Finite Group Databasebeta

← C2×D5×#625 #13088988

Subgroups — C2×D5×#625 #13088988

Order 1440.

998 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C22×C3×D3×D5C212yesmaximal, complemented
2C2×#67157C211yesmaximal, characteristic, complemented
3C2×#67150C212yesmaximal, complemented
4C2×C5×#625C211yesmaximal, characteristic, complemented
5C2×D5×C3⋊Dic3C211yesmaximal, characteristic, complemented
6D5×#625C218yesmaximal, direct product factor
7C2×D5×#67—32nomaximal
8C22×C3×D15C2212yescomplemented
9C22×C3×C5×D3C2212yescomplemented
10C22×C32×D5C2211yescharacteristic
11C2×C3×D3×D5—28no—
12C2×C3×D3×D5C2218yescomplemented
13C2×C3⋊Dic15C2211yescharacteristic, complemented
14C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
15#67157C2214yescomplemented
16#67150C2218yescomplemented
17C5×#625C2214yescomplemented
18D5×C3⋊Dic3C2214yescomplemented
19C22×#625—51nomaximal, Hall
20C23×C3×D5—32no—
21C22×D3×D5D312yescomplemented
22C2×#1220—32no—
23C2×C5×#67—32no—
24D5×#67—316no—
25C2×#1202—32no—
26C2×#1201—32no—
27C2×D5×Dic3—32no—
28C2×D5×Dic3—61no—
29C22×C32×C5C2311yescharacteristic
30C2×C3×D15—24no—
31C2×C3×D15C2314yescomplemented
32C2×C3×C5×D3—24no—
33C2×C3×C5×D3C2314yescomplemented
34C2×C32×D5D414yescomplemented
35C2×C32×D5C2312yes—
36C3×D3×D5—216no—
37C3⋊Dic15C2312yescomplemented
38C5×C3⋊Dic3C2312yescomplemented
39C2×D4×D5—91noHall
40C23×C3×D3—52no—
41C22×C3⋊Dic3—51no—
42C2×#625—52no—
43C2×#625—51no—
44C2×#625—58no—
45C2×#625D511yescharacteristic, direct product factor
46C23×C3×C5—32no—
47C22×D15C2×D312yescomplemented
48C22×C5×D3C2×D312yescomplemented
49C22×C3×D5—21no—
50C22×C3×D5—32no—
51C22×C3×D5—38no—
52C22×C3×D5—68no—
53C22×C3×D5C2×D312yes—
54C2×D3×D5—28no—
55C2×D3×D5C2×D318yescomplemented
56#1220—38no—
57C2×Dic15—32no—
58C2×Dic15—61no—
59C5×#67—38no—
60C2×C5×Dic3—32no—
61C2×C5×Dic3—61no—
62#1202—38no—
63#1201—38no—
64D5×Dic3—38no—
65D5×Dic3—64no—
66C22×#67—152no—
67C2×C32×C5C2×D412yescomplemented
68C2×C32×C5C2411yescharacteristic
69C3×D15—24no—
70C3×C5×D3—24no—
71C32×D5C2×D414yescomplemented
72C23×D5—32no—
73C2×C5×D4—91no—
74C2×C22⋊D5—92no—
75D4×D5—98no—
76C2×D20—91no—
77C2×C4×D5—91no—
78C23×C32—51no—
79C22×C3×D3—52no—
80C22×C3×D3—58no—
81C22×C3×D3—108no—
82C22×C3×D3C2×D512yescomplemented
83C2×C3⋊Dic3—51no—
84C2×C3⋊Dic3—54no—
85C2×C3⋊Dic3C2×D511yescharacteristic, complemented
86#625—58no—
87#625—54no—
88#625C2×D514yesdirect product factor
89C22×C3×C5—21no—
90C22×C3×C5—34no—
91C22×C3×C5—64no—
92C22×C3×C5C22×D312yes—
