Finite Group Databasebeta

← #11754823

Subgroups — #11754823

Order 960.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 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×D4×D15C211yesmaximal, characteristic, complemented
2C2×#6766C211yesmaximal, characteristic, complemented
3C5×#936C211yesmaximal, characteristic, complemented
4C3×#4086C211yesmaximal, characteristic, complemented
5#91572C214yesmaximal, complemented
6#91550C214yesmaximal, complemented
7#91458C211yesmaximal, characteristic, complemented
8#91386C211yesmaximal, characteristic, complemented
9#91375C211yesmaximal, characteristic, complemented
10#66008—31nomaximal, Hall
11C23×D15C2212yes—
12C2×C3×C5×D4C2211yescharacteristic
13C2×#1220C2212yes—
14D4×D15—24no—
15C2×D60C2211yescharacteristic
16C2×C4×D15C2211yescharacteristic
17C2×C5×Dic6C2211yescharacteristic, complemented
18C2×C3×C22⋊D5C2214yescomplemented
19C2×C3×D20C2211yescharacteristic, complemented
20C2×#1202C2212yes—
21C2×#1202C2214yescomplemented
22C2×D5×Dic3C2212yescomplemented
23#6766—24no—
24C5×#217C2214yescomplemented
25C4×C5×Dic3C2211yescharacteristic, complemented
26C3×C4×Dic5C2211yescharacteristic, complemented
27#6669C2212yescomplemented
28#6667C2212yes—
29#6667C2214yescomplemented
30#5715—51nomaximal, Hall
31C2×#713—31no—
32C2×D4×D5—31no—
33C5×#124—31no—
34#4093—31no—
35#4086D311yescharacteristic, complemented
36#4030—34no—
37#4028—34no—
38#4020—31no—
39#4018—31no—
40C23×C3×C5C2312yes—
41C22×D15—24no—
42C22×D15—44no—
43C22×D15C2312yes—
44C22×D15C2311yescharacteristic
45C22×C3×D5C2312yes—
46C3×C5×D4—22no—
47C2×C3×C4×C5C2311yescharacteristic
48#1220—24no—
49C2×Dic15C2311yescharacteristic
50D60—22no—
51C4×D15D414yescomplemented
52C2×C5×Dic3C2312yes—
53C2×C5×Dic3C2312yes—
54C5×Dic6—22no—
55C3×C22⋊D5—44no—
56C2×C3×Dic5C2312yes—
57C3×D20—22no—
58#1202—24no—
59#1202—44no—
60D5×Dic3—24no—
61C2×#237—51no—
62C2×D3×D4—51no—
63C3×#127—51no—
64#940—51no—
65#939—54no—
66#937—54no—
67#936D511yescharacteristic, complemented
68#875—51no—
69#872—51no—
70C23×D5—32no—
71C2×C5×Q8—31no—
72C2×C5×D4C2×D311yescharacteristic
73C2×C22⋊D5—32no—
74C2×C22⋊D5C2×D314yescomplemented
75#713—64no—
76D4×D5—64no—
77C2×D20—32no—
78C2×D20—31no—
79C2×D20—34no—
80C2×D20C2×D311yescharacteristic, complemented
81C2×C4×D5—31no—
82C2×C4×D5—32no—
83C5×C22⋊C4—34no—
84C42×C5—31no—
85#685—32no—
86#685—34no—
87#683—32no—
88C4×Dic5C2×D311yescharacteristic, complemented
89#535—151noSylow
90C22×C3×C5—22no—
91C22×C3×C5—24no—
92C22×C3×C5C2411yescharacteristic
93C2×D15—22no—
94C2×D15—22no—
95C2×D15—24no—
96C2×D15—48no—
97C2×D15C2×D412yes—
98C2×C3×D5—22no—
99C2×C3×D5—24no—
100C3×C4×C5C2×D412yes—
101Dic15C2×D412yes—
102C3×Dic5—22no—
103C5×Dic3—22no—
104C5×Dic3—22no—
105C23×D3—52no—
106C2×C3×D4—54no—
107C2×C3×D4—51no—
108C2×C3×D4C2×D511yescharacteristic
109C2×#67—52no—
110C2×#67—52no—
111C2×#67—54no—
112C22×Dic3—52no—
113#237—104no—
114D3×D4—104no—
115C2×D12—51no—
116C2×C4×D3—51no—
117C2×Dic6C2×D511yescharacteristic, complemented
118C3×C42—51no—
119#217C2×D514yescomplemented
120(C2×D3)⋊C4—52no—
121(C2×D3)⋊C4—54no—
122Dic3⋊C4—52no—
123C4×Dic3C2×D511yescharacteristic, complemented
124C23×C5C22×D312yes—
125C22×D5—31no—
126C22×D5—32no—
127C22×D5—64no—
128C22×D5—124no—
129C22×D5C22×D312yes—
130C5×Q8—62no—
131C5×D4—22no—
132C2×C4×C5—32no—
133C2×C4×C5—32no—
134C2×C4×C5C22×D311yescharacteristic
135C22⋊D5—44no—
136C22⋊D5—64no—
137C2×Dic5—31no—
138C2×Dic5C22×D312yes—
139D20—22no—
