Finite Group Databasebeta

← #11752028

Subgroups — #11752028

Order 960.

468 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 108 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×D5×Dic3C211yesmaximal, characteristic
2C2×#6763C211yesmaximal, characteristic
3C2×C3×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4C2×Dic3⋊AGL(1,𝔽5)C212yesmaximal, complemented
5(C2×AGL(1,𝔽5))⋊Dic3C212yesmaximal, complemented
6#65765—31nomaximal, Hall
7C23×C3×D5C2211yescharacteristic
8C22×ΓL(1,𝔽16)C2212yes—
9C22×C3×AGL(1,𝔽5)C2212yes—
10C22×Dic15C411yescharacteristic, complemented
11C22×C5×Dic3C411yescharacteristic, complemented
12C2×D5×Dic3—24no—
13C2×D5×Dic3C411yescharacteristic, complemented
14C2×D5×Dic3C411yescharacteristic, complemented
15C2×D5×Dic3C2211yescharacteristic
16C2×D5×Dic3C2211yescharacteristic
17#6763—22no—
18C3×C22⋊AGL(1,𝔽5)—22no—
19Dic3⋊AGL(1,𝔽5)—24no—
20#4942—51nomaximal, Hall
21C22×C4×D5—31no—
22C2×C22⋊AGL(1,𝔽5)—31no—
23C2×C22⋊AGL(1,𝔽5)D311yescharacteristic
24C2×C4⋊AGL(1,𝔽5)—32no—
25(C2×AGL(1,𝔽5))⋊C4—32no—
26C23×C3×C5C2×C411yescharacteristic
27C22×C3×D5—22no—
28C22×C3×D5—22no—
29C22×C3×D5C2×C411yescharacteristic
30C22×C3×D5D412yes—
31C22×C3×D5Q812yes—
32C22×C3×D5C2311yescharacteristic
33C2×ΓL(1,𝔽16)—22no—
34C2×ΓL(1,𝔽16)—22no—
35C2×ΓL(1,𝔽16)—22no—
36C2×C3×AGL(1,𝔽5)—22no—
37C2×C3×AGL(1,𝔽5)—22no—
38C2×C3×AGL(1,𝔽5)—22no—
39C2×Dic15—22no—
40C2×Dic15C2×C411yescharacteristic
41C2×Dic15C2×C411yescharacteristic
42C2×C5×Dic3—22no—
43C2×C5×Dic3C2×C411yescharacteristic
44C2×C5×Dic3C2×C411yescharacteristic
45D5×Dic3—22no—
46D5×Dic3—22no—
47D5×Dic3—22no—
48D5×Dic3D414yes—
49C23×Dic3—51no—
50C2×C3×C22⋊C4—51no—
51C2×#217—51no—
52C2×Dic3⋊C4—52no—
53C3⋊#95—52no—
54C23×D5C2×D311yescharacteristic
55C22×AGL(1,𝔽5)—32no—
56C22×AGL(1,𝔽5)C2×D312yes—
57C22×C4×C5—31no—
58C22×Dic5—31no—
59C2×C4×D5—31no—
60C2×C4×D5—31no—
61C2×C4×D5—31no—
62C2×C4×D5—31no—
63C2×C4×D5—64no—
64C22⋊AGL(1,𝔽5)—22no—
65C22⋊AGL(1,𝔽5)—62no—
66C4⋊AGL(1,𝔽5)—64no—
67#380—151noSylow
68C22×C3×C5—21no—
69C22×C3×C5—21no—
70C22×C3×C5C4⋊C412yescomplemented
71C22×C3×C5C22×C411yescharacteristic
72C2×C3×D5—21no—
73C2×C3×D5—21no—
74C2×C3×D5—24no—
75C2×C3×D5—24no—
76C2×C3×D5C4⋊C412yescomplemented
77C2×C3×D5C2×D412yes—
78C2×C3×D5C2×D411yescharacteristic
79C2×C3×D5C2×Q811yescharacteristic
80C2×C3×D5#4112yes—
81ΓL(1,𝔽16)—42no—
82C3×AGL(1,𝔽5)—42no—
83Dic15—21no—
84Dic15C22⋊C412yescomplemented
85C5×Dic3—21no—