93C2×D15—24no—
94C2×D15C22×D314yescomplemented
95C2×C5×D3—24no—
96C2×C5×D3C22×D314yescomplemented
97C2×C3×D5—24no—
98C2×C3×D5—22no—
99C2×C3×D5—34no—
100C2×C3×D5—64no—
101C2×C3×D5—616no—
102C2×C3×D5#6718yescomplemented
103C2×C3×D5C22×D314yes—
104D3×D5—216no—
105Dic15—34no—
106Dic15—62no—
107C5×Dic3—34no—
108C5×Dic3—62no—
109C24×C3—152no—
110C23×D3—52no—
111C2×#67—32no—
112C2×#67—152no—
113C2×#67—1516no—
114C2×#67—152no—
115C2×#67—152no—
116C22×Dic3—152no—
117C22×Dic3—301no—
118C32×C5C22×D411yescharacteristic, complemented, Hall
119C23×C5—32no—
120C22×D5—32no—
121C22×D5—38no—
122C22×D5—68no—
123C22×D5D3211yescharacteristic
124C5×D4—94no—
125C2×C4×C5—91no—
126C22⋊D5—98no—
127C2×Dic5—91no—
128D20—94no—
129C4×D5—94no—
130C22×C32—54no—
131C22×C32—52no—
132C22×C32C22×D511yescharacteristic
133C2×C3×D3—24no—
134C2×C3×D3—54no—
135C2×C3×D3—1016no—
136C2×C3×D3—104no—
137C2×C3×D3C22×D514yescomplemented
138C3⋊Dic3—52no—
139C3⋊Dic3C22×D512yescomplemented
140C22×D4—451noSylow
141C2×C3×C5—21no—
142C2×C3×C5—22no—
143C2×C3×C5—64no—
144C2×C3×C5C2×#6714yescomplemented
145C2×C3×C5C23×D312yes—
146D15—24no—
147C3×D5—24no—
148C3×D5—64no—
149C3×D5C2×#6718yescomplemented
150C5×D3—24no—
151C23×C3—32no—
152C23×C3—52no—
153C23×C3—101no—
154C23×C3—158no—
155C23×C3—152no—
156C23×C3—308no—
157C22×D3—52no—
158C22×D3—58no—
159C22×D3—108no—
160C22×D3D3×D512yescomplemented
161#67—38no—
162#67—158no—
163#67—158no—
164#67—158no—
165C2×Dic3—32no—
166C2×Dic3—61no—
167C2×Dic3—158no—
168C2×Dic3—152no—
169C2×Dic3—304no—
170C2×Dic3—301no—
171C22×C5—34no—
172C22×C5—64no—
173C22×C5C2×D3211yescharacteristic
174C2×D5—34no—
175C2×D5—64no—
176C2×D5—616no—
177C2×D5#62514yesdirect product factor
178C2×D5C2×D3212yes—
179C4×C5—92no—
180Dic5—92no—
181C2×C32—54no—
182C2×C32D4×D512yescomplemented
183C2×C32C23×D511yescharacteristic
184C3×D3—24no—
185C3×D3—104no—
186C24—152no—
187C2×D4—91no—
188C2×D4—451no—
189C2×D4—458no—
190C2×D4—452no—
191C22×C4—451no—
192C3×C5—21no—
193C3×C5C22×#6712yescomplemented
194C22×C3—21no—
195C22×C3—34no—
196C22×C3—58no—
197C22×C3—54no—
198C22×C3—64no—
199C22×C3—102no—
200C22×C3—104no—
201C22×C3—154no—
202C22×C3—3016no—
203C22×C3—304no—
204C22×C3C2×D3×D512yes—
205C2×D3—24no—
206C2×D3—54no—
207C2×D3—1016no—
208C2×D3—104no—
209C2×D3C2×D3×D514yescomplemented
210Dic3—34no—
211Dic3—62no—
212Dic3—154no—
213Dic3—302no—
214C2×C5—64no—
215C2×C5C2×#62512yescomplemented
216C2×C5C22×D3211yescharacteristic
217D5—64no—
218D5C2×#62514yesdirect product factor
