Finite Group Databasebeta

← C2×#10624022 #13088987

Subgroups — C2×#10624022 #13088987

Order 1440.

998 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 228 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
2C2×Dic3×D15C212yesmaximal, complemented
3C2×C5×#624C211yesmaximal, characteristic, complemented
4C2×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
5C2×C3×#1200C212yesmaximal, complemented
6#10624022C218yesmaximal, direct product factor
7C2×C4×D3×D5—32nomaximal
8C22×C5×C3⋊D3C2211yescharacteristic
9C22×C3×D15C2212yes—
10C2×(C3⋊D3)⋊D5C418yescomplemented
11C2×(C3⋊D3)⋊D5C2214yes—
12C2×C3⋊Dic15C2211yescharacteristic, complemented
13C2×C3×C5×Dic3C2212yescomplemented
14C2×C32×Dic5C2211yescharacteristic, complemented
15Dic3×D15C2218yescomplemented
16C5×#624C2214yescomplemented
17C3⋊D3×Dic5C2214yescomplemented
18C3×#1200C2218yescomplemented
19C2×C4×D32—51nomaximal, Hall
20C22×D3×D5—32no—
21C2×C4×D15—32no—
22C2×C4×C5×D3—32no—
23C2×C3×C4×D5—32no—
24C2×#1200D312yescomplemented
25C2×D3×Dic5—32no—
26C2×D3×Dic5—61no—
27C2×D5×Dic3—32no—
28C4×D3×D5—316no—
29C22×C32×C5C2311yescharacteristic
30C2×C5×C3⋊D3C2×C414yescomplemented
31C2×C5×C3⋊D3C2312yes—
32C2×C3×D15C2×C418yescomplemented
33C2×C3×D15C2314yes—
34(C3⋊D3)⋊D5C2×C4116yescomplemented
35C3⋊Dic15C2312yescomplemented
36C3×C5×Dic3C2314yescomplemented
37C32×Dic5C2312yescomplemented
38C22×C4×D5—91noHall
39C22×D32—51no—
40C2×C4×C3⋊D3—51no—
41C2×C3×C4×D3—52no—
42C2×#624D511yescharacteristic, complemented
43C2×D3×Dic3—52no—
44C4×D32—58no—
45C22×D15C2×D312yes—
46C22×C5×D3—32no—
47C22×C5×D3—61no—
48C22×C3×D5—32no—
49C2×D3×D5—316no—
50C2×D3×D5—38no—
51C2×C3×C4×C5—32no—
52C2×Dic15—32no—
53C2×Dic15—61no—
54C4×D15—38no—
55C2×C5×Dic3C2×D312yescomplemented
56C4×C5×D3—38no—
57C2×C3×Dic5—21no—
58C2×C3×Dic5C2×D312yescomplemented
59C3×C4×D5—38no—
60#1200C2×D318yescomplemented
61D3×Dic5—38no—
62D3×Dic5—64no—
63D5×Dic3—38no—
64C22×C4×D3—152no—
65C2×C32×C5C22×C412yescomplemented
66C2×C32×C5C2411yescharacteristic
67C5×C3⋊D3C22×C414yescomplemented
68C3×D15C22×C418yescomplemented
69C23×D5—91no—
70C22×C4×C5—91no—
71C22×Dic5—91no—
72C2×C4×D5—32no—
73C2×C4×D5—98no—
74C2×C4×D5—92no—
75C22×C3⋊D3C2×D511yescharacteristic
76C22×C3×D3—52no—
77C2×D32—58no—
78C2×D32—54no—
79C2×C32×C4—51no—
80C2×C3⋊Dic3—51no—
81C4×C3⋊D3—54no—
82C2×C3×Dic3C2×D512yescomplemented
83C3×C4×D3—58no—
84#624C2×D514yescomplemented
85D3×Dic3—58no—
86C22×C3×C5—21no—
