Finite Group Databasebeta

← #11754563

Subgroups — #11754563

Order 960.

660 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 168 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×#235C211yesmaximal, characteristic, complemented
2C22×C3×C4×D5C211yesmaximal, characteristic, complemented
3#91815C211yesmaximal, characteristic, complemented
4#91563C212yesmaximal, complemented
5#91539C212yesmaximal, complemented
6#91449C212yesmaximal, complemented
7C4×#1201C214yesmaximal, complemented
8#91359C212yesmaximal, complemented
9#66162—31nomaximal, Hall
10C23×C3×D5C2211yescharacteristic
11C22×C3×C4×C5C2211yescharacteristic
12C2×C5×#67C2212yes—
13C5×#235—24no—
14C2×C5×D12C2211yescharacteristic, complemented
15C2×C4×C5×D3C2212yescomplemented
16C2×C5×Dic6C2211yescharacteristic, complemented
17C22×C3×Dic5C2211yescharacteristic
18C2×C3×C4×D5—24no—
19C2×C3×C4×D5C2212yes—
20C2×C3×C4×D5C2212yes—
21C2×#1201C2214yes—
22#6719C2212yes—
23C4⋊Dic15C2212yescomplemented
24C4×Dic15C2212yescomplemented
25#6669C2214yescomplemented
26(C2×D3)⋊Dic5C2214yescomplemented
27#6665C2214yes—
28#5791—51nomaximal, Hall
29C2×C5×#41—31no—
30C22×C4×D5D311yescharacteristic, complemented
31#4092—32no—
32#4084—32no—
33#4083—32no—
34#4081—32no—
35C4×C22⋊D5—34no—
36#4072—31no—
37C23×C3×C5C2311yescharacteristic
38C22×C5×D3C2312yes—
39C22×C3×D5—22no—
40C22×C3×D5—24no—
41C22×C3×D5C2312yes—
42C2×C3×C4×C5D414yescomplemented
43C2×C3×C4×C5C2311yescharacteristic
44C2×C3×C4×C5C2311yescharacteristic
45C2×Dic15C2314yes—
46C5×#67—24no—
47C2×C5×Dic3C2312yes—
48C5×D12—22no—
49C4×C5×D3—24no—
50C5×Dic6—22no—
51C2×C3×Dic5—22no—
52C2×C3×Dic5C2312yes—
53C3×C4×D5—24no—
54C3×C4×D5—24no—
55#1201—28no—
56C23×C3×C4—51no—
57C2×#235D511yescharacteristic, complemented
58#953—52no—
59#930—52no—
60#929—52no—
61C4×#67—54no—
62#926—51no—
63#924—52no—
64C23×D5C2×D311yescharacteristic
65C5×#41—64no—
66C2×C5×Q8—31no—
67C2×C5×D4—32no—
68C2×C5×D4—31no—
69C22×C4×C5—32no—
70C22×C4×C5C2×D311yescharacteristic
71C2×C22⋊D5—34no—
72C22×Dic5C2×D311yescharacteristic
73C2×C4×D5—24no—
74C2×C4×D5C2×D312yes—
75C2×C4×D5C2×D312yes—
76C22⋊Dic5—32no—
77C22⋊Dic5—34no—
78#685—34no—
79C4⋊Dic5—32no—
80#683—34no—
81C4×Dic5—32no—
82#525—151noSylow
83C22×C3×C5—22no—
84C22×C3×C5C2×D412yes—
85C22×C3×C5C2411yescharacteristic
86C2×C5×D3—22no—
87C2×C5×D3—24no—
88C2×C3×D5—28no—
89C2×C3×D5—22no—
90C2×C3×D5—22no—
91C2×C3×D5#4114yescomplemented
92C3×C4×C5C2×D412yes—
93C3×C4×C5C2×D412yes—
94Dic15—24no—
95C3×Dic5—22no—
96C5×Dic3—22no—
97C24×C3—51no—
98C22×C3×C4—51no—
99C22×C3×C4—52no—
100C22×C3×C4—52no—
101C22×C3×C4—104no—
102C22×C3×C4C2×D511yescharacteristic
103C2×#67—54no—
104C2×#67C2×D512yes—
105#235—24no—
106C2×D12C2×D511yescharacteristic, complemented
107C2×C4×D3C2×D512yescomplemented
108C2×Dic6C2×D511yescharacteristic, complemented
109#217—52no—
110#217—54no—
111(C2×D3)⋊C4—54no—
112C4⋊Dic3—52no—
113Dic3⋊C4—54no—
114C4×Dic3—52no—
115C23×C5—32no—
116C23×C5C22×D311yescharacteristic
117C22×D5—22no—
118C22×D5—24no—
119C22×D5C22×D312yes—
120C5×Q8—62no—
121C5×D4—64no—
122C5×D4—62no—
123C2×C4×C5—32no—
124C2×C4×C5—64no—
125C2×C4×C5#6714yescomplemented
126C2×C4×C5C22×D311yescharacteristic
127C2×C4×C5C22×D311yescharacteristic
128C22⋊D5—68no—
129C2×Dic5—22no—
130C2×Dic5—34no—
131C2×Dic5C22×D312yes—
132C4×D5—24no—
133C4×D5—24no—
134C2×#41—31no—
135C23×C4—51no—
136#123—152no—
137C22⋊Q8—152no—
138#121—152no—
139#120—152no—
140C4×D4—154no—
141#117—151no—
