Finite Group Databasebeta

← C3×#65699 #11752232

Subgroups — C3×#65699 #11752232

Order 960.

468 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C23×C3×AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×C3×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C2×C3×C4⋊AGL(1,𝔽5)C212yesmaximal, complemented
4C3×D5×C22⋊C4C211yesmaximal, characteristic
5C3×(C2×AGL(1,𝔽5))⋊C4C212yesmaximal, complemented
6#65699C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×D5C2211yescharacteristic
8C22×C3×AGL(1,𝔽5)—22no—
9C22×C3×AGL(1,𝔽5)—24no—
10C22×C3×AGL(1,𝔽5)C2211yescharacteristic
11C22×C3×AGL(1,𝔽5)C2211yescharacteristic
12C22×C3×AGL(1,𝔽5)C2212yes—
13C2×C3×C4×D5C2212yes—
14C3×C22⋊AGL(1,𝔽5)—22no—
15C3×C4⋊AGL(1,𝔽5)—24no—
16C3×C5×C22⋊C4C411yescharacteristic, complemented
17C3×C22⋊Dic5C411yescharacteristic, complemented
18C3×#685C412yescomplemented
19C3×#385—51nomaximal, Hall
20C23×AGL(1,𝔽5)C2×C311yescharacteristic
21C2×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
22C2×C4⋊AGL(1,𝔽5)C2×C312yescomplemented
23D5×C22⋊C4C2×C311yescharacteristic
24(C2×AGL(1,𝔽5))⋊C4C2×C312yescomplemented
25C23×C3×C5C2×C411yescharacteristic
26C22×C3×D5—22no—
27C22×C3×D5—22no—
28C22×C3×D5C2×C411yescharacteristic
29C22×C3×D5D412yes—
30C22×C3×D5Q812yes—
31C22×C3×D5C2311yescharacteristic
32C2×C3×AGL(1,𝔽5)—28no—
33C2×C3×AGL(1,𝔽5)—22no—
34C2×C3×AGL(1,𝔽5)—22no—
35C2×C3×AGL(1,𝔽5)—22no—
36C2×C3×AGL(1,𝔽5)—21no—
37C2×C3×AGL(1,𝔽5)—21no—
38C2×C3×AGL(1,𝔽5)—22no—
39C2×C3×AGL(1,𝔽5)D414yes—
40C2×C3×C4×C5C2×C412yescomplemented
41C2×C3×Dic5C2×C412yescomplemented
42C3×C4×D5—22no—
43C3×C4×D5—22no—
44C23×C3×C4—51no—
45C2×C3×C4⋊C4—52no—
46C2×C3×C22⋊C4—51no—
47C2×C3×C22⋊C4—51no—
48C3×#95—52no—
49C23×D5C22×C311yescharacteristic
50C22×AGL(1,𝔽5)—22no—
51C22×AGL(1,𝔽5)—24no—
52C22×AGL(1,𝔽5)C22×C311yescharacteristic
53C22×AGL(1,𝔽5)C22×C311yescharacteristic
54C22×AGL(1,𝔽5)C22×C312yes—
55C2×C4×D5C22×C312yes—
56C22⋊AGL(1,𝔽5)—22no—
57C4⋊AGL(1,𝔽5)—24no—
58C5×C22⋊C4C3×C411yescharacteristic, complemented
59C22⋊Dic5C3×C411yescharacteristic, complemented
60#685C3×C412yescomplemented
61#385—51noSylow
62C22×C3×C5—22no—
63C22×C3×C5C4⋊C412yescomplemented
64C22×C3×C5C22×C411yescharacteristic
65C2×C3×D5—22no—
66C2×C3×D5—22no—
67C2×C3×D5—24no—
68C2×C3×D5—22no—
69C2×C3×D5C4⋊C412yescomplemented
70C2×C3×D5C2×D412yes—
71C2×C3×D5C2×D411yescharacteristic
72C2×C3×D5C2×Q811yescharacteristic
73C2×C3×D5#4112yes—
74C3×AGL(1,𝔽5)—24no—
75C3×AGL(1,𝔽5)—41no—
76C3×AGL(1,𝔽5)—41no—
77C3×C4×C5—22no—
