Finite Group Databasebeta

← C2×#10624034 #13089012

Subgroups — C2×#10624034 #13089012

Order 1440.

998 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 172 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)⋊D5C211yesmaximal, characteristic, complemented
2C22×C5×D32C211yesmaximal, characteristic, complemented
3C2×#67150C212yesmaximal, complemented
4C2×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
5C2×C3×#1203C212yesmaximal, complemented
6#10624034C218yesmaximal, direct product factor
7C2×D3×C22⋊D5—32nomaximal
8C22×C5×C3⋊D3C2211yescharacteristic
9C22×C3×D15C2212yescomplemented
10C22×C3×C5×D3C2212yescomplemented
11C2×(C3⋊D3)⋊D5—24no—
12C2×(C3⋊D3)⋊D5C2214yescomplemented
13C2×C5×D32—24no—
14C2×C5×D32C2214yescomplemented
15C2×C3⋊Dic15C2211yescharacteristic, complemented
16C2×C32×Dic5C2211yescharacteristic, complemented
17#67150C2218yescomplemented
18C3⋊D3×Dic5C2214yescomplemented
19C3×#1203C2218yescomplemented
20C2×#3802—51nomaximal, Hall
21C23×C5×D3—32no—
22C22×D3×D5—32no—
23C2×#1220—32no—
24C2×C3×C22⋊D5—32no—
25D3×C22⋊D5—316no—
26C2×#1203D312yescomplemented
27C2×#1201—32no—
28C2×D3×Dic5—32no—
29C2×D3×Dic5—61no—
30C22×C32×C5C2311yescharacteristic
31C2×C5×C3⋊D3D414yescomplemented
32C2×C5×C3⋊D3C2312yes—
33C2×C3×D15—24no—
34C2×C3×D15C2314yescomplemented
35C2×C3×C5×D3—24no—
36C2×C3×C5×D3C2314yescomplemented
37(C3⋊D3)⋊D5—28no—
38C5×D32—28no—
39C3⋊Dic15C2312yescomplemented
40C32×Dic5C2312yescomplemented
41C22×C22⋊D5—91noHall
42C22×D32—51no—
43C22×D32D511yescharacteristic, complemented
44C2×C4×C3⋊D3—51no—
45C2×C3×D12—52no—
46C2×#625—52no—
47#3802—58no—
48C23×C3×C5—32no—
49C22×D15C2×D312yescomplemented
50C22×C5×D3—32no—
51C22×C5×D3—38no—
52C22×C5×D3—68no—
53C22×C5×D3—61no—
54C22×C5×D3C2×D312yescomplemented
55C22×C3×D5—32no—
56C2×D3×D5—38no—
57C2×D3×D5—68no—
58#1220—38no—
59C2×Dic15—32no—
60C2×Dic15—61no—
61C3×C22⋊D5—38no—
62C2×C3×Dic5—21no—
63C2×C3×Dic5C2×D312yescomplemented
64#1203C2×D318yescomplemented
65#1201—38no—
66D3×Dic5—38no—
67D3×Dic5—64no—
68C2×D3×D4—152no—
69C2×C32×C5C2×D412yescomplemented
70C2×C32×C5C2411yescharacteristic
71C5×C3⋊D3C2×D414yescomplemented
72C3×D15—24no—
73C3×C5×D3—24no—
74C24×C5—91no—
75C23×D5—91no—
76C2×C22⋊D5—32no—
77C2×C22⋊D5—98no—
78C2×C22⋊D5—92no—
79C22×Dic5—91no—
80C22×C3⋊D3C2×D511yescharacteristic
81C22×C3×D3—52no—
82C22×C3×D3C2×D512yescomplemented
83C2×D32—24no—
84C2×D32—54no—
85C2×D32—104no—
86C2×D32C2×D514yescomplemented
87C2×C32×C4—51no—
88C2×C3⋊Dic3—51no—