86C5×Dic3C22⋊C412yescomplemented
87C24×C3—51no—
88C22×C3×C4—52no—
89C22×Dic3—51no—
90C22×Dic3—51no—
91C22×Dic3—51no—
92C22×Dic3—51no—
93C22×Dic3—51no—
94C22×Dic3—52no—
95C22×Dic3—104no—
96C22×Dic3AGL(1,𝔽5)11yescharacteristic, complemented
97C3×C22⋊C4—102no—
98#217—102no—
99Dic3⋊C4—104no—
100C23×C5C4×D311yescharacteristic
101C22×D5—22no—
102C22×D5—22no—
103C22×D5Dic612yes—
104C22×D5C4×D311yescharacteristic
105C22×D5#6712yes—
106C22×D5C22×D311yescharacteristic
107C2×AGL(1,𝔽5)—22no—
108C2×AGL(1,𝔽5)—22no—
109C2×AGL(1,𝔽5)—22no—
110C2×AGL(1,𝔽5)—62no—
111C2×AGL(1,𝔽5)—62no—
112C2×AGL(1,𝔽5)—62no—
113C2×C4×C5—31no—
114C2×C4×C5—31no—
115C2×C4×C5—62no—
116C2×Dic5—31no—
117C2×Dic5—31no—
118C2×Dic5—62no—
119C4×D5—34no—
120C4×D5—62no—
121C4×D5—62no—
122C4×D5—62no—
123C23×C4—151no—
124C2×C4⋊C4—152no—
125C2×C22⋊C4—51no—
126C2×C22⋊C4—151no—
127#95—152no—
128C2×C3×C5—22no—
129C2×C3×C5C2×C22⋊C411yescharacteristic
130C2×C3×C5C2×C4⋊C411yescharacteristic
131C2×C3×C5#11711yescharacteristic
132C3×D5—22no—
133C3×D5#12112yes—
134C3×D5C22⋊Q812yes—
135C23×C3—51no—
136C23×C3—52no—
137C23×C3—52no—
138C23×C3—51no—
139C23×C3—102no—
140C23×C3—102no—
141C23×C3C2×AGL(1,𝔽5)11yescharacteristic
142C2×C3×C4—102no—
143C2×C3×C4—102no—
144C2×C3×C4—102no—
145C2×Dic3—22no—
146C2×Dic3—51no—
147C2×Dic3—54no—
148C2×Dic3—51no—
149C2×Dic3—102no—
150C2×Dic3—102no—
151C2×Dic3—102no—
152C2×Dic3—102no—
153C2×Dic3—102no—
154C2×Dic3—102no—
155C2×Dic3—102no—
156C2×Dic3C2×AGL(1,𝔽5)11yescharacteristic
157C2×Dic3C2×AGL(1,𝔽5)11yescharacteristic
158C22×C5—21no—
159C22×C5—21no—
160C22×C5Dic3⋊C412yescomplemented
161C22×C5C2×C4×D311yescharacteristic
162C2×D5—21no—
163C2×D5—21no—
164C2×D5—24no—
165C2×D5—24no—
166C2×D5Dic3⋊C412yescomplemented
167C2×D5C2×Dic611yescharacteristic
168C2×D5D3×D412yes—
169C2×D5#23712yes—
170C2×D5C2×#6711yescharacteristic
171AGL(1,𝔽5)—42no—
172AGL(1,𝔽5)—122no—
173C4×C5—32no—
174C4×C5—61no—
175Dic5—32no—
176Dic5—61no—
177C24—51no—
178C22×C4—31no—
179C22×C4—52no—
180C22×C4—151no—
181C22×C4—151no—
182C22×C4—151no—
183C22×C4—151no—
184C22×C4—151no—
185C22×C4—152no—
186C22×C4—304no—
187C4⋊C4—304no—
188C22⋊C4—102no—
189C22⋊C4—302no—
190C3×C5#38011yescharacteristic, complemented, Hall
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—51no—
194C22×C3—51no—
195C22×C3—52no—
196C22×C3—52no—
197C22×C3—52no—
198C22×C3—104no—
199C22×C3—104no—
200C22×C3—101no—
201C22×C3—101no—
202C22×C3C4⋊AGL(1,𝔽5)12yescomplemented
