Finite Group Databasebeta

← C4×D3×#1044 #72230636

Subgroups — C4×D3×#1044 #72230636

Order 2400.

540 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 102 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×D3×#1044C212yesmaximal
2C4×D3×C5⋊D5C211yesmaximal, characteristic
3C4×C3⋊#1044C211yesmaximal, characteristic, complemented
4C3×C4×#1044C211yesmaximal, characteristic, complemented
5Dic3×#1044C212yesmaximal, complemented
6C2×C4×#1044—31nomaximal, Hall
7C2×D3×C5⋊D5C2211yescharacteristic
8C2×C3⋊#1044C2212yes—
9C2×C3×#1044C2212yes—
10D3×#1044C418yesdirect product factor
11C4×C3⋊(C5⋊D5)C411yescharacteristic, complemented
12C4×C52×D3C411yescharacteristic, complemented
13C3×C4×C5⋊D5C2211yescharacteristic
14#10597113C411yescharacteristic, complemented
15D3×#1039C411yescharacteristic, complemented
16Dic3×C5⋊D5C2211yescharacteristic
17C4×D3×AGL(1,𝔽5)—52nomaximal
18C22×#1044—32no—
19C2×C4×C5⋊D5—31no—
20C4×#1044—32no—
21C4×#1044—31no—
22C4×#1044D311yescharacteristic, direct product factor
23C2×C3⋊(C5⋊D5)C2×C411yescharacteristic
24C2×C52×D3C2×C411yescharacteristic
25C2×C3×C5⋊D5C2311yescharacteristic
26D3×C5⋊D5C2×C414yes—
27C3⋊#1044C2×C414yescomplemented
28C3×#1044C2×C414yescomplemented
29C3×C4×C52C2×C411yescharacteristic, complemented
30C3⋊#1039C2×C411yescharacteristic, complemented
31C52×Dic3C2×C411yescharacteristic, complemented
32C3×#1039C2×C411yescharacteristic, complemented
33C2×D3×AGL(1,𝔽5)—54no—
34C4×D3×D5—52no—
35C4×D3×D5—102no—
36C4×ΓL(1,𝔽16)—52no—
37C3×C4×AGL(1,𝔽5)—52no—
38Dic3×AGL(1,𝔽5)—54no—
39C22×C5⋊D5—31no—
40C2×#1044—32no—
41C2×#1044—38no—
42C2×#1044C2×D312yes—
43C2×C4×C52—31no—
44C2×#1039—31no—
45C4×C5⋊D5—31no—
46C4×C5⋊D5—31no—
47C4×C5⋊D5—31no—
48C4×C5⋊D5C2×D311yescharacteristic
49C2×C4×AGL(1,𝔽5)—152no—
50C2×C3×C52C22×C411yescharacteristic
51C3⋊(C5⋊D5)C4212yescomplemented
52C52×D3C4212yescomplemented
53C3×C5⋊D5C22×C412yes—
54C2×D3×D5—52no—
55C2×D3×D5—102no—
56C2×ΓL(1,𝔽16)—54no—
57C2×C3×AGL(1,𝔽5)—54no—
58D3×AGL(1,𝔽5)—516no—
59C4×D15—52no—
60C4×D15—102no—
61C4×C5×D3—22no—
62C4×C5×D3AGL(1,𝔽5)12yescomplemented
63C3×C4×D5—52no—
64C3×C4×D5—102no—
65#1200—52no—
66#1200—102no—
67D3×Dic5—52no—
68D3×Dic5—102no—
69D5×Dic3—52no—
70D5×Dic3—102no—
71C22×C52—31no—
72C2×C5⋊D5—31no—
73C2×C5⋊D5—34no—
74C2×C5⋊D5C22×D311yescharacteristic
75#1044—34no—
76#1044C4×D314yesdirect product factor
77C4×C52—31no—
78C4×C52C4×D311yescharacteristic, complemented
79#1039—31no—
80#1039C4×D311yescharacteristic, complemented
81C42×D3—251noHall
82C22×AGL(1,𝔽5)—154no—
83C2×C4×D5—152no—
84C2×C4×D5—302no—
85C4×AGL(1,𝔽5)—52no—
86C4×AGL(1,𝔽5)—152no—
87C4×AGL(1,𝔽5)—154no—
88C3×C52C2×C4211yescharacteristic, complemented, Hall
89C2×D15—52no—
90C2×D15—102no—
91C2×C5×D3—22no—
92C2×C5×D3C2×AGL(1,𝔽5)12yes—
93C2×C3×D5—52no—
94C2×C3×D5—102no—
95D3×D5—58no—
96D3×D5—108no—
97ΓL(1,𝔽16)—58no—
98C3×AGL(1,𝔽5)—58no—
99C3×C4×C5—22no—
100C3×C4×C5C2×AGL(1,𝔽5)12yescomplemented
101Dic15—52no—
102Dic15—102no—
103C3×Dic5—52no—
104C3×Dic5—102no—
105C5×Dic3—22no—
106C5×Dic3C2×AGL(1,𝔽5)12yescomplemented
107C2×C52—32no—
108C2×C52C2×C4×D311yescharacteristic
109C5⋊D5—32no—
110C5⋊D5C2×C4×D312yes—
111C2×C4×D3—252no—
112C2×C4×D3—251no—
113C3×C42—251no—
114C4×Dic3—252no—
115C4×Dic3—251no—
116C22×D5—152no—
117C22×D5—302no—
118C2×AGL(1,𝔽5)—54no—
119C2×AGL(1,𝔽5)—1516no—
120C2×AGL(1,𝔽5)—154no—
121C2×C4×C5—32no—
122C2×C4×C5—62no—
123C2×Dic5—152no—
124C2×Dic5—302no—
125C4×D5—52no—