87C22×C3×C5C22×D312yes—
88C2×D15C4×D318yescomplemented
89C2×D15C22×D314yes—
90C2×C5×D3—38no—
91C2×C5×D3—34no—
92C2×C5×D3—64no—
93C2×C5×D3—62no—
94C2×C3×D5—38no—
95C2×C3×D5—34no—
96D3×D5—332no—
97C3×C4×C5—34no—
98Dic15—34no—
99Dic15—62no—
100C3×Dic5—22no—
101C3×Dic5C22×D314yescomplemented
102C5×Dic3C22×D314yescomplemented
103C23×D3—152no—
104C22×C3×C4—152no—
105C22×Dic3—152no—
106C2×C4×D3—32no—
107C2×C4×D3—52no—
108C2×C4×D3—1516no—
109C2×C4×D3—152no—
110C2×C4×D3—152no—
111C2×C4×D3—301no—
112C32×C5C23×C411yescharacteristic, complemented, Hall
113C23×C5—91no—
114C22×D5—32no—
115C22×D5—94no—
116C22×D5—98no—
117C2×C4×C5—32no—
118C2×C4×C5—94no—
119C2×Dic5—91no—
120C2×Dic5—94no—
121C2×Dic5D3211yescharacteristic, complemented
122C4×D5—38no—
123C4×D5—98no—
124C22×C32C22×D511yescharacteristic
125C2×C3⋊D3C4×D514yescomplemented
126C2×C3⋊D3C22×D512yes—
127C2×C3×D3—58no—
128C2×C3×D3—54no—
129D32—516no—
130C32×C4—52no—
131C3⋊Dic3—52no—
132C3×Dic3C22×D514yescomplemented
133C23×C4—451noSylow
134C2×C3×C5—21no—
135C2×C3×C5—22no—
136C2×C3×C5C2×C4×D314yescomplemented
137C2×C3×C5C23×D312yes—
138D15C2×C4×D318yescomplemented
139C3×D5—38no—
140C5×D3—38no—
141C5×D3—64no—
142C23×C3—152no—
143C22×D3—32no—
144C22×D3—52no—
145C22×D3—61no—
146C22×D3—158no—
147C22×D3—1516no—
148C2×C3×C4—32no—
149C2×C3×C4—52no—
150C2×C3×C4—101no—
151C2×C3×C4—158no—
152C2×Dic3—152no—
153C2×Dic3—158no—
154C2×Dic3—301no—
155C2×Dic3D3×D512yescomplemented
156C4×D3—38no—
157C4×D3—58no—
158C4×D3—158no—
159C4×D3—158no—
160C4×D3—304no—
161C22×C5—92no—
162C22×C5—94no—
163C22×C5C2×D3211yescharacteristic
164C2×D5—34no—
165C2×D5—38no—
166C2×D5—916no—
167C4×C5—34no—
168Dic5—92no—
169Dic5C2×D3212yescomplemented
170C2×C32C2×C4×D512yescomplemented
171C2×C32C23×D511yescharacteristic
172C3⋊D3C2×C4×D514yescomplemented
173C3×D3—58no—
174C24—451no—
175C22×C4—91no—
176C22×C4—152no—
177C22×C4—452no—
178C22×C4—451no—
179C22×C4—458no—
180C3×C5—21no—
181C3×C5C22×C4×D312yescomplemented
182C22×C3—21no—
183C22×C3—158no—
184C22×C3—154no—
185C22×C3C2×D3×D512yes—
186C2×D3—38no—
187C2×D3—34no—
188C2×D3—58no—
189C2×D3—54no—
190C2×D3—62no—
191C2×D3—64no—
192C2×D3—1532no—
193C3×C4—34no—
194C3×C4—54no—
195C3×C4—102no—
196Dic3—154no—
197Dic3—302no—
198Dic3C2×D3×D514yescomplemented
199C2×C5—94no—
200C2×C5C4×D3212yescomplemented
201C2×C5C22×D3211yescharacteristic
202D5—38no—
203C32C22×C4×D511yescharacteristic, complemented, Sylow
