Finite Group Databasebeta

← C2×C3×#4007 #11752299

Subgroups — C2×C3×#4007 #11752299

Order 960.

404 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 120 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×C3×D4×D5C211yesmaximal, characteristic
2C2×C3×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C2×C3×D5⋊C8C211yesmaximal, characteristic, complemented
4C3×#4007C214yesmaximal, direct product factor
5C2×#4007C311yesmaximal, characteristic, direct product factor, Hall
6C23×C3×D5—21no—
7C22×C3×AGL(1,𝔽5)—21no—
8C2×C3×C5×D4C411yescharacteristic, complemented
9C2×C3×C22⋊D5—21no—
10C3×D4×D5—21no—
11C3×D4×D5—21no—
12C3×D4×D5C412yescomplemented
13C3×D4×D5C2212yes—
14C2×C3×D20C411yescharacteristic, complemented
15C2×C3×C4×D5C2211yescharacteristic
16C2×C3×#170—21no—
17C3×C4⋊AGL(1,𝔽5)—21no—
18C3×C4⋊AGL(1,𝔽5)C2212yescomplemented
19C3×D5⋊C8—21no—
20C3×D5⋊C8C2212yescomplemented
21C2×C3×#102—51nomaximal, Hall
22C2×D4×D5C2×C311yescharacteristic
23C2×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
24C2×D5⋊C8C2×C311yescharacteristic, complemented
25#4007C2×C314yesdirect product factor
26C23×C3×C5—21no—
27C22×C3×D5—21no—
28C22×C3×D5—22no—
29C22×C3×D5—22no—
30C22×C3×D5—42no—
31C22×C3×D5—42no—
32C22×C3×D5D411yescharacteristic
33C2×C3×AGL(1,𝔽5)—22no—
34C2×C3×AGL(1,𝔽5)—42no—
35C3×C5×D4—21no—
36C3×C5×D4C2×C412yescomplemented
37C2×C3×C4×C5C2×C411yescharacteristic
38C3×C22⋊D5—22no—
39C3×C22⋊D5—22no—
40C2×C3×Dic5D411yescharacteristic
41C3×D20—21no—
42C3×D20C2×C412yescomplemented
43C3×C4×D5C2×C411yescharacteristic
44C3×C4×D5D411yescharacteristic
45C3×C4×D5D411yescharacteristic
46C3×C4×D5C2311yescharacteristic
47C3×#170—22no—
48C22×C3×D4—51no—
49C22×C3×C8—51no—
50C2×C3×C4⋊C4—51no—
51C3×#102—54no—
52C23×D5—21no—
53C22×AGL(1,𝔽5)—21no—
54C2×C5×D4C3×C411yescharacteristic, complemented
55C2×C22⋊D5—21no—
56D4×D5—21no—
57D4×D5—21no—
58D4×D5C3×C412yescomplemented
59D4×D5C22×C312yes—
60C2×D20C3×C411yescharacteristic, complemented
61C2×C4×D5C22×C311yescharacteristic
62C2×#170—21no—
63C4⋊AGL(1,𝔽5)—21no—
64C4⋊AGL(1,𝔽5)C22×C312yescomplemented
65D5⋊C8—21no—
66D5⋊C8C22×C312yescomplemented
67C2×#102—51noSylow
68C22×C3×C5—22no—
69C22×C3×C5—42no—
70C22×C3×C5C22⋊C411yescharacteristic
71C2×C3×D5—22no—
72C2×C3×D5—42no—
73C2×C3×D5—44no—
74C2×C3×D5—42no—
75C2×C3×D5—42no—
76C2×C3×D5C22⋊C411yescharacteristic
77C2×C3×D5D812yes—
78C2×C3×D5SD1612yes—
79C2×C3×D5C2×D411yescharacteristic
80C3×AGL(1,𝔽5)—42no—
81C3×C4×C5C22⋊C411yescharacteristic
82C3×C4×C5C22×C411yescharacteristic
83C3×Dic5C22⋊C411yescharacteristic
84C3×Dic5C2×D411yescharacteristic
85C24×C3—101no—
86C2×C3×D4—52no—
87C2×C3×D4—52no—
88C2×C3×D4—51no—
89C2×C3×D4—101no—
