Finite Group Databasebeta

← #11755068

Subgroups — #11755068

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×#237C211yesmaximal, characteristic, complemented
2C2×D3×D20C211yesmaximal, characteristic, complemented
3#91829C211yesmaximal, characteristic, complemented
4C3×#4086C211yesmaximal, characteristic, complemented
5#91577C214yesmaximal, complemented
6#91569C214yesmaximal, complemented
7#91375C211yesmaximal, characteristic, complemented
8#91371C211yesmaximal, characteristic, complemented
9#91368C211yesmaximal, characteristic, complemented
10#66164—31nomaximal, Hall
11C22×D3×D5C2212yes—
12C2×C3×C5×D4C2211yescharacteristic, complemented
13C2×#1220C2214yescomplemented
14C2×D60C2211yescharacteristic
15C2×C5×#67C2212yes—
16C22×C5×Dic3C2212yescomplemented
17C5×#237—24no—
18C2×C4×C5×D3C2211yescharacteristic
19C2×C5×Dic6C2211yescharacteristic, complemented
20C2×C3×C22⋊D5C2214yescomplemented
21C2×C3×D20C2211yescharacteristic
22C2×#1202C2212yes—
23D3×D20—24no—
24C4×Dic15C2211yescharacteristic, complemented
25C3×C4×Dic5C2211yescharacteristic, complemented
26Dic3⋊Dic5C2212yescomplemented
27#6667C2214yescomplemented
28(C2×D3)⋊Dic5C2212yes—
29#6665C2214yescomplemented
30#5715—51nomaximal, Hall
31C2×C5×#41—31no—
32C22×D20—31no—
33#4093—31no—
34#4093—31no—
35#4086—31no—
36#4086D311yescharacteristic, complemented
37#4083—34no—
38#4076—34no—
39#4072—31no—
40C23×C3×C5C2312yes—
41C22×D15C2312yes—
42C22×C5×D3C2311yescharacteristic
43C22×C3×D5C2312yes—
44C2×D3×D5—24no—
45C2×D3×D5—44no—
46C3×C5×D4—22no—
47C2×C3×C4×C5C2311yescharacteristic
48#1220—44no—
49C2×Dic15C2312yes—
50D60—22no—
51C5×#67—24no—
52C2×C5×Dic3—24no—
53C2×C5×Dic3C2312yes—
54C2×C5×Dic3C2311yescharacteristic
55C4×C5×D3D414yescomplemented
56C5×Dic6—22no—
57C3×C22⋊D5—44no—
58C2×C3×Dic5C2312yes—
59C3×D20—22no—
60#1202—24no—
61C2×#237D511yescharacteristic, complemented
62C2×D3×D4—51no—
63C3×#127—51no—
64#940—51no—
65#939—54no—
66#937—54no—
67#936—51no—
68#875—51no—
69#872—51no—
70C23×D5—32no—
71C5×#41—64no—
72C2×C5×Q8—31no—
73C2×C5×D4—32no—
74C2×C5×D4C2×D311yescharacteristic, complemented
75C22×C4×C5—31no—
76C22×C4×C5—32no—
77C2×C22⋊D5—34no—
78C2×C22⋊D5C2×D314yescomplemented
79C2×D20—32no—
80C2×D20—31no—
81C2×D20—64no—
82C2×D20C2×D311yescharacteristic
83C22⋊Dic5—32no—
84#685—34no—
85#685—34no—
86C4⋊Dic5—32no—
87C4×Dic5—31no—
88C4×Dic5C2×D311yescharacteristic, complemented
89#535—151noSylow
90C22×C3×C5—22no—
91C22×C3×C5—24no—
92C22×C3×C5C2411yescharacteristic
93C2×D15—22no—
94C2×D15—24no—
95C2×C5×D3—22no—
96C2×C5×D3C2×D412yes—
97C2×C3×D5—22no—
98C2×C3×D5—24no—
99D3×D5—48no—
100C3×C4×C5C2×D412yes—
101Dic15—22no—
102C3×Dic5—22no—
103C5×Dic3—22no—
104C5×Dic3C2×D412yes—
105C23×D3—52no—
106C2×C3×D4—54no—
107C2×C3×D4—51no—
108C2×C3×D4C2×D511yescharacteristic, complemented
109C2×#67—54no—
110C2×#67—52no—
111C2×#67C2×D512yes—
112C22×Dic3C2×D512yescomplemented
113#237—24no—
114D3×D4—104no—
115C2×D12—51no—
116C2×C4×D3C2×D511yescharacteristic
117C2×Dic6C2×D511yescharacteristic, complemented
118C3×C42—51no—
119#217—54no—
120(C2×D3)⋊C4—52no—
121(C2×D3)⋊C4—54no—
122Dic3⋊C4—52no—
123C4×Dic3—51no—
124C23×C5—31no—
125C23×C5C22×D312yes—
126C22×D5—32no—
127C22×D5—64no—
128C22×D5—124no—
129C22×D5C22×D312yes—
130C5×Q8—62no—
131C5×D4—22no—
132C5×D4—64no—
133C2×C4×C5—31no—
134C2×C4×C5—34no—
135C2×C4×C5—32no—
136C2×C4×C5—64no—
137C2×C4×C5C22×D311yescharacteristic
138C22⋊D5—44no—
139C22⋊D5—124no—
140C2×Dic5—32no—
141C2×Dic5C22×D312yes—
142D20—22no—
143D20—64no—
144D20—62no—
145C2×#41—31no—
146C22×D4—151no—
147#127—51no—
148#127—151no—
149#124—151no—
