Finite Group Databasebeta

← C3×#65765 #11752298

Subgroups — C3×#65765 #11752298

Order 960.

468 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C22×C3×C4×D5C211yesmaximal, characteristic
2C2×C3×C22⋊AGL(1,𝔽5)C212yesmaximal
3C2×C3×C4⋊AGL(1,𝔽5)C212yesmaximal, complemented
4C3×(C2×AGL(1,𝔽5))⋊C4C212yesmaximal, complemented
5#65765C311yesmaximal, characteristic, direct product factor, Hall
6C23×C3×D5C2211yescharacteristic
7C22×C3×AGL(1,𝔽5)C2214yes—
8C22×C3×C4×C5C411yescharacteristic, complemented
9C22×C3×Dic5C411yescharacteristic, complemented
10C2×C3×C4×D5—24no—
11C2×C3×C4×D5C411yescharacteristic, complemented
12C2×C3×C4×D5C411yescharacteristic, complemented
13C2×C3×C4×D5C2211yescharacteristic
14C2×C3×C4×D5C2211yescharacteristic
15C3×C22⋊AGL(1,𝔽5)—24no—
16C3×C4⋊AGL(1,𝔽5)—24no—
17C3×#380—51nomaximal, Hall
18C22×C4×D5C2×C311yescharacteristic
19C2×C22⋊AGL(1,𝔽5)C2×C312yes—
20C2×C4⋊AGL(1,𝔽5)C2×C312yescomplemented
21(C2×AGL(1,𝔽5))⋊C4C2×C312yescomplemented
22C23×C3×C5C2×C411yescharacteristic
23C22×C3×D5—22no—
24C22×C3×D5—22no—
25C22×C3×D5C2×C411yescharacteristic
26C22×C3×D5D412yes—
27C22×C3×D5Q812yes—
28C22×C3×D5C2311yescharacteristic
29C2×C3×AGL(1,𝔽5)—24no—
30C2×C3×AGL(1,𝔽5)—24no—
31C2×C3×AGL(1,𝔽5)—24no—
32C2×C3×C4×C5—22no—
33C2×C3×C4×C5C2×C411yescharacteristic
34C2×C3×C4×C5C2×C411yescharacteristic
35C2×C3×Dic5—22no—
36C2×C3×Dic5C2×C411yescharacteristic
37C2×C3×Dic5C2×C411yescharacteristic
38C3×C4×D5—22no—
39C3×C4×D5—22no—
40C3×C4×D5—22no—
41C3×C4×D5D414yes—
42C23×C3×C4—51no—
43C2×C3×C4⋊C4—52no—
44C2×C3×C22⋊C4—52no—
45C3×#95—52no—
46C23×D5C22×C311yescharacteristic
47C22×AGL(1,𝔽5)C22×C314yes—
48C22×C4×C5C3×C411yescharacteristic, complemented
49C22×Dic5C3×C411yescharacteristic, complemented
50C2×C4×D5—24no—
51C2×C4×D5C3×C411yescharacteristic, complemented
52C2×C4×D5C3×C411yescharacteristic, complemented
53C2×C4×D5C22×C311yescharacteristic
54C2×C4×D5C22×C311yescharacteristic
55C22⋊AGL(1,𝔽5)—24no—
56C4⋊AGL(1,𝔽5)—24no—
57#380—51noSylow
58C22×C3×C5—21no—
59C22×C3×C5—21no—
60C22×C3×C5C4⋊C412yescomplemented
61C22×C3×C5C22×C411yescharacteristic
62C2×C3×D5—21no—
63C2×C3×D5—21no—
64C2×C3×D5—24no—
65C2×C3×D5—24no—
66C2×C3×D5C4⋊C412yescomplemented
67C2×C3×D5C2×D411yescharacteristic
68C2×C3×D5C2×D412yes—
69C2×C3×D5C2×Q811yescharacteristic
70C2×C3×D5#4112yes—
71C3×AGL(1,𝔽5)—44no—
72C3×C4×C5—21no—
73C3×C4×C5C22⋊C412yescomplemented
74C3×Dic5—21no—
75C3×Dic5C22⋊C412yescomplemented
76C24×C3—51no—
77C22×C3×C4—51no—
78C22×C3×C4—51no—
79C22×C3×C4—51no—
80C22×C3×C4—51no—
81C22×C3×C4—51no—
82C22×C3×C4—54no—
83C22×C3×C4—104no—