90C2×C3×D4—101no—
91C2×C3×D4—101no—
92C2×C3×D4AGL(1,𝔽5)11yescharacteristic, complemented
93C22×C3×C4—51no—
94C22×C3×C4—101no—
95C2×C3×C8—52no—
96C2×C3×C8—101no—
97C2×C3×C8—101no—
98C3×C4⋊C4—52no—
99C3×C4⋊C4—101no—
100C23×C5—21no—
101C22×D5—21no—
102C22×D5—22no—
103C22×D5—22no—
104C22×D5—42no—
105C22×D5—42no—
106C22×D5C3×D411yescharacteristic
107C2×AGL(1,𝔽5)—22no—
108C2×AGL(1,𝔽5)—42no—
109C5×D4—21no—
110C5×D4C2×C3×C412yescomplemented
111C2×C4×C5C2×C3×C411yescharacteristic
112C22⋊D5—22no—
113C22⋊D5—22no—
114C2×Dic5C3×D411yescharacteristic
115D20—21no—
116D20C2×C3×C412yescomplemented
117C4×D5C2×C3×C411yescharacteristic
118C4×D5C3×D411yescharacteristic
119C4×D5C3×D411yescharacteristic
120C4×D5C23×C311yescharacteristic
121#170—22no—
122C22×D4—51no—
123C22×C8—51no—
124C2×C4⋊C4—51no—
125#102—54no—
126C2×C3×C5—42no—
127C2×C3×C5#10212yescomplemented
128C2×C3×C5C2×C22⋊C411yescharacteristic
129C3×D5—42no—
130C3×D5#10212yescomplemented
131C3×D5C2×D811yescharacteristic
132C3×D5C2×SD1611yescharacteristic
133C23×C3—21no—
134C23×C3—51no—
135C23×C3—102no—
136C23×C3—102no—
137C23×C3—101no—
138C23×C3—202no—
139C23×C3—202no—
140C3×D4—21no—
141C3×D4—52no—
142C3×D4—102no—
143C3×D4—102no—
144C3×D4—101no—
145C3×D4C2×AGL(1,𝔽5)12yescomplemented
146C2×C3×C4—51no—
147C2×C3×C4—51no—
148C2×C3×C4—51no—
149C2×C3×C4—51no—
150C2×C3×C4—51no—
151C2×C3×C4—102no—
152C2×C3×C4—202no—
153C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
154C3×C8—102no—
155C22×C5—22no—
156C22×C5—42no—
157C22×C5C3×C22⋊C411yescharacteristic
158C2×D5—22no—
159C2×D5—42no—
160C2×D5—44no—
161C2×D5—42no—
162C2×D5—42no—
163C2×D5C3×C22⋊C411yescharacteristic
164C2×D5C3×D812yes—
165C2×D5C3×SD1612yes—
166C2×D5C2×C3×D411yescharacteristic
167AGL(1,𝔽5)—42no—
168C4×C5C3×C22⋊C411yescharacteristic
169C4×C5C22×C3×C411yescharacteristic
170Dic5C3×C22⋊C411yescharacteristic
171Dic5C2×C3×D411yescharacteristic
172C24—101no—
173C2×D4—52no—
174C2×D4—52no—
175C2×D4—51no—
176C2×D4—101no—
177C2×D4—101no—
178C2×D4—101no—
179C2×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
180C22×C4—51no—
181C22×C4—101no—
182C2×C8—52no—
183C2×C8—101no—
184C2×C8—101no—
185C4⋊C4—52no—
186C4⋊C4—101no—
187C3×C5C2×#10211yescharacteristic, complemented, Hall
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—51no—
191C22×C3—51no—
192C22×C3—52no—
193C22×C3—52no—
194C22×C3—102no—
195C22×C3—202no—
196C22×C3—204no—
197C22×C3—202no—
198C22×C3—202no—
199C22×C3C22⋊AGL(1,𝔽5)11yescharacteristic
200C3×C4—51no—
201C3×C4—51no—
202C3×C4—202no—
203C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
204C3×C4C22×AGL(1,𝔽5)11yescharacteristic
205C2×C5—42no—
206C2×C5C3×#10212yescomplemented
207C2×C5C2×C3×C22⋊C411yescharacteristic