126C4×D5—102no—
127C4×D5—152no—
128C4×D5—152no—
129C4×D5—152no—
130C4×D5—302no—
131C4×D5—302no—
132C4×D5—302no—
133C2×C42—751noSylow
134C2×C3×C5—22no—
135C2×C3×C5C22×AGL(1,𝔽5)12yes—
136D15—54no—
137D15—104no—
138C3×D5—54no—
139C3×D5—104no—
140C5×D3—24no—
141C5×D3C4×AGL(1,𝔽5)14yescomplemented
142C52C42×D311yescharacteristic, complemented, Sylow
143C22×D3—251no—
144C2×C3×C4—252no—
145C2×C3×C4—251no—
146C2×Dic3—252no—
147C2×Dic3—251no—
148C4×D3—258no—
149C4×D3—251no—
150C4×D3—251no—
151C4×D3—251no—
152C4×D3#104411yescharacteristic, direct product factor
153C22×C5—32no—
154C22×C5—62no—
155C2×D5—52no—
156C2×D5—102no—
157C2×D5—158no—
158C2×D5—152no—
159C2×D5—308no—
160C2×D5—302no—
161AGL(1,𝔽5)—58no—
162AGL(1,𝔽5)—158no—
163C4×C5—22no—
164C4×C5—32no—
165C4×C5—62no—
166C4×C5D3×AGL(1,𝔽5)12yescomplemented
167Dic5—52no—
168Dic5—102no—
169Dic5—152no—
170Dic5—302no—
171C22×C4—751no—
172C22×C4—752no—
173C42—251no—
174C42—752no—
175C42—751no—
176C3×C5—22no—
177C3×C5C2×C4×AGL(1,𝔽5)12yescomplemented
178C22×C3—251no—
179C2×D3—251no—
180C2×D3—254no—
181C2×D3C2×#104411yescharacteristic
182C3×C4—254no—
183C3×C4—251no—
184C3×C4C2×#104411yescharacteristic, complemented
185Dic3—254no—
186Dic3—251no—
187Dic3C2×#104411yescharacteristic, complemented
188C2×C5—22no—
189C2×C5—34no—
190C2×C5—64no—
191C2×C5C2×D3×AGL(1,𝔽5)12yes—
192D5—54no—
193D5—104no—
194D5—154no—
195D5—304no—
196C23—751no—
197C2×C4—31no—
198C2×C4—252no—
199C2×C4—251no—
200C2×C4—751no—
201C2×C4—751no—
202C2×C4—751no—
203C2×C4—758no—
204C2×C4—752no—
205C2×C4—751no—
206C2×C3—252no—
207C2×C3C22×#104411yescharacteristic
208D3—252no—
209D3C4×#104412yesdirect product factor
210C5—22no—
211C5C4×D3×AGL(1,𝔽5)12yescomplemented
212C22—31no—
213C22—251no—
214C22—754no—
215C22—751no—
216C4—31no—
217C4—254no—
218C4—251no—
219C4—754no—
220C4—751no—
221C4D3×#104411yescharacteristic, direct product factor
222C3C2×C4×#104411yescharacteristic, complemented, Sylow
223C2—32no—
224C2—252no—
225C2—752no—
226C2C2×D3×#104411yescharacteristic
subgroupG/H
center, hypercenterC4D3×#1044
commutator, hypocenterC3×C52C2×C42
FrattiniC2C2×D3×#1044
FittingC3×C4×C52C2×C4
socleC2×C3×C52C22×C4
prime radicalC2×D3×C5⋊D5C22
cosocleC2×C3×C5⋊D5C23
LCSG ▷ C3×C52
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
3. C4×C3⋊#1044C2semidirect
4. C3×C4×#1044C2semidirect
5. Dic3×#1044C2semidirect
10. D3×#1044C4direct (N × H)
11. C4×C3⋊(C5⋊D5)C4semidirect
12. C4×C52×D3C4semidirect
14. #10597113C4semidirect
15. D3×#1039C4semidirect
22. C4×#1044D3direct (N × H)
27. C3⋊#1044C2×C4semidirect
28. C3×#1044C2×C4semidirect
29. C3×C4×C52C2×C4semidirect
30. C3⋊#1039C2×C4semidirect
31. C52×Dic3C2×C4semidirect
32. C3×#1039C2×C4semidirect
51. C3⋊(C5⋊D5)C42semidirect
52. C52×D3C42semidirect
62. C4×C5×D3AGL(1,𝔽5)semidirect
76. #1044C4×D3direct (N × H)
78. C4×C52C4×D3semidirect
80. #1039C4×D3semidirect
88. C3×C52C2×C42semidirect
100. C3×C4×C5C2×AGL(1,𝔽5)semidirect
106. C5×Dic3C2×AGL(1,𝔽5)semidirect
141. C5×D3C4×AGL(1,𝔽5)semidirect
142. C52C42×D3semidirect
152. C4×D3#1044direct (N × H)
166. C4×C5D3×AGL(1,𝔽5)semidirect
177. C3×C5C2×C4×AGL(1,𝔽5)semidirect
184. C3×C4C2×#1044semidirect
187. Dic3C2×#1044semidirect
209. D3C4×#1044direct (N × H)
211. C5C4×D3×AGL(1,𝔽5)semidirect
221. C4D3×#1044direct (N × H)
222. C3C2×C4×#1044semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#10442
C4×D3×C5⋊D52
C4×C3⋊#10442
C3×C4×#10442
Dic3×#10442
C2×C4×#10443
C4×D3×AGL(1,𝔽5)5