204C23—91no—
205C23—152no—
206C23—454no—
207C23—458no—
208C2×C4—32no—
209C2×C4—51no—
210C2×C4—94no—
211C2×C4—158no—
212C2×C4—458no—
213C2×C4—454no—
214C2×C4—451no—
215C2×C3—22no—
216C2×C3—21no—
217C2×C3—158no—
218C2×C3C4×D3×D514yescomplemented
219C2×C3C22×D3×D512yes—
220D3—38no—
221D3—58no—
222D3—64no—
223C5C2×C4×D3211yescharacteristic, complemented, Sylow
224C22—94no—
225C22—92no—
226C22—158no—
227C22—154no—
228C22—4516no—
229C22D32×D511yescharacteristic
230C4—34no—
231C4—52no—
232C4—452no—
233C3—21no—
234C3C2×C4×D3×D512yescomplemented
235C2—94no—
236C2—158no—
237C2#1062402212yesdirect product factor
238C2C2×D32×D511yescharacteristic
subgroupG/H
center, hypercenterC22D32×D5
commutator, hypocenterC32×C5C23×C4
FrattiniC2C2×D32×D5
Fitting, socleC22×C32×C5C23
prime radicalC22×(C3⋊D3)⋊D5C2
cosocleC2×C32×C5C24
LCSG ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×D15C2semidirect
3. C2×C5×#624C2semidirect
4. C2×C3⋊D3×Dic5C2semidirect
5. C2×C3×#1200C2semidirect
6. #10624022C2direct (N × H)
10. C2×(C3⋊D3)⋊D5C4semidirect
12. C2×C3⋊Dic15C22semidirect
13. C2×C3×C5×Dic3C22semidirect
14. C2×C32×Dic5C22semidirect
15. Dic3×D15C22semidirect
16. C5×#624C22semidirect
17. C3⋊D3×Dic5C22semidirect
18. C3×#1200C22semidirect
24. C2×#1200D3semidirect
30. C2×C5×C3⋊D3C2×C4semidirect
32. C2×C3×D15C2×C4semidirect
34. (C3⋊D3)⋊D5C2×C4semidirect
35. C3⋊Dic15C23semidirect
36. C3×C5×Dic3C23semidirect
37. C32×Dic5C23semidirect
42. C2×#624D5semidirect
55. C2×C5×Dic3C2×D3semidirect
58. C2×C3×Dic5C2×D3semidirect
60. #1200C2×D3semidirect
65. C2×C32×C5C22×C4semidirect
67. C5×C3⋊D3C22×C4semidirect
68. C3×D15C22×C4semidirect
82. C2×C3×Dic3C2×D5semidirect
84. #624C2×D5semidirect
88. C2×D15C4×D3semidirect
101. C3×Dic5C22×D3semidirect
102. C5×Dic3C22×D3semidirect
112. C32×C5C23×C4semidirect
121. C2×Dic5D32semidirect
125. C2×C3⋊D3C4×D5semidirect
132. C3×Dic3C22×D5semidirect
136. C2×C3×C5C2×C4×D3semidirect
138. D15C2×C4×D3semidirect
155. C2×Dic3D3×D5semidirect
169. Dic5C2×D32semidirect
170. C2×C32C2×C4×D5semidirect
172. C3⋊D3C2×C4×D5semidirect
181. C3×C5C22×C4×D3semidirect
198. Dic3C2×D3×D5semidirect
200. C2×C5C4×D32semidirect
203. C32C22×C4×D5semidirect
218. C2×C3C4×D3×D5semidirect
223. C5C2×C4×D32semidirect
234. C3C2×C4×D3×D5semidirect
237. C2#10624022direct (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×Dic3×D152
C2×C5×#6242
C2×C3⋊D3×Dic52
C2×C3×#12002
#106240222
C2×C4×D3×D53
C2×C4×D325