203C22×C3C22×AGL(1,𝔽5)11yescharacteristic
204C3×C4—202no—
205Dic3—21no—
206Dic3—52no—
207Dic3—101no—
208Dic3—202no—
209Dic3C22⋊AGL(1,𝔽5)12yescomplemented
210C2×C5—22no—
211C2×C5#87711yescharacteristic
212C2×C5D3×C22⋊C411yescharacteristic
213C2×C5C2×Dic3⋊C411yescharacteristic
214D5—22no—
215D5#87812yes—
216D5#93912yes—
217C23—51no—
218C23—52no—
219C23—52no—
220C23—51no—
221C23—102no—
222C23—102no—
223C23D3×AGL(1,𝔽5)11yescharacteristic
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—62no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—154no—
231C2×C4—151no—
232C2×C4—151no—
233C2×C4—302no—
234C2×C4—302no—
235C2×C4—302no—
236C2×C4—302no—
237C2×C4—302no—
238C2×C4—302no—
239C2×C4—302no—
240C2×C3—22no—
241C2×C3—52no—
242C2×C3—52no—
243C2×C3—102no—
244C2×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
245C2×C3#413011yescharacteristic
246C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
247C5#494211yescharacteristic, complemented, Sylow
248C22—21no—
249C22—21no—
250C22—52no—
251C22—51no—
252C22—51no—
253C22—52no—
254C22—52no—
255C22—104no—
256C22—104no—
257C22—101no—
258C22—101no—
259C22Dic3⋊AGL(1,𝔽5)12yescomplemented
260C22C2×D3×AGL(1,𝔽5)11yescharacteristic
261C4—32no—
262C4—61no—
263C4—152no—
264C4—202no—
265C4—301no—
266C4—602no—
267C3#6576511yescharacteristic, complemented, Sylow
268C2—22no—
269C2—52no—
270C2—52no—
271C2—102no—
272C2#9192311yescharacteristic
273C2C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
274C2D3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C22×C4
FittingC23×C3×C5C2×C4
prime radicalC23×C3×D5C22
cosocleC22×C3×D5C23
hypercenterC23D3×AGL(1,𝔽5)
hypocenterC3×C5#380
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊AGL(1,𝔽5)C2semidirect
5. (C2×AGL(1,𝔽5))⋊Dic3C2semidirect
10. C22×Dic15C4semidirect
11. C22×C5×Dic3C4semidirect
13. C2×D5×Dic3C4semidirect
14. C2×D5×Dic3C4semidirect
70. C22×C3×C5C4⋊C4semidirect
76. C2×C3×D5C4⋊C4semidirect
84. Dic15C22⋊C4semidirect
86. C5×Dic3C22⋊C4semidirect
96. C22×Dic3AGL(1,𝔽5)semidirect
160. C22×C5Dic3⋊C4semidirect
166. C2×D5Dic3⋊C4semidirect
190. C3×C5#380semidirect
202. C22×C3C4⋊AGL(1,𝔽5)semidirect
209. Dic3C22⋊AGL(1,𝔽5)semidirect
247. C5#4942semidirect
259. C22Dic3⋊AGL(1,𝔽5)semidirect
267. C3#65765semidirect

Maximal subgroups

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

subgroupindex
C22×D5×Dic32
C2×#67632
C2×C3×C22⋊AGL(1,𝔽5)2
C2×Dic3⋊AGL(1,𝔽5)2
(C2×AGL(1,𝔽5))⋊Dic32
#657653
#49425