89C4×C3⋊D3—54no—
90C3×D12—58no—
91#625—58no—
92C22×C3×C5—21no—
93C22×C3×C5—34no—
94C22×C3×C5—64no—
95C22×C3×C5C22×D312yes—
96C2×D15—24no—
97C2×D15C22×D314yescomplemented
98C2×C5×D3—24no—
99C2×C5×D3—38no—
100C2×C5×D3—34no—
101C2×C5×D3—64no—
102C2×C5×D3—62no—
103C2×C5×D3—616no—
104C2×C5×D3C22×D314yescomplemented
105C2×C3×D5—34no—
106C2×C3×D5—64no—
107D3×D5—616no—
108Dic15—34no—
109Dic15—62no—
110C3×Dic5—22no—
111C3×Dic5C22×D314yescomplemented
112C23×D3—32no—
113C23×D3—152no—
114C2×C3×D4—152no—
115C2×#67—152no—
116C2×#67—152no—
117D3×D4—1516no—
118C2×D12—52no—
119C2×C4×D3—152no—
120C2×C4×D3—301no—
121C32×C5C22×D411yescharacteristic, complemented, Hall
122C23×C5—32no—
123C23×C5—94no—
124C23×C5—91no—
125C23×C5—184no—
126C22×D5—32no—
127C22×D5—94no—
128C22×D5—184no—
129C22⋊D5—38no—
130C22⋊D5—98no—
131C2×Dic5—91no—
132C2×Dic5—94no—
133C2×Dic5D3211yescharacteristic, complemented
134C22×C32C22×D511yescharacteristic
135C2×C3⋊D3C22⋊D514yescomplemented
136C2×C3⋊D3C22×D512yes—
137C2×C3×D3—24no—
138C2×C3×D3—54no—
139C2×C3×D3—104no—
140C2×C3×D3C22×D514yescomplemented
141D32—28no—
142D32—108no—
143C32×C4—52no—
144C3⋊Dic3—52no—
145C22×D4—451noSylow
146C2×C3×C5—21no—
147C2×C3×C5—22no—
148C2×C3×C5—64no—
149C2×C3×C5D3×D414yescomplemented
150C2×C3×C5C23×D312yes—
151D15—24no—
152C3×D5—64no—
153C5×D3—24no—
154C5×D3—38no—
155C5×D3—64no—
156C23×C3—32no—
157C23×C3—152no—
158C22×D3—32no—
159C22×D3—38no—
160C22×D3—52no—
161C22×D3—61no—
162C22×D3—68no—
163C22×D3—158no—
164C22×D3—308no—
165C22×D3D3×D512yescomplemented
166C3×D4—158no—
167C2×C3×C4—52no—
168C2×C3×C4—101no—
169#67—158no—
170#67—158no—
171C2×Dic3—152no—
172C2×Dic3—301no—
173D12—58no—
174C4×D3—158no—
175C4×D3—304no—
176C22×C5—34no—
177C22×C5—64no—
178C22×C5—92no—
179C22×C5—94no—
180C22×C5—188no—
181C22×C5C2×D3211yescharacteristic
182C2×D5—34no—
183C2×D5—64no—
184C2×D5—188no—
185Dic5—92no—
186Dic5C2×D3212yescomplemented
187C2×C32C2×C22⋊D512yescomplemented
188C2×C32C23×D511yescharacteristic
189C3⋊D3C2×C22⋊D514yescomplemented
190C3×D3—24no—
191C3×D3—104no—
192C24—91no—
193C24—451no—
194C2×D4—152no—
195C2×D4—458no—
196C2×D4—452no—
197C22×C4—451no—
198C3×C5—21no—
199C3×C5C2×D3×D412yescomplemented
200C22×C3—21no—
201C22×C3—34no—
202C22×C3—64no—
203C22×C3—154no—
204C22×C3—304no—
205C22×C3C2×D3×D512yes—
206C2×D3—24no—
207C2×D3—38no—
208C2×D3—34no—
209C2×D3—54no—
210C2×D3—62no—
211C2×D3—64no—
212C2×D3—616no—
213C2×D3—104no—
214C2×D3—3016no—
215C2×D3C2×D3×D514yescomplemented
216C3×C4—54no—
