Finite Group Databasebeta

← #11751854

Subgroups — #11751854

Order 960.

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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×C4×D3×D5C211yesmaximal, characteristic
2C2×Dic3⋊AGL(1,𝔽5)C212yesmaximal, complemented
3C2×(C2×D3)⋊AGL(1,𝔽5)C212yesmaximal
4#91235C211yesmaximal, characteristic, complemented
5C3×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic, complemented
6#65765—31nomaximal, Hall
7C22×D3×D5C2211yescharacteristic
8C22×ΓL(1,𝔽16)C2212yes—
9C22×C3×AGL(1,𝔽5)C2212yes—
10C2×C4×D15C411yescharacteristic, complemented
11C2×C4×C5×D3C411yescharacteristic, complemented
12C2×C3×C4×D5C2211yescharacteristic
13C2×#1200C411yescharacteristic, complemented
14C2×D3×Dic5C411yescharacteristic, complemented
15C2×D5×Dic3C2211yescharacteristic
16C4×D3×D5—24no—
17Dic3⋊AGL(1,𝔽5)—24no—
18(C2×D3)⋊AGL(1,𝔽5)—24no—
19#4667—51nomaximal, Hall
20C22×C4×D5—31no—
21C2×C22⋊AGL(1,𝔽5)—32no—
22C2×C4⋊AGL(1,𝔽5)—32no—
23(C2×AGL(1,𝔽5))⋊C4—31no—
24(C2×AGL(1,𝔽5))⋊C4D311yescharacteristic, complemented
25C22×D15C2×C411yescharacteristic
26C22×C5×D3C2×C411yescharacteristic
27C22×C3×D5C2311yescharacteristic
28C2×D3×D5—22no—
29C2×D3×D5—22no—
30C2×D3×D5D412yes—
31C2×D3×D5Q812yes—
32C2×ΓL(1,𝔽16)—22no—
33C2×ΓL(1,𝔽16)—22no—
34C2×ΓL(1,𝔽16)—22no—
35C2×C3×AGL(1,𝔽5)—22no—
36C2×C3×AGL(1,𝔽5)—22no—
37C2×C3×AGL(1,𝔽5)—22no—
38C2×C3×C4×C5C2×C411yescharacteristic
39C2×Dic15C2×C411yescharacteristic
40C4×D15—22no—
41C2×C5×Dic3C2×C411yescharacteristic
42C4×C5×D3—22no—
43C2×C3×Dic5C2×C411yescharacteristic
44C3×C4×D5—22no—
45#1200—22no—
46D3×Dic5—22no—
47D5×Dic3D414yes—
48C22×C4×D3—51no—
49C2×(C2×D3)⋊C4—52no—
50C2×Dic3⋊C4—52no—
51C3×#95—51no—
52C3⋊#95—51no—
53C23×D5—31no—
54C22×AGL(1,𝔽5)—32no—
55C22×AGL(1,𝔽5)C2×D312yes—
56C22×C4×C5—31no—
57C22×Dic5—31no—
58C2×C4×D5—31no—
59C2×C4×D5—31no—
60C2×C4×D5—31no—
61C2×C4×D5—64no—
62C2×C4×D5C2×D311yescharacteristic
63C22⋊AGL(1,𝔽5)—64no—
64C4⋊AGL(1,𝔽5)—64no—
65#380—151noSylow
66C22×C3×C5C22×C411yescharacteristic
67C2×D15—21no—
68C2×D15—21no—
69C2×D15C4⋊C412yescomplemented
70C2×C5×D3—21no—
71C2×C5×D3—21no—
72C2×C5×D3C4⋊C412yescomplemented
73C2×C3×D5C2×D411yescharacteristic
74C2×C3×D5C2×D412yes—
75C2×C3×D5C2×Q811yescharacteristic
76C2×C3×D5#4112yes—
77D3×D5—24no—
78D3×D5—24no—
79ΓL(1,𝔽16)—42no—
80C3×AGL(1,𝔽5)—42no—
81C3×C4×C5—21no—
82Dic15C22⋊C412yescomplemented
83C3×Dic5—21no—
84C5×Dic3C22⋊C412yescomplemented
85C23×D3—51no—
86C22×C3×C4—51no—
87C22×C3×C4—52no—
88C22×Dic3—51no—
89C22×Dic3—52no—
90C2×C4×D3—51no—
91C2×C4×D3—51no—
92C2×C4×D3—51no—
93C2×C4×D3—104no—
94C2×C4×D3AGL(1,𝔽5)11yescharacteristic, complemented
95(C2×D3)⋊C4—104no—
96Dic3⋊C4—104no—
97C23×C5—31no—
98C22×D5—31no—
99C22×D5—32no—
100C22×D5—32no—
101C22×D5—62no—
102C22×D5—62no—
103C22×D5C22×D311yescharacteristic
104C2×AGL(1,𝔽5)—22no—
105C2×AGL(1,𝔽5)—22no—
106C2×AGL(1,𝔽5)—22no—
107C2×AGL(1,𝔽5)—62no—
108C2×AGL(1,𝔽5)—62no—
109C2×AGL(1,𝔽5)—62no—
110C2×C4×C5—31no—
111C2×C4×C5—62no—
112C2×C4×C5C4×D311yescharacteristic
113C2×Dic5—31no—
114C2×Dic5—62no—
115C2×Dic5C4×D311yescharacteristic
116C4×D5—22no—
117C4×D5—34no—
118C4×D5—62no—
119C4×D5—62no—
120C23×C4—151no—
121C2×C4⋊C4—152no—
122C2×C22⋊C4—152no—
123#95—51no—
124#95—151no—
125C2×C3×C5C2×C22⋊C411yescharacteristic
126C2×C3×C5C2×C4⋊C411yescharacteristic
127C2×C3×C5#11711yescharacteristic
128D15—22no—
129C3×D5#12112yes—
130C3×D5C22⋊Q812yes—
131C5×D3—22no—
132C23×C3—51no—
133C22×D3—51no—
134C22×D3—52no—
135C22×D3—52no—
136C22×D3—102no—
137C22×D3—102no—
138C22×D3C2×AGL(1,𝔽5)11yescharacteristic
