Finite Group Databasebeta

← C2×C5×#560⋊C3 #14511360

Subgroups — C2×C5×#560⋊C3 #14511360

Order 1920.

608 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C5×#560⋊C3C214yesmaximal, direct product factor
2C2×C5×#127⋊C3C211yesmaximal, characteristic, complemented
3C2×C5×#126⋊C3C212yesmaximal, complemented
4C2×C5×#560C311yesmaximal, characteristic, complemented, Hall
5C23×C5×A4—41nomaximal
6C5×#127⋊C3C2212yescomplemented
7C5×#126⋊C3C2214yescomplemented
8C2×C5×C42⋊C3C2211yescharacteristic, complemented
9C2×#560⋊C3C511yesmaximal, characteristic, direct product factor, Hall
10C5×#560—34no—
11C5×#560C2×C314yescomplemented
12C2×C5×#127C2×C311yescharacteristic, complemented
13C2×C5×#126C2×C312yescomplemented
14C2×C5×#123—31no—
15C2×C5×#121—32no—
16C2×C5×#120—31no—
17C22×C5×A4—42no—
18C22×C5×A4—41no—
19C22×C5×A4—44no—
20C5×C42⋊C3C2311yescharacteristic, complemented
21#560⋊C3C2×C514yesdirect product factor
22C2×#127⋊C3C2×C511yescharacteristic, complemented
23C2×#126⋊C3C2×C512yescomplemented
24C25×C5A411yescharacteristic
25C22×C5×D4—31no—
26C22×C5×D4—32no—
27C22×C5×D4—31no—
28C23×C4×C5—31no—
29C5×#127—32no—
30C5×#127C22×C312yescomplemented
31C5×#126—34no—
32C5×#126C22×C314yescomplemented
33C5×#123—34no—
34C5×#123—34no—
35C5×#121—38no—
36C5×#121—38no—
37C5×#120—34no—
38C5×#120—34no—
39C2×C5×C4⋊C4—32no—
40C2×C5×C22⋊C4—31no—
41C2×C5×C22⋊C4—32no—
42C2×C5×C22⋊C4—31no—
43C2×C42×C5C22×C311yescharacteristic, complemented
44C2×#560C3×C511yescharacteristic, complemented, Sylow
45C23×C3×C5—161no—
46C2×C5×A4—41no—
47C2×C5×A4—44no—
48C2×C5×A4—42no—
49C23×A4—41no—
50#127⋊C3C22×C512yescomplemented
51#126⋊C3C22×C514yescomplemented
52C2×C42⋊C3C22×C511yescharacteristic, complemented
53C24×C5—31no—
54C24×C5—121no—
55C24×C5—121no—
56C24×C5C2×A412yes—
57C24×C5C2×A411yescharacteristic
58C24×C5C2×A414yes—
59C2×C5×D4—32no—
60C2×C5×D4—32no—
61C2×C5×D4—38no—
62C2×C5×D4—32no—
63C2×C5×D4—32no—
64C2×C5×D4—122no—
65C2×C5×D4—122no—
66C2×C5×D4—121no—
67C2×C5×D4—121no—
68C2×C5×D4—121no—
69C2×C5×D4—121no—
70C22×C4×C5—31no—
71C22×C4×C5—32no—
72C22×C4×C5—34no—
73C22×C4×C5—121no—
74C22×C4×C5—121no—
75C5×C4⋊C4—34no—
76C5×C4⋊C4—34no—
77C5×C22⋊C4—34no—
78C5×C22⋊C4—34no—
79C5×C22⋊C4—34no—
80C5×C22⋊C4—34no—
81C42×C5—31no—
82C42×C5C23×C311yescharacteristic, complemented
83#560—34no—
84#560C2×C3×C514yescomplemented
85C2×#127C2×C3×C511yescharacteristic, complemented
86C2×#126C2×C3×C512yescomplemented
87C2×#123—31no—
88C2×#121—32no—
89C2×#120—31no—
90C22×C3×C5—164no—
91C22×C3×C5—162no—
92C22×C3×C5—161no—
93C5×A4—41no—
94C22×A4—42no—
95C22×A4—41no—
96C22×A4—44no—
97C42⋊C3C23×C511yescharacteristic, complemented
98C23×C5—32no—
99C23×C5—41no—
100C23×C5—62no—
101C23×C5—61no—
102C23×C5—62no—
103C23×C5—61no—
104C23×C5—61no—
105C23×C5—61no—
106C23×C5—62no—
107C23×C5—62no—
108C23×C5—121no—
109C23×C5—124no—
110C23×C5—124no—
111C23×C5C22×A411yescharacteristic
112C23×C5C22×A414yes—
113C23×C5C22×A412yes—
114C5×D4—128no—
115C5×D4—122no—
116C5×D4—122no—
117C5×D4—122no—
118C5×D4—122no—
119C2×C4×C5—31no—
120C2×C4×C5—31no—
121C2×C4×C5—34no—
122C2×C4×C5—62no—
123C2×C4×C5—62no—
124C2×C4×C5—61no—
125C2×C4×C5—61no—
126C2×C4×C5—124no—
127C25C5×A411yescharacteristic
128C22×D4—31no—
129C22×D4—32no—
130C22×D4—31no—
131C23×C4—31no—
132#127—32no—
133#127C22×C3×C512yescomplemented
134#126—34no—
135#126C22×C3×C514yescomplemented
136#123—34no—
137#123—34no—
138#121—38no—
139#121—38no—
140#120—34no—
141#120—34no—
142C2×C4⋊C4—32no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—32no—
145C2×C22⋊C4—31no—