217C3×C4—102no—
218Dic3—154no—
219Dic3—302no—
220C2×C5—64no—
221C2×C5—94no—
222C2×C5#380212yescomplemented
223C2×C5C22×D3211yescharacteristic
224D5—64no—
225C32C22×C22⋊D511yescharacteristic, complemented, Sylow
226C23—32no—
227C23—94no—
228C23—91no—
229C23—152no—
230C23—184no—
231C23—454no—
232C23—904no—
233D4—158no—
234D4—458no—
235C2×C4—51no—
236C2×C4—454no—
237C2×C4—451no—
238C2×C3—22no—
239C2×C3—21no—
240C2×C3—64no—
241C2×C3—304no—
242C2×C3D3×C22⋊D514yescomplemented
243C2×C3C22×D3×D512yes—
244D3—24no—
245D3—38no—
246D3—64no—
247D3—104no—
248C5C2×#380211yescharacteristic, complemented, Sylow
249C22—34no—
250C22—64no—
251C22—94no—
252C22—92no—
253C22—154no—
254C22—188no—
255C22—304no—
256C22—908no—
257C22D32×D511yescharacteristic
258C4—52no—
259C4—452no—
260C3—21no—
261C3C2×D3×C22⋊D512yescomplemented
262C2—64no—
263C2—94no—
264C2—304no—
265C2#1062403412yesdirect product factor
266C2C2×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. C22×C5×D32C2semidirect
3. C2×#67150C2semidirect
4. C2×C3⋊D3×Dic5C2semidirect
5. C2×C3×#1203C2semidirect
6. #10624034C2direct (N × H)
9. C22×C3×D15C22semidirect
10. C22×C3×C5×D3C22semidirect
12. C2×(C3⋊D3)⋊D5C22semidirect
14. C2×C5×D32C22semidirect
15. C2×C3⋊Dic15C22semidirect
16. C2×C32×Dic5C22semidirect
17. #67150C22semidirect
18. C3⋊D3×Dic5C22semidirect
19. C3×#1203C22semidirect
26. C2×#1203D3semidirect
31. C2×C5×C3⋊D3D4semidirect
34. C2×C3×D15C23semidirect
36. C2×C3×C5×D3C23semidirect
39. C3⋊Dic15C23semidirect
40. C32×Dic5C23semidirect
43. C22×D32D5semidirect
49. C22×D15C2×D3semidirect
54. C22×C5×D3C2×D3semidirect
63. C2×C3×Dic5C2×D3semidirect
64. #1203C2×D3semidirect
69. C2×C32×C5C2×D4semidirect
71. C5×C3⋊D3C2×D4semidirect
82. C22×C3×D3C2×D5semidirect
86. C2×D32C2×D5semidirect
97. C2×D15C22×D3semidirect
104. C2×C5×D3C22×D3semidirect
111. C3×Dic5C22×D3semidirect
121. C32×C5C22×D4semidirect
133. C2×Dic5D32semidirect
135. C2×C3⋊D3C22⋊D5semidirect
140. C2×C3×D3C22×D5semidirect
149. C2×C3×C5D3×D4semidirect
165. C22×D3D3×D5semidirect
186. Dic5C2×D32semidirect
187. C2×C32C2×C22⋊D5semidirect
189. C3⋊D3C2×C22⋊D5semidirect
199. C3×C5C2×D3×D4semidirect
215. C2×D3C2×D3×D5semidirect
222. C2×C5#3802semidirect
225. C32C22×C22⋊D5semidirect
242. C2×C3D3×C22⋊D5semidirect
248. C5C2×#3802semidirect
261. C3C2×D3×C22⋊D5semidirect
265. C2#10624034direct (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
C22×C5×D322
C2×#671502
C2×C3⋊D3×Dic52
C2×C3×#12032
#106240342
C2×D3×C22⋊D53
C2×#38025