Finite Group Databasebeta

← C2×#10624031 #13089007

Subgroups — C2×#10624031 #13089007

Order 1440.

1078 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1078 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×D5×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#67151C212yesmaximal, complemented
4C2×C5×#624C211yesmaximal, characteristic, complemented
5C2×C3×#1202C212yesmaximal, complemented
6#10624031C218yesmaximal, direct product factor
7C2×D3×D20—32nomaximal
8C22×C3⋊D15C2211yescharacteristic, complemented
9C22×C5×C3⋊D3C2211yescharacteristic
10C22×C3×D15C2212yescomplemented
11C22×C32×D5C2211yescharacteristic, complemented
12C2×(C3⋊D3)⋊D5—24no—
13C2×(C3⋊D3)⋊D5C2214yescomplemented
14C2×D5×C3⋊D3—24no—
15C2×D5×C3⋊D3C2214yescomplemented
16C2×C3×C5×Dic3C2212yescomplemented
17#67151C2218yescomplemented
18C5×#624C2214yescomplemented
19C3×#1202C2218yescomplemented
20C2×#3811—51nomaximal, Hall
21C22×D3×D5—32no—
22C22×D3×D5—32no—
23C22×D3×D5—61no—
24C2×D60—32no—
25C2×C4×C5×D3—32no—
26C2×C3×D20—32no—
27C2×#1202—32no—
28C2×#1202D312yescomplemented
29D3×D20—316no—
30C22×C32×C5C2311yescharacteristic
31C2×C3⋊D15—22no—
32C2×C3⋊D15C2312yescomplemented
33C2×C5×C3⋊D3D414yescomplemented
34C2×C5×C3⋊D3C2312yes—
35C2×C3×D15—24no—
36C2×C3×D15C2314yescomplemented
37C2×C32×D5—22no—
38C2×C32×D5C2312yescomplemented
39(C3⋊D3)⋊D5—28no—
40D5×C3⋊D3—28no—
41C3×C5×Dic3C2314yescomplemented
42C22×D20—91noHall
43C23×C3⋊D3—51no—
44C22×D32—51no—
45C2×C3×#67—52no—
46#3811—58no—
47C2×#626—52no—
48C2×#624D511yescharacteristic, complemented
49C22×D15—32no—
50C22×D15—61no—
51C22×D15C2×D312yescomplemented
52C22×C5×D3—32no—
53C22×C5×D3—61no—
54C22×C3×D5—21no—
55C22×C3×D5—32no—
56C22×C3×D5C2×D312yescomplemented
57C2×D3×D5—38no—
58C2×D3×D5—38no—
59C2×D3×D5—68no—
60C2×D3×D5—64no—
61C2×D3×D5—68no—
62C2×D3×D5—68no—
63C2×C3×C4×C5—32no—
64D60—38no—
65C2×C5×Dic3C2×D312yescomplemented
66C4×C5×D3—38no—
67C3×D20—38no—
68#1202—38no—
69#1202C2×D318yescomplemented
70C2×D3×D4—152no—
71C2×C32×C5C2×D412yescomplemented
72C2×C32×C5C2411yescharacteristic
73C3⋊D15—22no—
74C5×C3⋊D3C2×D414yescomplemented
75C3×D15—24no—
76C32×D5—22no—
77C23×D5—91no—
78C23×D5—91no—
79C22×C4×C5—91no—
80C2×D20—32no—
81C2×D20—98no—
82C2×D20—92no—
83C23×C32—51no—
84C22×C3⋊D3—51no—
85C22×C3⋊D3—54no—
86C22×C3⋊D3—104no—
87C22×C3⋊D3C2×D511yescharacteristic
88C22×C3×D3—52no—
89C2×D32—54no—
90C2×D32—104no—
91C3×#67—58no—
92C2×C3×Dic3C2×D512yescomplemented
93#626—58no—
94#624C2×D514yescomplemented
95C22×C3×C5—21no—
96C22×C3×C5C22×D312yes—
97C2×D15—24no—
