5C2312yes—
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×Dic3C2311yescharacteristic
52C5×D12—24no—
53C5×D12—22no—
54C4×C5×D3—24no—
55C4×C5×D3D414yescomplemented
56C2×C3×Dic5C2312yes—
57C3×D20—22no—
58#1203—44no—
59#1202—24no—
60#1201—44no—
61C2×#239D511yescharacteristic, complemented
62C2×D3×D4—51no—
63C3×#124—51no—
64#947—51no—
65#940—51no—
66#884—54no—
67#882—54no—
68#874—51no—
69#872—51no—
70C23×D5—32no—
71C5×#41—64no—
72C2×C5×Q8C2×D311yescharacteristic, complemented
73C2×C5×D4—32no—
74C2×C5×D4—31no—
75C22×C4×C5—31no—
76C22×C4×C5—32no—
77C2×C22⋊D5—34no—
78C2×C22⋊D5—34no—
79C2×D20—32no—
80C2×D20—31no—
81C2×D20—64no—
82C2×D20C2×D311yescharacteristic
83C22⋊Dic5—32no—
84#685—34no—
85#685C2×D314yescomplemented
86C4⋊Dic5—32no—
87C4×Dic5—31no—
88C4×Dic5C2×D311yescharacteristic, complemented
89#535—151noSylow
90C22×C3×C5C2411yescharacteristic
91C2×D15—22no—
92C2×D15—24no—
93C2×C5×D3—22no—
94C2×C5×D3—22no—
95C2×C5×D3—24no—
96C2×C5×D3C2×D412yes—
97C2×C3×D5—22no—
98C2×C3×D5—24no—
99D3×D5—48no—
100C3×C4×C5—22no—
101C3×C4×C5C2×D412yes—
102Dic15—22no—
103C3×Dic5—22no—
104C5×Dic3C2×D412yes—
105C23×D3—52no—
106C2×C3×Q8C2×D511yescharacteristic, complemented
107C2×C3×D4—51no—
108C2×#67—54no—
109C2×#67—52no—
110#239—24no—
111D3×D4—104no—
112C2×D12—51no—
113C2×D12—54no—
114C2×D12C2×D512yes—
115C2×D12C2×D511yescharacteristic, complemented
116C2×C4×D3C2×D511yescharacteristic
117C2×C4×D3C2×D512yescomplemented
118C3×C22⋊C4—54no—
119C3×C42—51no—
120(C2×D3)⋊C4—52no—
121(C2×D3)⋊C4—54no—
122Dic3⋊C4—52no—
123C4×Dic3—51no—
124C23×C5—31no—
125C23×C5—32no—
126C22×D5—32no—
127C22×D5—64no—
128C22×D5—124no—
129C22×D5C22×D312yes—
130C5×Q8—22no—
131C5×D4—64no—
132C5×D4—62no—
133C2×C4×C5—31no—
134C2×C4×C5—34no—
135C2×C4×C5—64no—
136C2×C4×C5C22×D312yes—
137C2×C4×C5C22×D311yescharacteristic
138C22⋊D5—124no—
139C22⋊D5—124no—
140C2×Dic5—32no—
141C2×Dic5C22×D312yes—
142D20—22no—
143D20—64no—
144D20—62no—
145C2×#41—31no—
146C22×D4—151no—
147#127—151no—
148#127—151no—
149#124—51no—
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—42no—
158C5×D3—22no—
159C5×D3—42no—
160C23×C3—52no—
161C22×D3—52no—
162C22×D3—104no—
163C22×D3—204no—
164C22×D3C22×D512yes—
165C22×D3C22×D511yescharacteristic
166C3×Q8—22no—
167C3×D4—102no—
168C2×C3×C4—52no—
169C2×C3×C4C22×D512yes—
170C2×C3×C4C22×D511yescharacteristic
171#67—104no—
172#67—204no—
173C2×Dic3—52no—
174C2×Dic3C22×D511yescharacteristic
175D12—24no—
176D12—22no—
177D12—102no—
178D12—204no—
179C4×D3—24no—
180C4×D3C22⋊D514yescomplemented
181C22×C5—32no—
182C22×C5—62no—
183C22×C5—62no—
184C22×C5—64no—
185C22×C5C23×D311yescharacteristic
186C2×D5—22no—
187C2×D5—24no—
188C2×D5—62no—
189C2×D5—64no—
190C2×D5—128no—
191C4×C5—22no—
192C4×C5—32no—