84C22×C3×C4AGL(1,𝔽5)11yescharacteristic, complemented
85C3×C4⋊C4—104no—
86C3×C22⋊C4—104no—
87C23×C5C2×C3×C411yescharacteristic
88C22×D5—22no—
89C22×D5—22no—
90C22×D5C2×C3×C411yescharacteristic
91C22×D5C3×D412yes—
92C22×D5C3×Q812yes—
93C22×D5C23×C311yescharacteristic
94C2×AGL(1,𝔽5)—24no—
95C2×AGL(1,𝔽5)—24no—
96C2×AGL(1,𝔽5)—24no—
97C2×C4×C5—22no—
98C2×C4×C5C2×C3×C411yescharacteristic
99C2×C4×C5C2×C3×C411yescharacteristic
100C2×Dic5—22no—
101C2×Dic5C2×C3×C411yescharacteristic
102C2×Dic5C2×C3×C411yescharacteristic
103C4×D5—22no—
104C4×D5—22no—
105C4×D5—22no—
106C4×D5C3×D414yes—
107C23×C4—51no—
108C2×C4⋊C4—52no—
109C2×C22⋊C4—52no—
110#95—52no—
111C2×C3×C5—22no—
112C2×C3×C5C2×C22⋊C411yescharacteristic
113C2×C3×C5C2×C4⋊C411yescharacteristic
114C2×C3×C5#11711yescharacteristic
115C3×D5—22no—
116C3×D5#12112yes—
117C3×D5C22⋊Q812yes—
118C23×C3—51no—
119C23×C3—52no—
120C23×C3—52no—
121C23×C3—51no—
122C23×C3—102no—
123C23×C3—102no—
124C23×C3C2×AGL(1,𝔽5)11yescharacteristic
125C2×C3×C4—22no—
126C2×C3×C4—51no—
127C2×C3×C4—54no—
128C2×C3×C4—51no—
129C2×C3×C4—102no—
130C2×C3×C4—102no—
131C2×C3×C4—102no—
132C2×C3×C4—102no—
133C2×C3×C4—104no—
134C2×C3×C4—104no—
135C2×C3×C4—104no—
136C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
137C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
138C22×C5—21no—
139C22×C5—21no—
140C22×C5C3×C4⋊C412yescomplemented
141C22×C5C22×C3×C411yescharacteristic
142C2×D5—21no—
143C2×D5—21no—
144C2×D5—24no—
145C2×D5—24no—
146C2×D5C3×C4⋊C412yescomplemented
147C2×D5C2×C3×D411yescharacteristic
148C2×D5C2×C3×D412yes—
149C2×D5C2×C3×Q811yescharacteristic
150C2×D5C3×#4112yes—
151AGL(1,𝔽5)—44no—
152C4×C5—21no—
153C4×C5C3×C22⋊C412yescomplemented
154Dic5—21no—
155Dic5C3×C22⋊C412yescomplemented
156C24—51no—
157C22×C4—51no—
158C22×C4—51no—
159C22×C4—51no—
160C22×C4—51no—
161C22×C4—51no—
162C22×C4—54no—
163C22×C4—104no—
164C22×C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
165C4⋊C4—104no—
166C22⋊C4—104no—
167C3×C5#38011yescharacteristic, complemented, Hall
168C22×C3—21no—
169C22×C3—21no—
170C22×C3—51no—
171C22×C3—51no—
172C22×C3—52no—
173C22×C3—52no—
174C22×C3—52no—
175C22×C3—104no—
176C22×C3—104no—
177C22×C3—101no—
178C22×C3—101no—
179C22×C3C4⋊AGL(1,𝔽5)12yescomplemented
180C22×C3C22×AGL(1,𝔽5)11yescharacteristic
181C3×C4—21no—
182C3×C4—52no—
183C3×C4—101no—
184C3×C4—204no—
185C3×C4C22⋊AGL(1,𝔽5)12yescomplemented
186C2×C5—22no—
187C2×C5C2×C3×C22⋊C411yescharacteristic
188C2×C5C2×C3×C4⋊C411yescharacteristic
189C2×C5C3×#11711yescharacteristic
190D5—22no—
191D5C3×#12112yes—
192D5C3×C22⋊Q812yes—
193C23—51no—