78C3×Dic5—22no—
79C24×C3—51no—
80C22×C3×C4—52no—
81C22×C3×C4—52no—
82C22×C3×C4—51no—
83C22×C3×C4—51no—
84C22×C3×C4—104no—
85C22×C3×C4—102no—
86C3×C4⋊C4—104no—
87C3×C22⋊C4—51no—
88C3×C22⋊C4—52no—
89C3×C22⋊C4—102no—
90C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
91C23×C5C2×C3×C411yescharacteristic
92C22×D5—22no—
93C22×D5—22no—
94C22×D5C2×C3×C411yescharacteristic
95C22×D5C3×D412yes—
96C22×D5C3×Q812yes—
97C22×D5C23×C311yescharacteristic
98C2×AGL(1,𝔽5)—28no—
99C2×AGL(1,𝔽5)—22no—
100C2×AGL(1,𝔽5)—22no—
101C2×AGL(1,𝔽5)—22no—
102C2×AGL(1,𝔽5)—21no—
103C2×AGL(1,𝔽5)—21no—
104C2×AGL(1,𝔽5)—22no—
105C2×AGL(1,𝔽5)C3×D414yes—
106C2×C4×C5C2×C3×C412yescomplemented
107C2×Dic5C2×C3×C412yescomplemented
108C4×D5—22no—
109C4×D5—22no—
110C23×C4—51no—
111C2×C4⋊C4—52no—
112C2×C22⋊C4—51no—
113C2×C22⋊C4—51no—
114#95—52no—
115C2×C3×C5—22no—
116C2×C3×C5C2×C4⋊C411yescharacteristic
117C2×C3×C5C4×D412yes—
118C3×D5—22no—
119C3×D5#12011yescharacteristic
120C3×D5C22⋊Q812yes—
121C3×D5#12311yescharacteristic
122C23×C3—51no—
123C23×C3—52no—
124C23×C3—52no—
125C23×C3—51no—
126C23×C3—102no—
127C23×C3—102no—
128C23×C3C2×AGL(1,𝔽5)11yescharacteristic
129C2×C3×C4—54no—
130C2×C3×C4—52no—
131C2×C3×C4—102no—
132C2×C3×C4—102no—
133C2×C3×C4—102no—
134C2×C3×C4—102no—
135C2×C3×C4—108no—
136C2×C3×C4—101no—
137C2×C3×C4—101no—
138C2×C3×C4—102no—
139C2×C3×C4—102no—
140C2×C3×C4C2×AGL(1,𝔽5)12yescomplemented
141C22×C5—22no—
142C22×C5C3×C4⋊C412yescomplemented
143C22×C5C22×C3×C411yescharacteristic
144C2×D5—22no—
145C2×D5—22no—
146C2×D5—24no—
147C2×D5—22no—
148C2×D5C3×C4⋊C412yescomplemented
149C2×D5C2×C3×D412yes—
150C2×D5C2×C3×D411yescharacteristic
151C2×D5C2×C3×Q811yescharacteristic
152C2×D5C3×#4112yes—
153AGL(1,𝔽5)—24no—
154AGL(1,𝔽5)—41no—
155AGL(1,𝔽5)—41no—
156C4×C5—22no—
157Dic5—22no—
158C24—51no—
159C22×C4—52no—
160C22×C4—52no—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4—104no—
164C22×C4—102no—
165C4⋊C4—104no—
166C22⋊C4—51no—
167C22⋊C4—52no—
168C22⋊C4—102no—
169C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic, complemented
170C3×C5#38511yescharacteristic, complemented, Hall
171C22×C3—22no—
172C22×C3—51no—
173C22×C3—51no—
174C22×C3—52no—
175C22×C3—52no—
176C22×C3—52no—
177C22×C3—102no—
178C22×C3—102no—
179C22×C3—104no—
180C22×C3—102no—
181C22×C3C4⋊AGL(1,𝔽5)12yescomplemented
182C22×C3C22×AGL(1,𝔽5)11yescharacteristic
183C3×C4—22no—
184C3×C4—104no—
185C3×C4—102no—
186C3×C4—201no—
187C3×C4—201no—
188C2×C5—22no—
189C2×C5C2×C3×C4⋊C411yescharacteristic