140D20—64no—
141D20—62no—
142D20—124no—
143C4×D5—34no—
144C4×D5—64no—
145C2×#41—151no—
146C22×D4—151no—
147#127—51no—
148#127—151no—
149#124—31no—
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—22no—
158D15—42no—
159C3×D5—42no—
160C23×C3—52no—
161C23×C3C22×D512yes—
162C22×D3—51no—
163C22×D3—52no—
164C22×D3—104no—
165C22×D3—204no—
166C3×D4—22no—
167C3×D4—102no—
168C3×D4—204no—
169C2×C3×C4—52no—
170C2×C3×C4C22×D511yescharacteristic
171#67—104no—
172#67—104no—
173#67—204no—
174C2×Dic3—51no—
175C2×Dic3—104no—
176C2×Dic3C22×D512yes—
177C2×Dic3C22×D512yes—
178D12—102no—
179C4×D3—54no—
180Dic6—22no—
181C22×C5—22no—
182C22×C5—24no—
183C22×C5C23×D311yescharacteristic
184C2×D5—22no—
185C2×D5—24no—
186C2×D5—32no—
187C2×D5—62no—
188C2×D5—62no—
189C2×D5—64no—
190C2×D5—128no—
191C4×C5—62no—
192C4×C5—62no—
193C4×C5D3×D412yes—
194Dic5—22no—
195Dic5—32no—
196C24—152no—
197#41—304no—
198C2×Q8—31no—
199C2×D4—54no—
200C2×D4—51no—
201C2×D4—152no—
202C2×D4—154no—
203C2×D4—152no—
204C2×D4—151no—
205C2×D4—304no—
206C2×D4D3×D511yescharacteristic
207C22×C4—152no—
208C22×C4—151no—
209C4⋊C4—152no—
210C22⋊C4—34no—
211C22⋊C4—152no—
212C22⋊C4—154no—
213C42—31no—
214C42—51no—
215C3×C5#53511yescharacteristic, complemented, Hall
216C22×C3—22no—
217C22×C3—24no—
218C22×C3—102no—
219C22×C3—104no—
220C22×C3C23×D511yescharacteristic
221C2×D3—52no—
222C2×D3—102no—
223C2×D3—102no—
224C2×D3—104no—
225C2×D3—208no—
226C3×C4—102no—
227C3×C4D4×D512yes—
228Dic3—22no—
229Dic3—22no—
230Dic3—52no—
231C2×C5—42no—
232C2×C5C2×D3×D411yescharacteristic
233C2×C5#100412yes—
234D5—42no—
235D5—62no—
236D5—122no—
237C23—52no—
238C23—151no—
239C23—152no—
240C23—304no—
241C23—604no—
242C23C2×D3×D512yes—
243Q8—62no—
244D4—22no—
245D4—102no—
246D4—204no—
247D4—304no—
248D4—304no—
249D4—302no—
250D4—604no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—52no—
254C2×C4—151no—
255C2×C4—154no—
256C2×C4—304no—
257C2×C4C2×D3×D511yescharacteristic
258C2×C3—42no—
259C2×C3—202no—
260C2×C3C2×D4×D511yescharacteristic
261C2×C3#414912yes—
262D3—102no—
263D3—202no—
264C5#571511yescharacteristic, complemented, Sylow
265C22—24no—
266C22—22no—
267C22—104no—
268C22—102no—
269C22—152no—
270C22—302no—
271C22—304no—
272C22—302no—
273C22—608no—
274C22C22×D3×D511yescharacteristic
275C4—62no—
276C4—62no—
277C4—102no—
278C4—152no—
279C4#679012yes—
280C3#6600811yescharacteristic, complemented, Sylow
281C2—42no—
282C2—202no—
283C2—302no—
284C2—602no—
285C2#9202612yes—
286C2C2×#679011yescharacteristic
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×D4×D15C2semidirect
2. C2×#6766C2semidirect
3. C5×#936C2semidirect
4. C3×#4086C2semidirect
5. #91572C2semidirect
6. #91550C2semidirect
7. #91458C2semidirect
8. #91386C2semidirect
9. #91375C2semidirect
17. C2×C5×Dic6C22semidirect
18. C2×C3×C22⋊D5C22semidirect
19. C2×C3×D20C22semidirect
21. C2×#1202C22semidirect
22. C2×D5×Dic3C22semidirect
24. C5×#217C22semidirect
25. C4×C5×Dic3C22semidirect
26. C3×C4×Dic5C22semidirect
27. #6669C22semidirect
29. #6667C22semidirect
35. #4086D3semidirect
51. C4×D15D4semidirect
67. #936D5semidirect
74. C2×C22⋊D5C2×D3semidirect
80. C2×D20C2×D3semidirect
88. C4×Dic5C2×D3semidirect
117. C2×Dic6C2×D5semidirect
119. #217C2×D5semidirect
123. C4×Dic3C2×D5semidirect
215. C3×C5#535semidirect
264. C5#5715semidirect
280. C3#66008semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D152
C2×#67662
C5×#9362
C3×#40862
#915722
#915502
#914582
#913862
#913752
#660083
#57155