98C2×D15—34no—
99C2×D15—62no—
100C2×D15—64no—
101C2×D15—64no—
102C2×D15C22×D314yescomplemented
103C2×C5×D3—38no—
104C2×C5×D3—34no—
105C2×C5×D3—64no—
106C2×C5×D3—62no—
107C2×C3×D5—24no—
108C2×C3×D5—24no—
109C2×C3×D5—22no—
110C2×C3×D5—34no—
111C2×C3×D5—64no—
112C2×C3×D5C22×D314yescomplemented
113D3×D5—616no—
114D3×D5—616no—
115D3×D5—616no—
116C3×C4×C5—34no—
117C5×Dic3C22×D314yescomplemented
118C23×D3—152no—
119C23×D3—152no—
120C23×D3—301no—
121C2×C3×D4—152no—
122C2×#67—52no—
123C2×#67—152no—
124D3×D4—1516no—
125C2×D12—152no—
126C2×C4×D3—32no—
127C32×C5C22×D411yescharacteristic, complemented, Hall
128C23×C5—91no—
129C22×D5—32no—
130C22×D5—94no—
131C22×D5—91no—
132C22×D5—94no—
133C22×D5—184no—
134C22×D5—184no—
135C22×D5D3211yescharacteristic, complemented
136C2×C4×C5—32no—
137C2×C4×C5—94no—
138D20—38no—
139D20—98no—
140C22×C32—52no—
141C22×C32—102no—
142C22×C32C22×D511yescharacteristic
143C2×C3⋊D3—52no—
144C2×C3⋊D3—102no—
145C2×C3⋊D3—108no—
146C2×C3⋊D3D2014yescomplemented
147C2×C3⋊D3C22×D512yes—
148C2×C3×D3—54no—
149C2×C3×D3—104no—
150D32—108no—
151C3×Dic3C22×D514yescomplemented
152C22×D4—451noSylow
153C2×C3×C5—21no—
154C2×C3×C5—22no—
155C2×C3×C5D3×D414yescomplemented
156C2×C3×C5C23×D312yes—
157D15—24no—
158D15—64no—
159D15—64no—
160C3×D5—24no—
161C3×D5—24no—
162C3×D5—64no—
163C5×D3—38no—
164C5×D3—64no—
165C23×C3—52no—
166C23×C3—101no—
167C23×C3—152no—
168C22×D3—32no—
169C22×D3—52no—
170C22×D3—61no—
171C22×D3—158no—
172C22×D3—158no—
173C22×D3—152no—
174C22×D3—304no—
175C22×D3—301no—
176C22×D3—308no—
177C22×D3—308no—
178C22×D3—308no—
179C3×D4—158no—
180C2×C3×C4—32no—
181#67—58no—
182#67—158no—
183C2×Dic3D3×D512yescomplemented
184D12—158no—
185C4×D3—38no—
186C22×C5—92no—
187C22×C5—94no—
188C22×C5C2×D3211yescharacteristic
189C2×D5—22no—
190C2×D5—34no—
191C2×D5—64no—
192C2×D5—92no—
193C2×D5—188no—
194C2×D5—188no—
195C2×D5—182no—
196C2×D5C2×D3212yescomplemented
197C4×C5—34no—
198C2×C32—102no—
199C2×C32C2×D2012yescomplemented
200C2×C32C23×D511yescharacteristic
201C3⋊D3—102no—
202C3⋊D3C2×D2014yescomplemented
203C3×D3—104no—
204C24—451no—
205C24—451no—
206C2×D4—152no—
207C2×D4—452no—
208C2×D4—458no—
209C22×C4—91no—
210C3×C5—21no—
211C3×C5C2×D3×D412yescomplemented
212C22×C3—21no—
213C22×C3—54no—
214C22×C3—102no—
215C22×C3—104no—
216C22×C3—104no—
217C22×C3—154no—
218C22×C3—304no—
219C22×C3C2×D3×D512yes—
220C2×D3—38no—
221C2×D3—34no—
222C2×D3—54no—
223C2×D3—64no—
224C2×D3—62no—
225C2×D3—104no—
226C2×D3—154no—
227C2×D3—302no—
228C2×D3—3016no—
229C2×D3—3016no—
230C2×D3—3016no—
