Finite Group Databasebeta

← C2×D52×A4 #72231280

Subgroups — C2×D52×A4 #72231280

Order 2400.

790 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 84 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×A4×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×C5×D5×A4C212yesmaximal, complemented
3D52×A4C214yesmaximal, direct product factor
4C23×D52C311yesmaximal, characteristic, complemented, Hall
5C2×C52×A4C2211yescharacteristic, complemented
6C2×C3×D52—41nomaximal
7A4×C5⋊D5C2212yescomplemented
8C5×D5×A4C2214yescomplemented
9C22×D5×A4—52nomaximal
10C23×C5⋊D5C2×C311yescharacteristic, complemented
11C23×C5×D5C2×C312yescomplemented
12C22×D52—31no—
13C22×D52—31no—
14C22×D52—32no—
15C22×D52—34no—
16C22×D52C2×C314yescomplemented
17C2×C3×C5⋊D5—41no—
18C2×C3×C5×D5—42no—
19C52×A4C2311yescharacteristic, complemented
20C3×D52—44no—
21C22×C5×A4—52no—
22C2×D5×A4—52no—
23C2×D5×A4—58no—
24C2×D5×A4—102no—
25C2×D5×A4D512yesdirect product factor
26C23×C52C22×C311yescharacteristic, complemented
27C22×C5⋊D5—31no—
28C22×C5⋊D5—31no—
29C22×C5⋊D5—32no—
30C22×C5⋊D5C22×C312yescomplemented
31C22×C5×D5—32no—
32C22×C5×D5—32no—
33C22×C5×D5—34no—
34C22×C5×D5C22×C314yescomplemented
35C2×D52—32no—
36C2×D52—38no—
37C2×D52—34no—
38C2×D52—31no—
39C2×D52—32no—
40C2×D52—34no—
41C2×D52—34no—
42C2×D52—38no—
43C2×D52—34no—
44C2×D52A411yescharacteristic, direct product factor
45C24×D5—52no—
46C2×C3×C52—41no—
47C3×C5⋊D5—42no—
48C3×C5×D5—44no—
49C22×C3×D5—202no—
50C2×C5×A4—22no—
51C2×C5×A4—54no—
52C2×C5×A4C2×D512yescomplemented
53D5×A4—54no—
54D5×A4—104no—
55D5×A4C2×D514yesdirect product factor
56C22×C52—31no—
57C22×C52—31no—
58C22×C52C23×C311yescharacteristic, complemented
59C2×C5⋊D5—32no—
60C2×C5⋊D5—31no—
61C2×C5⋊D5—32no—
62C2×C5⋊D5—32no—
63C2×C5⋊D5—32no—
64C2×C5⋊D5C2×A411yescharacteristic, complemented
65C2×C5×D5—34no—
66C2×C5×D5—34no—
67C2×C5×D5—34no—
68C2×C5×D5—34no—
69C2×C5×D5—32no—
70C2×C5×D5C2×A412yescomplemented
71D52—38no—
72D52—38no—
73D52—34no—
74D52C2×A414yesdirect product factor
75C23×A4—251noHall
76C24×C5—52no—
77C23×D5—52no—
78C23×D5—58no—
79C23×D5—102no—
80C23×D5—152no—
81C23×D5—152no—
82C23×D5—158no—
83C23×D5—152no—
84C23×D5—152no—
85C23×D5C3×D512yescomplemented
86C3×C52—41noHall
87C22×C3×C5—202no—
88C2×C3×D5—42no—
89C2×C3×D5—208no—
90C2×C3×D5—202no—
91C2×C3×D5—402no—
92C5×A4—22no—
93C5×A4C22×D512yescomplemented
94C2×C52—31no—
95C2×C52—31no—
96C2×C52C22×A411yescharacteristic, complemented
97C5⋊D5—32no—
98C5⋊D5C22×A412yescomplemented
99C5×D5—34no—
100C5×D5C22×A414yescomplemented
101C22×A4—52no—
102C22×A4—254no—
103C22×A4—251no—
104C23×C5—22no—
105C23×C5—54no—
106C23×C5—152no—
107C23×C5—154no—
108C23×C5—152no—
109C23×C5C2×C3×D512yescomplemented
110C22×D5—32no—
111C22×D5—32no—
112C22×D5—34no—
113C22×D5—52no—
114C22×D5—54no—
115C22×D5—104no—
116C22×D5—158no—
117C22×D5—152no—
118C22×D5—158no—
119C22×D5—152no—
120C22×D5—158no—
121C22×D5—158no—
122C22×D5—154no—
123C22×D5—158no—
124C22×D5—158no—
125C22×D5—158no—
126C22×D5—158no—
127C22×D5—152no—
128C22×D5—152no—
129C22×D5—154no—
130C22×D5—152no—
131C22×D5—302no—
132C22×D5—304no—
133C22×D5—302no—
134C22×D5C2×C3×D514yescomplemented
135C25—251noSylow
136C2×C3×C5—42no—
137C2×C3×C5—82no—
138C2×C3×C5—204no—
139C3×D5—44no—
140C3×D5—204no—
141C3×D5—404no—
142C52C23×A411yescharacteristic, complemented, Sylow
143C23×C3—1001no—
144C2×A4—54no—
145C2×A4—252no—