139C2×C3×C4—51no—
140C2×C3×C4—102no—
141C2×C3×C4—102no—
142C2×C3×C4—102no—
143C2×C3×C4—102no—
144C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
145C2×Dic3—51no—
146C2×Dic3—54no—
147C2×Dic3—102no—
148C2×Dic3—102no—
149C2×Dic3—102no—
150C2×Dic3C2×AGL(1,𝔽5)11yescharacteristic
151C4×D3—22no—
152C4×D3—102no—
153C4×D3—102no—
154C4×D3—102no—
155C22×C5—32no—
156C22×C5—61no—
157C22×C5—61no—
158C22×C5C2×C4×D311yescharacteristic
159C2×D5—32no—
160C2×D5—61no—
161C2×D5—61no—
162C2×D5—64no—
163C2×D5—64no—
164C2×D5#23512yes—
165C2×D5D3×D411yescharacteristic
166C2×D5D3×D412yes—
167C2×D5D3×Q811yescharacteristic
168AGL(1,𝔽5)—42no—
169AGL(1,𝔽5)—122no—
170C4×C5—21no—
171C4×C5—32no—
172Dic5—21no—
173Dic5—32no—
174C24—151no—
175C22×C4—31no—
176C22×C4—51no—
177C22×C4—52no—
178C22×C4—151no—
179C22×C4—151no—
180C22×C4—151no—
181C22×C4—151no—
182C22×C4—152no—
183C22×C4—304no—
184C4⋊C4—304no—
185C22⋊C4—304no—
186C3×C5#38011yescharacteristic, complemented, Hall
187C22×C3—51no—
188C22×C3—51no—
189C22×C3—52no—
190C22×C3—52no—
191C22×C3C22×AGL(1,𝔽5)11yescharacteristic
192C2×D3—21no—
193C2×D3—21no—
194C2×D3—52no—
195C2×D3—104no—
196C2×D3—104no—
197C2×D3—101no—
198C2×D3—101no—
199C2×D3C4⋊AGL(1,𝔽5)12yescomplemented
200C3×C4—21no—
201C3×C4—101no—
202C3×C4—202no—
203Dic3—52no—
204Dic3—202no—
205Dic3C22⋊AGL(1,𝔽5)12yescomplemented
206C2×C5—62no—
207C2×C5#87211yescharacteristic
208C2×C5D3×C22⋊C411yescharacteristic
209C2×C5D3×C4⋊C411yescharacteristic
210D5—62no—
211D5#88412yes—
212D5#89612yes—
213C23—31no—
214C23—51no—
215C23—152no—
216C23—152no—
217C23—151no—
218C23—302no—
219C23—302no—
220C2×C4—31no—
221C2×C4—51no—
222C2×C4—62no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—102no—
226C2×C4—102no—
227C2×C4—154no—
228C2×C4—151no—
229C2×C4—302no—
230C2×C4—302no—
231C2×C4—302no—
232C2×C4—302no—
233C2×C4—302no—
234C2×C4—302no—
235C2×C4D3×AGL(1,𝔽5)11yescharacteristic
236C2×C3—52no—
237C2×C3—52no—
238C2×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
239C2×C3#413011yescharacteristic
240C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
241D3—22no—
242D3—102no—
243C5#466711yescharacteristic, complemented, Sylow
244C22—32no—
245C22—52no—
246C22—51no—
247C22—51no—
248C22—52no—
249C22—61no—
250C22—61no—
251C22—152no—
252C22—304no—
253C22—304no—
254C22—301no—
255C22—301no—
256C22C2×D3×AGL(1,𝔽5)11yescharacteristic
257C4—21no—
258C4—32no—
259C4—101no—
260C4—152no—
261C4—202no—
262C4—602no—
263C3#6576511yescharacteristic, complemented, Sylow
264C2—52no—
265C2—52no—
266C2—62no—
267C2—302no—
268C2#9190911yescharacteristic
269C2D3×C4⋊AGL(1,𝔽5)11yescharacteristic
270C2D3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C22×C4
FittingC2×C3×C4×C5C2×C4
prime radicalC22×D3×D5C22
cosocleC22×C3×D5C23
hypercenterC2×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#380
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3⋊AGL(1,𝔽5)C2semidirect
4. #91235C2semidirect
5. C3×(C2×AGL(1,𝔽5))⋊C4C2semidirect
10. C2×C4×D15C4semidirect
11. C2×C4×C5×D3C4semidirect
13. C2×#1200C4semidirect
14. C2×D3×Dic5C4semidirect
24. (C2×AGL(1,𝔽5))⋊C4D3semidirect
69. C2×D15C4⋊C4semidirect
72. C2×C5×D3C4⋊C4semidirect
82. Dic15C22⋊C4semidirect
84. C5×Dic3C22⋊C4semidirect
94. C2×C4×D3AGL(1,𝔽5)semidirect
186. C3×C5#380semidirect
199. C2×D3C4⋊AGL(1,𝔽5)semidirect
205. Dic3C22⋊AGL(1,𝔽5)semidirect
243. C5#4667semidirect
263. C3#65765semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D52
C2×Dic3⋊AGL(1,𝔽5)2
C2×(C2×D3)⋊AGL(1,𝔽5)2
#912352
C3×(C2×AGL(1,𝔽5))⋊C42
#657653
#46675