231C2×D3—304no—
232C2×D3—304no—
233C3×C4—34no—
234Dic3C2×D3×D514yescomplemented
235C2×C5—94no—
236C2×C5#381112yescomplemented
237C2×C5C22×D3211yescharacteristic
238D5—22no—
239D5—64no—
240D5—182no—
241C32C22×D2011yescharacteristic, complemented, Sylow
242C23—51no—
243C23—91no—
244C23—152no—
245C23—454no—
246C23—454no—
247C23—451no—
248C23—904no—
249C23—904no—
250D4—158no—
251D4—458no—
252C2×C4—32no—
253C2×C4—94no—
254C2×C3—22no—
255C2×C3—21no—
256C2×C3—104no—
257C2×C3—104no—
258C2×C3—304no—
259C2×C3D3×D2014yescomplemented
260C2×C3C22×D3×D512yes—
261D3—38no—
262D3—64no—
263D3—104no—
264D3—304no—
265D3—304no—
266C5C2×#381111yescharacteristic, complemented, Sylow
267C22—52no—
268C22—94no—
269C22—92no—
270C22—102no—
271C22—154no—
272C22—304no—
273C22—452no—
274C22—908no—
275C22—908no—
276C22—902no—
277C22D32×D511yescharacteristic
278C4—34no—
279C3—21no—
280C3C2×D3×D2012yescomplemented
281C2—94no—
282C2—102no—
283C2—304no—
284C2—902no—
285C2#1062403112yesdirect product factor
286C2C2×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×D5×C3⋊D3C2semidirect
3. C2×#67151C2semidirect
4. C2×C5×#624C2semidirect
5. C2×C3×#1202C2semidirect
6. #10624031C2direct (N × H)
8. C22×C3⋊D15C22semidirect
10. C22×C3×D15C22semidirect
11. C22×C32×D5C22semidirect
13. C2×(C3⋊D3)⋊D5C22semidirect
15. C2×D5×C3⋊D3C22semidirect
16. C2×C3×C5×Dic3C22semidirect
17. #67151C22semidirect
18. C5×#624C22semidirect
19. C3×#1202C22semidirect
28. C2×#1202D3semidirect
32. C2×C3⋊D15C23semidirect
33. C2×C5×C3⋊D3D4semidirect
36. C2×C3×D15C23semidirect
38. C2×C32×D5C23semidirect
41. C3×C5×Dic3C23semidirect
48. C2×#624D5semidirect
51. C22×D15C2×D3semidirect
56. C22×C3×D5C2×D3semidirect
65. C2×C5×Dic3C2×D3semidirect
69. #1202C2×D3semidirect
71. C2×C32×C5C2×D4semidirect
74. C5×C3⋊D3C2×D4semidirect
92. C2×C3×Dic3C2×D5semidirect
94. #624C2×D5semidirect
102. C2×D15C22×D3semidirect
112. C2×C3×D5C22×D3semidirect
117. C5×Dic3C22×D3semidirect
127. C32×C5C22×D4semidirect
135. C22×D5D32semidirect
146. C2×C3⋊D3D20semidirect
151. C3×Dic3C22×D5semidirect
155. C2×C3×C5D3×D4semidirect
183. C2×Dic3D3×D5semidirect
196. C2×D5C2×D32semidirect
199. C2×C32C2×D20semidirect
202. C3⋊D3C2×D20semidirect
211. C3×C5C2×D3×D4semidirect
234. Dic3C2×D3×D5semidirect
236. C2×C5#3811semidirect
241. C32C22×D20semidirect
259. C2×C3D3×D20semidirect
266. C5C2×#3811semidirect
280. C3C2×D3×D20semidirect
285. C2#10624031direct (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×D5×C3⋊D32
C2×#671512
C2×C5×#6242
C2×C3×#12022
#106240312
C2×D3×D203
C2×#38115