150#124—151no—
151#121—154no—
152#120—154no—
153#117—151no—
154C2×C3×C5—42no—
155C2×C3×C5C22×D411yescharacteristic
156C2×C3×C5H(5,𝔽2)12yes—
157D15—42no—
158C3×D5—42no—
159C5×D3—22no—
160C23×C3—52no—
161C23×C3C22×D512yes—
162C22×D3—52no—
163C22×D3—104no—
164C22×D3—204no—
165C22×D3C22×D511yescharacteristic
166C3×D4—22no—
167C3×D4—102no—
168C3×D4—204no—
169C2×C3×C4—52no—
170C2×C3×C4C22×D511yescharacteristic
171#67—24no—
172#67—104no—
173#67—204no—
174C2×Dic3—24no—
175C2×Dic3—52no—
176C2×Dic3C22×D512yes—
177C2×Dic3C22×D511yescharacteristic
178D12—102no—
179C4×D3C22⋊D514yescomplemented
180Dic6—22no—
181C22×C5—22no—
182C22×C5—24no—
183C22×C5—32no—
184C22×C5—62no—
185C22×C5C23×D311yescharacteristic
186C2×D5—22no—
187C2×D5—24no—
188C2×D5—62no—
189C2×D5—64no—
190C2×D5—128no—
191C4×C5—32no—
192C4×C5—62no—
193C4×C5D3×D412yes—
194Dic5—22no—
195Dic5—62no—
196C24—152no—
197#41—64no—
198C2×Q8—31no—
199C2×D4—32no—
200C2×D4—54no—
201C2×D4—51no—
202C2×D4—152no—
203C2×D4—154no—
204C2×D4—151no—
205C2×D4—304no—
206C2×D4D3×D511yescharacteristic, complemented
207C22×C4—31no—
208C22×C4—32no—
209C4⋊C4—152no—
210C22⋊C4—154no—
211C22⋊C4—154no—
212C22⋊C4—152no—
213C42—51no—
214C42—151no—
215C3×C5#53511yescharacteristic, complemented, Hall
216C22×C3—24no—
217C22×C3—22no—
218C22×C3—104no—
219C22×C3—102no—
220C22×C3C23×D511yescharacteristic
221C2×D3—22no—
222C2×D3—104no—
223C2×D3—102no—
224C2×D3—208no—
225C2×D3C2×C22⋊D512yes—
226C3×C4—102no—
227C3×C4C2×C22⋊D512yes—
228Dic3—22no—
229Dic3—102no—
230Dic3C2×C22⋊D512yes—
231C2×C5—42no—
232C2×C5—62no—
233C2×C5C2×D3×D411yescharacteristic
234C2×C5#100412yes—
235D5—42no—
236D5—122no—
237C23—31no—
238C23—52no—
239C23—152no—
240C23—304no—
241C23—604no—
242C23C2×D3×D512yes—
243Q8—62no—
244D4—22no—
245D4—64no—
246D4—102no—
247D4—204no—
248D4—304no—
249D4—302no—
250D4—604no—
251C2×C4—31no—
252C2×C4—34no—
253C2×C4—32no—
254C2×C4—52no—
255C2×C4—64no—
256C2×C4—152no—
257C2×C4C2×D3×D511yescharacteristic
258C2×C3—42no—
259C2×C3—202no—
260C2×C3#414912yes—
261C2×C3C22×C22⋊D511yescharacteristic
262D3—22no—
263D3—202no—
264C5#571511yescharacteristic, complemented, Sylow
265C22—24no—
266C22—22no—
267C22—32no—
268C22—62no—
269C22—104no—
270C22—102no—
271C22—304no—
272C22—302no—
273C22—608no—
274C22C22×D3×D511yescharacteristic
275C4—32no—
276C4—62no—
277C4—102no—
278C4—302no—
279C4D3×C22⋊D512yes—
280C3#6616411yescharacteristic, complemented, Sylow
281C2—42no—
282C2—62no—
283C2—202no—
284C2—602no—
285C2#9202612yes—
286C2C2×D3×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#535
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. C2×C5×#237C2semidirect
2. C2×D3×D20C2semidirect
3. #91829C2semidirect
4. C3×#4086C2semidirect
5. #91577C2semidirect
6. #91569C2semidirect
7. #91375C2semidirect
8. #91371C2semidirect
9. #91368C2semidirect
12. C2×C3×C5×D4C22semidirect
13. C2×#1220C22semidirect
16. C22×C5×Dic3C22semidirect
19. C2×C5×Dic6C22semidirect
20. C2×C3×C22⋊D5C22semidirect
24. C4×Dic15C22semidirect
25. C3×C4×Dic5C22semidirect
26. Dic3⋊Dic5C22semidirect
27. #6667C22semidirect
29. #6665C22semidirect
36. #4086D3semidirect
55. C4×C5×D3D4semidirect
61. C2×#237D5semidirect
74. C2×C5×D4C2×D3semidirect
78. C2×C22⋊D5C2×D3semidirect
88. C4×Dic5C2×D3semidirect
108. C2×C3×D4C2×D5semidirect
112. C22×Dic3C2×D5semidirect
117. C2×Dic6C2×D5semidirect
179. C4×D3C22⋊D5semidirect
206. C2×D4D3×D5semidirect
215. C3×C5#535semidirect
264. C5#5715semidirect
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×#2372
C2×D3×D202
#918292
C3×#40862
#915772
#915692
#913752
#913712
#913682
#661643
#57155