219C32C2×D4×D511yescharacteristic, complemented, Sylow
220C23—32no—
221C23—51no—
222C23—158no—
223C23—152no—
224C23—308no—
225D4—94no—
226D4—454no—
227D4—458no—
228C2×C4—91no—
229C2×C4—454no—
230C2×C4—451no—
231C2×C3—22no—
232C2×C3—21no—
233C2×C3—58no—
234C2×C3—64no—
235C2×C3—104no—
236C2×C3—304no—
237C2×C3D5×#6714yescomplemented
238C2×C3C22×D3×D512yes—
239D3—24no—
240D3—104no—
241C5C22×#62511yescharacteristic, complemented, Sylow
242C22—34no—
243C22—54no—
244C22—52no—
245C22—64no—
246C22—154no—
247C22—3016no—
248C22—304no—
249C22D32×D511yescharacteristic
250C4—92no—
251C4—452no—
252C3—21no—
253C3C2×D5×#6712yescomplemented
254C2—54no—
255C2—64no—
256C2—304no—
257C2D5×#62512yesdirect product factor
258C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC22D32×D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C2×D32×D5
Fitting, socleC22×C32×C5C23
hypocenterC32×C5C22×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×D3×D5C2semidirect
2. C2×#67157C2semidirect
3. C2×#67150C2semidirect
4. C2×C5×#625C2semidirect
5. C2×D5×C3⋊Dic3C2semidirect
6. D5×#625C2direct (N × H)
8. C22×C3×D15C22semidirect
9. C22×C3×C5×D3C22semidirect
12. C2×C3×D3×D5C22semidirect
13. C2×C3⋊Dic15C22semidirect
14. C2×C5×C3⋊Dic3C22semidirect
15. #67157C22semidirect
16. #67150C22semidirect
17. C5×#625C22semidirect
18. D5×C3⋊Dic3C22semidirect
21. C22×D3×D5D3semidirect
31. C2×C3×D15C23semidirect
33. C2×C3×C5×D3C23semidirect
34. C2×C32×D5D4semidirect
37. C3⋊Dic15C23semidirect
38. C5×C3⋊Dic3C23semidirect
45. C2×#625D5direct (N × H)
47. C22×D15C2×D3semidirect
48. C22×C5×D3C2×D3semidirect
55. C2×D3×D5C2×D3semidirect
67. C2×C32×C5C2×D4semidirect
71. C32×D5C2×D4semidirect
82. C22×C3×D3C2×D5semidirect
85. C2×C3⋊Dic3C2×D5semidirect
88. #625C2×D5direct (N × H)
94. C2×D15C22×D3semidirect
96. C2×C5×D3C22×D3semidirect
102. C2×C3×D5#67semidirect
118. C32×C5C22×D4semidirect
137. C2×C3×D3C22×D5semidirect
139. C3⋊Dic3C22×D5semidirect
144. C2×C3×C5C2×#67semidirect
149. C3×D5C2×#67semidirect
160. C22×D3D3×D5semidirect
177. C2×D5#625direct (N × H)
182. C2×C32D4×D5semidirect
193. C3×C5C22×#67semidirect
209. C2×D3C2×D3×D5semidirect
215. C2×C5C2×#625semidirect
218. D5C2×#625direct (N × H)
219. C32C2×D4×D5semidirect
237. C2×C3D5×#67semidirect
241. C5C22×#625semidirect
253. C3C2×D5×#67semidirect
257. C2D5×#625direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D3×D52
C2×#671572
C2×#671502
C2×C5×#6252
C2×D5×C3⋊Dic32
D5×#6252
C2×D5×#673
C22×#6255