146C2×C42C22×C3×C511yescharacteristic, complemented
147C2×C3×C5—164no—
148C2×C3×C5—162no—
149C2×C3×C5—161no—
150C23×C3—161no—
151C2×A4—41no—
152C2×A4—44no—
153C2×A4—42no—
154C22×C5—31no—
155C22×C5—31no—
156C22×C5—44no—
157C22×C5—42no—
158C22×C5—41no—
159C22×C5—64no—
160C22×C5—62no—
161C22×C5—62no—
162C22×C5—64no—
163C22×C5—122no—
164C22×C5—121no—
165C22×C5—122no—
166C22×C5—121no—
167C22×C5—124no—
168C22×C5—121no—
169C22×C5C23×A411yescharacteristic
170C4×C5—64no—
171C4×C5—61no—
172C4×C5—61no—
173C24—31no—
174C24—121no—
175C24—121no—
176C24C2×C5×A412yes—
177C24C2×C5×A411yescharacteristic
178C24C2×C5×A414yes—
179C2×D4—32no—
180C2×D4—32no—
181C2×D4—38no—
182C2×D4—32no—
183C2×D4—32no—
184C2×D4—122no—
185C2×D4—122no—
186C2×D4—121no—
187C2×D4—121no—
188C2×D4—121no—
189C2×D4—121no—
190C22×C4—31no—
191C22×C4—32no—
192C22×C4—34no—
193C22×C4—121no—
194C22×C4—121no—
195C4⋊C4—34no—
196C4⋊C4—34no—
197C22⋊C4—34no—
198C22⋊C4—34no—
199C22⋊C4—34no—
200C22⋊C4—34no—
201C42—31no—
202C42C23×C3×C511yescharacteristic, complemented
203C3×C5—161noHall
204C22×C3—164no—
205C22×C3—162no—
206C22×C3—161no—
207A4—41no—
208C2×C5—31no—
209C2×C5—31no—
210C2×C5—44no—
211C2×C5—42no—
212C2×C5—122no—
213C2×C5#560⋊C311yescharacteristic, direct product factor
214C23—32no—
215C23—41no—
216C23—62no—
217C23—61no—
218C23—62no—
219C23—61no—
220C23—61no—
221C23—61no—
222C23—62no—
223C23—62no—
224C23—121no—
225C23—124no—
226C23—124no—
227C23C22×C5×A411yescharacteristic
228C23C22×C5×A414yes—
229C23C22×C5×A412yes—
230D4—128no—
231D4—122no—
232D4—122no—
233D4—122no—
234D4—122no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—34no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—124no—
243C2×C3—164no—
244C2×C3—162no—
245C2×C3—161no—
246C5C2×#560⋊C311yescharacteristic, direct product factor, Sylow
247C22—31no—
248C22—31no—
249C22—44no—
250C22—42no—
251C22—41no—
252C22—64no—
253C22—62no—
254C22—62no—
255C22—64no—
256C22—122no—
257C22—121no—
258C22—122no—
259C22—121no—
260C22—124no—
261C22—121no—
262C22C23×C5×A411yescharacteristic
263C4—64no—
264C4—61no—
265C4—61no—
266C3—161noSylow
267C2—31no—
268C2—31no—
269C2—44no—
270C2—42no—
271C2—122no—
272C2C5×#560⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C5#560⋊C3
commutator, cosocle, hypocenterC42C23×C3×C5
FrattiniC22C23×C5×A4
FittingC2×C5×#560C3
socleC23×C5C22×A4
LCSG ▷ C42
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
1. C5×#560⋊C3C2direct (N × H)
2. C2×C5×#127⋊C3C2semidirect
3. C2×C5×#126⋊C3C2semidirect
4. C2×C5×#560C3semidirect
6. C5×#127⋊C3C22semidirect
7. C5×#126⋊C3C22semidirect
8. C2×C5×C42⋊C3C22semidirect
9. C2×#560⋊C3C5direct (N × H)
11. C5×#560C2×C3semidirect
12. C2×C5×#127C2×C3semidirect
13. C2×C5×#126C2×C3semidirect
20. C5×C42⋊C3C23semidirect
21. #560⋊C3C2×C5direct (N × H)
22. C2×#127⋊C3C2×C5semidirect
23. C2×#126⋊C3C2×C5semidirect
30. C5×#127C22×C3semidirect
32. C5×#126C22×C3semidirect
43. C2×C42×C5C22×C3semidirect
44. C2×#560C3×C5semidirect
50. #127⋊C3C22×C5semidirect
51. #126⋊C3C22×C5semidirect
52. C2×C42⋊C3C22×C5semidirect
82. C42×C5C23×C3semidirect
84. #560C2×C3×C5semidirect
85. C2×#127C2×C3×C5semidirect
86. C2×#126C2×C3×C5semidirect
97. C42⋊C3C23×C5semidirect
133. #127C22×C3×C5semidirect
135. #126C22×C3×C5semidirect
146. C2×C42C22×C3×C5semidirect
202. C42C23×C3×C5semidirect
213. C2×C5#560⋊C3direct (N × H)
246. C5C2×#560⋊C3direct (N × H)
272. C2C5×#560⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#560⋊C32
C2×C5×#127⋊C32
C2×C5×#126⋊C32
C2×C5×#5603
C23×C5×A44
C2×#560⋊C35