208D5—42no—
209D5C3×#10212yescomplemented
210D5C2×C3×D811yescharacteristic
211D5C2×C3×SD1611yescharacteristic
212C23—21no—
213C23—51no—
214C23—102no—
215C23—102no—
216C23—101no—
217C23—202no—
218C23—202no—
219D4—21no—
220D4—52no—
221D4—102no—
222D4—102no—
223D4—101no—
224D4C2×C3×AGL(1,𝔽5)12yescomplemented
225C2×C4—51no—
226C2×C4—51no—
227C2×C4—51no—
228C2×C4—51no—
229C2×C4—51no—
230C2×C4—102no—
231C2×C4—202no—
232C2×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
233C8—102no—
234C2×C3—42no—
235C2×C3—51no—
236C2×C3—52no—
237C2×C3—51no—
238C2×C3—202no—
239C2×C3#400712yesdirect product factor
240C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
241C5C2×C3×#10211yescharacteristic, complemented, Sylow
242C22—22no—
243C22—42no—
244C22—52no—
245C22—52no—
246C22—51no—
247C22—51no—
248C22—102no—
249C22—202no—
250C22—204no—
251C22—202no—
252C22—202no—
253C22C3×C22⋊AGL(1,𝔽5)11yescharacteristic
254C4—51no—
255C4—51no—
256C4—202no—
257C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
258C4C22×C3×AGL(1,𝔽5)11yescharacteristic
259C3C2×#400711yescharacteristic, direct product factor, Sylow
260C2—42no—
261C2—51no—
262C2—52no—
263C2—51no—
264C2—202no—
265C2C3×#400712yesdirect product factor
266C2C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC22×C3C22⋊AGL(1,𝔽5)
commutatorC4×C5C22×C3×C4
FrattiniC4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×D4C4
socleC22×C3×C5C22⋊C4
prime radicalC2×C3×D4×D5C2
cosocleC4×D5C23×C3
hypercenterC2×C3×D4AGL(1,𝔽5)
hypocenterC5C2×C3×#102
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊AGL(1,𝔽5)C2semidirect
3. C2×C3×D5⋊C8C2semidirect
4. C3×#4007C2direct (N × H)
5. C2×#4007C3direct (N × H)
8. C2×C3×C5×D4C4semidirect
12. C3×D4×D5C4semidirect
14. C2×C3×D20C4semidirect
18. C3×C4⋊AGL(1,𝔽5)C22semidirect
20. C3×D5⋊C8C22semidirect
23. C2×C4⋊AGL(1,𝔽5)C2×C3semidirect
24. C2×D5⋊C8C2×C3semidirect
25. #4007C2×C3direct (N × H)
36. C3×C5×D4C2×C4semidirect
42. C3×D20C2×C4semidirect
54. C2×C5×D4C3×C4semidirect
58. D4×D5C3×C4semidirect
60. C2×D20C3×C4semidirect
64. C4⋊AGL(1,𝔽5)C22×C3semidirect
66. D5⋊C8C22×C3semidirect
92. C2×C3×D4AGL(1,𝔽5)semidirect
110. C5×D4C2×C3×C4semidirect
116. D20C2×C3×C4semidirect
127. C2×C3×C5#102semidirect
130. C3×D5#102semidirect
145. C3×D4C2×AGL(1,𝔽5)semidirect
179. C2×D4C3×AGL(1,𝔽5)semidirect
187. C3×C5C2×#102semidirect
206. C2×C5C3×#102semidirect
209. D5C3×#102semidirect
224. D4C2×C3×AGL(1,𝔽5)semidirect
239. C2×C3#4007direct (N × H)
241. C5C2×C3×#102semidirect
259. C3C2×#4007direct (N × H)
265. C2C3×#4007direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D52
C2×C3×C4⋊AGL(1,𝔽5)2
C2×C3×D5⋊C82
C3×#40072
C2×#40073
C2×C3×#1025