194C23—52no—
195C23—52no—
196C23—51no—
197C23—102no—
198C23—102no—
199C23C2×C3×AGL(1,𝔽5)11yescharacteristic
200C2×C4—22no—
201C2×C4—51no—
202C2×C4—54no—
203C2×C4—51no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—104no—
209C2×C4—104no—
210C2×C4—104no—
211C2×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
212C2×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
213C2×C3—22no—
214C2×C3—52no—
215C2×C3—52no—
216C2×C3—102no—
217C2×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
218C2×C3#413011yescharacteristic
219C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
220C5C3×#38011yescharacteristic, complemented, Sylow
221C22—21no—
222C22—21no—
223C22—51no—
224C22—51no—
225C22—52no—
226C22—52no—
227C22—52no—
228C22—104no—
229C22—104no—
230C22—101no—
231C22—101no—
232C22C3×C4⋊AGL(1,𝔽5)12yescomplemented
233C22C22×C3×AGL(1,𝔽5)11yescharacteristic
234C4—21no—
235C4—52no—
236C4—101no—
237C4—204no—
238C4C3×C22⋊AGL(1,𝔽5)12yescomplemented
239C3#6576511yescharacteristic, direct product factor, Sylow
240C2—22no—
241C2—52no—
242C2—52no—
243C2—102no—
244C2C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
245C2C3×#413011yescharacteristic
246C2C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C22×AGL(1,𝔽5)
commutatorC22×C5C22×C3×C4
FrattiniC22C22×C3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C4
socleC22×C3×C5C22×C4
prime radicalC23×C3×D5C22
cosocleC22×D5C23×C3
hypercenterC22×C3×C4AGL(1,𝔽5)
hypocenterC5C3×#380
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊AGL(1,𝔽5)C2semidirect
4. C3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
5. #65765C3direct (N × H)
8. C22×C3×C4×C5C4semidirect
9. C22×C3×Dic5C4semidirect
11. C2×C3×C4×D5C4semidirect
12. C2×C3×C4×D5C4semidirect
20. C2×C4⋊AGL(1,𝔽5)C2×C3semidirect
21. (C2×AGL(1,𝔽5))⋊C4C2×C3semidirect
48. C22×C4×C5C3×C4semidirect
49. C22×Dic5C3×C4semidirect
51. C2×C4×D5C3×C4semidirect
52. C2×C4×D5C3×C4semidirect
60. C22×C3×C5C4⋊C4semidirect
66. C2×C3×D5C4⋊C4semidirect
73. C3×C4×C5C22⋊C4semidirect
75. C3×Dic5C22⋊C4semidirect
84. C22×C3×C4AGL(1,𝔽5)semidirect
140. C22×C5C3×C4⋊C4semidirect
146. C2×D5C3×C4⋊C4semidirect
153. C4×C5C3×C22⋊C4semidirect
155. Dic5C3×C22⋊C4semidirect
164. C22×C4C3×AGL(1,𝔽5)semidirect
167. C3×C5#380semidirect
179. C22×C3C4⋊AGL(1,𝔽5)semidirect
185. C3×C4C22⋊AGL(1,𝔽5)semidirect
220. C5C3×#380semidirect
232. C22C3×C4⋊AGL(1,𝔽5)semidirect
238. C4C3×C22⋊AGL(1,𝔽5)semidirect
239. C3#65765direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D52
C2×C3×C22⋊AGL(1,𝔽5)2
C2×C3×C4⋊AGL(1,𝔽5)2
C3×(C2×AGL(1,𝔽5))⋊C42
#657653
C3×#3805