146C2×A4D5211yescharacteristic, direct product factor
147C22×C5—22no—
148C22×C5—32no—
149C22×C5—32no—
150C22×C5—52no—
151C22×C5—62no—
152C22×C5—62no—
153C22×C5—154no—
154C22×C5—154no—
155C22×C5—154no—
156C22×C5—154no—
157C22×C5—152no—
158C22×C5C22×C3×D512yescomplemented
159C2×D5—34no—
160C2×D5—32no—
161C2×D5—34no—
162C2×D5—34no—
163C2×D5—34no—
164C2×D5—52no—
165C2×D5—58no—
166C2×D5—102no—
167C2×D5—158no—
168C2×D5—158no—
169C2×D5—1516no—
170C2×D5—154no—
171C2×D5—154no—
172C2×D5—154no—
173C2×D5—152no—
174C2×D5—158no—
175C2×D5—154no—
176C2×D5—304no—
177C2×D5—304no—
178C2×D5—302no—
179C2×D5—304no—
180C2×D5—304no—
181C2×D5D5×A412yesdirect product factor
182C24—52no—
183C24—254no—
184C24—251no—
185C24—751no—
186C24—752no—
187C24—751no—
188C24—754no—
189C3×C5—42no—
190C3×C5—82no—
191C22×C3—202no—
192C22×C3—1001no—
193C22×C3—1004no—
194A4C2×D5211yescharacteristic, direct product factor
195C2×C5—22no—
196C2×C5—32no—
197C2×C5—32no—
198C2×C5—54no—
199C2×C5—62no—
200C2×C5—62no—
201C2×C5—154no—
202C2×C5C2×D5×A412yescomplemented
203D5—34no—
204D5—54no—
205D5—104no—
206D5—154no—
207D5—304no—
208D5C2×D5×A414yesdirect product factor
209C23—54no—
210C23—152no—
211C23—154no—
212C23—152no—
213C23—251no—
214C23—252no—
215C23—751no—
216C23—752no—
217C23—751no—
218C23—752no—
219C23—754no—
220C23—754no—
221C23—751no—
222C23—758no—
223C23—754no—
224C23—758no—
225C23—754no—
226C23—752no—
227C23C3×D5211yescharacteristic, complemented
228C2×C3—41no—
229C2×C3—204no—
230C2×C3—1002no—
231C5—22no—
232C5C22×D5×A412yescomplemented
233C22—31no—
234C22—31no—
235C22—52no—
236C22—154no—
237C22—154no—
238C22—154no—
239C22—154no—
240C22—152no—
241C22—254no—
242C22—251no—
243C22—752no—
244C22—752no—
245C22—752no—
246C22—752no—
247C22—758no—
248C22—754no—
249C22—758no—
250C22—751no—
251C22C2×C3×D5211yescharacteristic, complemented
252C3—41noSylow
253C2—31no—
254C2—31no—
255C2—54no—
256C2—154no—
257C2—252no—
258C2—752no—
259C2D52×A411yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D52×A4
commutator, cosocle, hypocenterC22×C52C23×C3
FrattiniC1G
Fitting, socleC23×C52C22×C3
LCSG ▷ C22×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×A4×C5⋊D5C2semidirect
2. C2×C5×D5×A4C2semidirect
3. D52×A4C2direct (N × H)
4. C23×D52C3semidirect
5. C2×C52×A4C22semidirect
7. A4×C5⋊D5C22semidirect
8. C5×D5×A4C22semidirect
10. C23×C5⋊D5C2×C3semidirect
11. C23×C5×D5C2×C3semidirect
16. C22×D52C2×C3semidirect
19. C52×A4C23semidirect
25. C2×D5×A4D5direct (N × H)
26. C23×C52C22×C3semidirect
30. C22×C5⋊D5C22×C3semidirect
34. C22×C5×D5C22×C3semidirect
44. C2×D52A4direct (N × H)
52. C2×C5×A4C2×D5semidirect
55. D5×A4C2×D5direct (N × H)
58. C22×C52C23×C3semidirect
64. C2×C5⋊D5C2×A4semidirect
70. C2×C5×D5C2×A4semidirect
74. D52C2×A4direct (N × H)
85. C23×D5C3×D5semidirect
93. C5×A4C22×D5semidirect
96. C2×C52C22×A4semidirect
98. C5⋊D5C22×A4semidirect
100. C5×D5C22×A4semidirect
109. C23×C5C2×C3×D5semidirect
134. C22×D5C2×C3×D5semidirect
142. C52C23×A4semidirect
146. C2×A4D52direct (N × H)
158. C22×C5C22×C3×D5semidirect
181. C2×D5D5×A4direct (N × H)
194. A4C2×D52direct (N × H)
202. C2×C5C2×D5×A4semidirect
208. D5C2×D5×A4direct (N × H)
227. C23C3×D52semidirect
232. C5C22×D5×A4semidirect
251. C22C2×C3×D52semidirect
259. C2D52×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×A4×C5⋊D52
C2×C5×D5×A42
D52×A42
C23×D523
C2×C3×D524
C22×D5×A45