142C2×C3×C5—22no—
143C2×C3×C5C22×D411yescharacteristic
144C2×C3×C5C2×#4112yes—
145C3×D5—24no—
146C5×D3—42no—
147C23×C3—52no—
148C23×C3—104no—
149C23×C3—102no—
150C23×C3C22×D511yescharacteristic
151C22×D3C22×D512yes—
152C2×C3×C4—52no—
153C2×C3×C4—102no—
154C2×C3×C4—104no—
155C2×C3×C4—104no—
156C2×C3×C4C22⋊D514yescomplemented
157C2×C3×C4C22×D511yescharacteristic
158C2×C3×C4C22×D511yescharacteristic
159#67—24no—
160#67—108no—
161C2×Dic3—54no—
162C2×Dic3C22×D512yes—
163D12—22no—
164C4×D3—24no—
165Dic6—22no—
166C22×C5—22no—
167C22×C5—64no—
168C22×C5—62no—
169C22×C5C2×#6712yes—
170C22×C5C23×D311yescharacteristic
171C2×D5—28no—
172C2×D5—22no—
173C2×D5—22no—
174C2×D5#23514yescomplemented
175C4×C5—62no—
176C4×C5C2×#6712yes—
177C4×C5C2×#6712yes—
178Dic5—22no—
179Dic5—64no—
180C24—51no—
181#41—64no—
182C2×Q8—31no—
183C2×D4—32no—
184C2×D4—31no—
185C2×D4—154no—
186C22×C4—32no—
187C22×C4—51no—
188C22×C4—52no—
189C22×C4—52no—
190C22×C4—104no—
191C22×C4D3×D511yescharacteristic
192C4⋊C4—154no—
193C4⋊C4—152no—
194C22⋊C4—154no—
195C22⋊C4—154no—
196C22⋊C4—152no—
197C42—152no—
198C3×C5#52511yescharacteristic, complemented, Hall
199C22×C3—22no—
200C22×C3—54no—
201C22×C3—108no—
202C22×C3—102no—
203C22×C3—102no—
204C22×C3C2×C22⋊D512yes—
205C22×C3C23×D511yescharacteristic
206C2×D3—24no—
207C2×D3—22no—
208C3×C4—102no—
209C3×C4C2×C22⋊D512yes—
210C3×C4C2×C22⋊D512yes—
211Dic3—22no—
212Dic3—104no—
213C2×C5—22no—
214C2×C5—122no—
215C2×C5C2×#23512yes—
216C2×C5C22×#6711yescharacteristic
217D5—24no—
218C23—32no—
219C23—52no—
220C23—104no—
221C23—102no—
222C23C2×D3×D511yescharacteristic
223Q8—62no—
224D4—64no—
225D4—62no—
226D4—308no—
227C2×C4—32no—
228C2×C4—52no—
229C2×C4—64no—
230C2×C4—102no—
231C2×C4—104no—
232C2×C4—104no—
233C2×C4—154no—
234C2×C4#120114yescomplemented
235C2×C4C2×D3×D511yescharacteristic
236C2×C4C2×D3×D511yescharacteristic
237C2×C3—22no—
238C2×C3—104no—
239C2×C3D5×#4112yes—
240C2×C3C22×C22⋊D511yescharacteristic
241D3—42no—
242C5#579111yescharacteristic, complemented, Sylow
243C22—22no—
244C22—54no—
245C22—64no—
246C22—62no—
247C22—108no—
248C22—102no—
249C22—102no—
250C22C2×#120112yes—
251C22C22×D3×D511yescharacteristic
252C4—62no—
253C4—102no—
254C4—304no—
255C4C2×#120112yes—
256C4C2×#120112yes—
257C3#6616211yescharacteristic, complemented, Sylow
258C2—22no—
259C2—104no—
260C2—122no—
261C2D5×#23512yes—
262C2C22×#120111yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FrattiniC22C22×D3×D5
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#525
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#235C2semidirect
2. C22×C3×C4×D5C2semidirect
3. #91815C2semidirect
4. #91563C2semidirect
5. #91539C2semidirect
6. #91449C2semidirect
7. C4×#1201C2semidirect
8. #91359C2semidirect
14. C2×C5×D12C22semidirect
15. C2×C4×C5×D3C22semidirect
16. C2×C5×Dic6C22semidirect
23. C4⋊Dic15C22semidirect
24. C4×Dic15C22semidirect
25. #6669C22semidirect
26. (C2×D3)⋊Dic5C22semidirect
30. C22×C4×D5D3semidirect
42. C2×C3×C4×C5D4semidirect
57. C2×#235D5semidirect
91. C2×C3×D5#41semidirect
106. C2×D12C2×D5semidirect
107. C2×C4×D3C2×D5semidirect
108. C2×Dic6C2×D5semidirect
125. C2×C4×C5#67semidirect
156. C2×C3×C4C22⋊D5semidirect
174. C2×D5#235semidirect
198. C3×C5#525semidirect
234. C2×C4#1201semidirect
242. C5#5791semidirect
257. C3#66162semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#2352
C22×C3×C4×D52
#918152
#915632
#915392
#914492
C4×#12012
#913592
#661623
#57915