193C4×C5D3×D412yes—
194Dic5—22no—
195Dic5—62no—
196C24—152no—
197#41—64no—
198C2×Q8D3×D511yescharacteristic, complemented
199C2×D4—32no—
200C2×D4—31no—
201C2×D4—51no—
202C2×D4—152no—
203C2×D4—154no—
204C2×D4—154no—
205C2×D4—151no—
206C2×D4—304no—
207C22×C4—31no—
208C22×C4—32no—
209C4⋊C4—152no—
210C22⋊C4—54no—
211C22⋊C4—152no—
212C22⋊C4—154no—
213C42—51no—
214C42—151no—
215C3×C5#53511yescharacteristic, complemented, Hall
216C22×C3—104no—
217C22×C3—102no—
218C22×C3C23×D511yescharacteristic
219C2×D3—22no—
220C2×D3—24no—
221C2×D3—22no—
222C2×D3—104no—
223C2×D3—102no—
224C2×D3—208no—
225C2×D3C2×C22⋊D512yes—
226C3×C4—22no—
227C3×C4—102no—
228C3×C4C2×C22⋊D512yes—
229Dic3—102no—
230Dic3C2×C22⋊D512yes—
231C2×C5—62no—
232C2×C5—122no—
233C2×C5C2×D3×D411yescharacteristic
234C2×C5#100912yes—
235D5—42no—
236D5—122no—
237C23—31no—
238C23—32no—
239C23—52no—
240C23—152no—
241C23—304no—
242C23—604no—
243Q8—22no—
244D4—64no—
245D4—62no—
246D4—102no—
247D4—304no—
248D4—302no—
249D4—604no—
250D4—604no—
251C2×C4—31no—
252C2×C4—34no—
253C2×C4—52no—
254C2×C4—64no—
255C2×C4—152no—
256C2×C4C2×D3×D512yes—
257C2×C4C2×D3×D511yescharacteristic
258C2×C3—202no—
259C2×C3#414912yes—
260C2×C3C22×C22⋊D511yescharacteristic
261D3—22no—
262D3—42no—
263D3—202no—
264C5#567411yescharacteristic, complemented, Sylow
265C22—32no—
266C22—62no—
267C22—64no—
268C22—62no—
269C22—104no—
270C22—102no—
271C22—304no—
272C22—302no—
273C22—608no—
274C22C22×D3×D511yescharacteristic
275C4—22no—
276C4—32no—
277C4—102no—
278C4—302no—
279C4D3×C22⋊D512yes—
280C3#6616411yescharacteristic, complemented, Sylow
281C2—62no—
282C2—122no—
283C2—202no—
284C2—602no—
285C2#9203512yes—
286C2C2×D3×C22⋊D511yescharacteristic
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×C5×#239C2semidirect
2. C2×D3×D20C2semidirect
3. #91836C2semidirect
4. C3×#4093C2semidirect
5. #91475C2semidirect
6. #91463C2semidirect
7. #91454C2semidirect
8. #91451C2semidirect
9. #91371C2semidirect
12. C2×C3×C5×Q8C22semidirect
16. C2×C5×D12C22semidirect
18. C2×C4×C5×D3C22semidirect
20. C2×#1203C22semidirect
22. C2×#1201C22semidirect
24. #6714C22semidirect
25. C4×Dic15C22semidirect
26. C3×#685C22semidirect
27. C3×C4×Dic5C22semidirect
28. Dic3⋊Dic5C22semidirect
34. #4093D3semidirect
55. C4×C5×D3D4semidirect
61. C2×#239D5semidirect
72. C2×C5×Q8C2×D3semidirect
85. #685C2×D3semidirect
88. C4×Dic5C2×D3semidirect
106. C2×C3×Q8C2×D5semidirect
115. C2×D12C2×D5semidirect
117. C2×C4×D3C2×D5semidirect
180. C4×D3C22⋊D5semidirect
198. C2×Q8D3×D5semidirect
215. C3×C5#535semidirect
264. C5#5674semidirect
280. C3#66164semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#2392
C2×D3×D202
#918362
C3×#40932
#914752
#914632
#914542
#914512
#913712
#661643
#56745