190C2×C5C3×C4×D412yes—
191D5—22no—
192D5C3×#12011yescharacteristic
193D5C3×C22⋊Q812yes—
194D5C3×#12311yescharacteristic
195C23—51no—
196C23—52no—
197C23—52no—
198C23—51no—
199C23—102no—
200C23—102no—
201C23C2×C3×AGL(1,𝔽5)11yescharacteristic
202C2×C4—54no—
203C2×C4—52no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—108no—
209C2×C4—101no—
210C2×C4—101no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4C2×C3×AGL(1,𝔽5)12yescomplemented
214C2×C3—22no—
215C2×C3—51no—
216C2×C3—51no—
217C2×C3—52no—
218C2×C3—102no—
219C2×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
220C2×C3D4×AGL(1,𝔽5)12yes—
221C5C3×#38511yescharacteristic, complemented, Sylow
222C22—22no—
223C22—51no—
224C22—51no—
225C22—52no—
226C22—52no—
227C22—52no—
228C22—102no—
229C22—102no—
230C22—104no—
231C22—102no—
232C22C3×C4⋊AGL(1,𝔽5)12yescomplemented
233C22C22×C3×AGL(1,𝔽5)11yescharacteristic
234C4—22no—
235C4—104no—
236C4—102no—
237C4—201no—
238C4—201no—
239C3#6569911yescharacteristic, direct product factor, Sylow
240C2—22no—
241C2—51no—
242C2—51no—
243C2—52no—
244C2—102no—
245C2C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
246C2C3×D4×AGL(1,𝔽5)12yes—
subgroupG/H
centerC22×C3C22×AGL(1,𝔽5)
commutatorC22×C5C22×C3×C4
FrattiniC22C22×C3×AGL(1,𝔽5)
FittingC3×C5×C22⋊C4C4
socleC22×C3×C5C22×C4
prime radicalC23×C3×D5C22
cosocleC22×D5C23×C3
hypercenterC3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#385
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊AGL(1,𝔽5)C2semidirect
5. C3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
6. #65699C3direct (N × H)
16. C3×C5×C22⋊C4C4semidirect
17. C3×C22⋊Dic5C4semidirect
18. C3×#685C4semidirect
22. C2×C4⋊AGL(1,𝔽5)C2×C3semidirect
24. (C2×AGL(1,𝔽5))⋊C4C2×C3semidirect
40. C2×C3×C4×C5C2×C4semidirect
41. C2×C3×Dic5C2×C4semidirect
58. C5×C22⋊C4C3×C4semidirect
59. C22⋊Dic5C3×C4semidirect
60. #685C3×C4semidirect
63. C22×C3×C5C4⋊C4semidirect
69. C2×C3×D5C4⋊C4semidirect
90. C3×C22⋊C4AGL(1,𝔽5)semidirect
106. C2×C4×C5C2×C3×C4semidirect
107. C2×Dic5C2×C3×C4semidirect
140. C2×C3×C4C2×AGL(1,𝔽5)semidirect
142. C22×C5C3×C4⋊C4semidirect
148. C2×D5C3×C4⋊C4semidirect
169. C22⋊C4C3×AGL(1,𝔽5)semidirect
170. C3×C5#385semidirect
181. C22×C3C4⋊AGL(1,𝔽5)semidirect
213. C2×C4C2×C3×AGL(1,𝔽5)semidirect
221. C5C3×#385semidirect
232. C22C3×C4⋊AGL(1,𝔽5)semidirect
239. C3#65699direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×AGL(1,𝔽5)2
C2×C3×C22⋊AGL(1,𝔽5)2
C2×C3×C4⋊AGL(1,𝔽5)2
C3×D5×C22⋊C42
C3×(C2×AGL(1,𝔽5))⋊C42
#656993
C3×#3855