Finite Group Databasebeta

← C2×#10624236 #13089432

Subgroups — C2×#10624236 #13089432

Order 1440.

1078 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C23×C5×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×(C3⋊D3)⋊D5C211yesmaximal, characteristic, complemented
3C2×C3×#1220C212yesmaximal, complemented
4#10624236C218yesmaximal, direct product factor
5C2×#67156C212yesmaximal, complemented
6C2×#67155C211yesmaximal, characteristic, complemented
7C2×D3×C22⋊D5—32nomaximal
8C23×C32×C5C2211yescharacteristic, complemented
9C22×C5×C3⋊D3—24no—
10C22×C5×C3⋊D3C2211yescharacteristic
11C22×C5×C3⋊D3C2214yescomplemented
12C22×C5×C3⋊D3C2211yescharacteristic, complemented
13C22×C3×D15C2212yescomplemented
14C2×(C3⋊D3)⋊D5—24no—
15C2×(C3⋊D3)⋊D5C2214yescomplemented
16C3×#1220C2218yescomplemented
17C2×C3×Dic15C2212yescomplemented
18#67156C2218yescomplemented
19#67155C2214yescomplemented
20C2×#3811—51nomaximal, Hall
21C23×C5×D3—32no—
22C23×C5×D3—61no—
23C22×D3×D5—32no—
24C2×#1220D312yescomplemented
25C2×C3×C22⋊D5—32no—
26D3×C22⋊D5—316no—
27C2×#1203—32no—
28C2×#1201—32no—
29C2×D3×Dic5—32no—
30C22×C32×C5—22no—
31C22×C32×C5C2311yescharacteristic
32C22×C32×C5C2312yescomplemented
33C2×C5×C3⋊D3—22no—
34C2×C5×C3⋊D3—28no—
35C2×C5×C3⋊D3D414yescomplemented
36C2×C5×C3⋊D3C2312yes—
37C2×C5×C3⋊D3C2312yescomplemented
38C2×C3×D15—24no—
39C2×C3×D15C2314yescomplemented
40(C3⋊D3)⋊D5—28no—
41C3×Dic15C2314yescomplemented
42C22×C22⋊D5—91noHall
43C23×C3⋊D3D511yescharacteristic, complemented
44C22×D32—51no—
45C2×C3×#67—52no—
46#3811—58no—
47C2×#626—52no—
48C2×#624—51no—
49C23×C3×C5—21no—
50C23×C3×C5C2×D312yescomplemented
51C22×D15C2×D312yescomplemented
52C22×C5×D3—32no—
53C22×C5×D3—32no—
54C22×C5×D3—38no—
55C22×C5×D3—68no—
56C22×C5×D3—61no—
57C22×C5×D3—64no—
58C22×C5×D3—61no—
59C22×C5×D3—68no—
60C22×C3×D5—32no—
61C2×D3×D5—38no—
62C2×D3×D5—68no—
63#1220C2×D318yescomplemented
64C2×Dic15C2×D312yescomplemented
65C3×C22⋊D5—38no—
66C2×C3×Dic5—32no—
67#1203—38no—
68#1201—38no—
69D3×Dic5—38no—
70C2×D3×D4—152no—
71C2×C32×C5—22no—
72C2×C32×C5C2×D412yescomplemented
73C2×C32×C5C2411yescharacteristic
74C5×C3⋊D3—22no—
75C5×C3⋊D3C2×D414yescomplemented
76C3×D15—24no—
77C24×C5—91no—
78C23×D5—91no—
79C2×C22⋊D5—32no—
80C2×C22⋊D5—92no—
81C2×C22⋊D5—98no—
82C22×Dic5—91no—
83C23×C32C2×D511yescharacteristic, complemented
84C22×C3⋊D3—24no—
85C22×C3⋊D3C2×D511yescharacteristic
86C22×C3⋊D3C2×D514yescomplemented
87C22×C3⋊D3C2×D511yescharacteristic, complemented
88C22×C3×D3—52no—
89C2×D32—54no—
90C2×D32—104no—
91C3×#67—58no—
92C2×C3×Dic3—52no—
93#626—58no—
94#624—54no—
95C22×C3×C5—24no—
96C22×C3×C5—24no—
97C22×C3×C5—21no—
98C22×C3×C5—22no—
99C22×C3×C5C22×D312yes—
100C22×C3×C5C22×D314yescomplemented
101C2×D15—24no—
102C2×D15C22×D314yescomplemented
103C2×C5×D3—38no—
104C2×C5×D3—34no—
105C2×C5×D3—34no—
106C2×C5×D3—64no—
107C2×C5×D3—62no—
108C2×C5×D3—62no—
109C2×C5×D3—616no—
110C2×C5×D3—616no—
111C2×C5×D3—64no—
112C2×C5×D3—64no—
113C2×C3×D5—34no—
114C2×C3×D5—64no—
115D3×D5—616no—
116Dic15C22×D314yescomplemented
117C3×Dic5—34no—
118C23×D3—32no—
119C23×D3—61no—
120C23×D3—152no—
121C2×C3×D4—152no—
122C2×#67—52no—
123C2×#67—152no—
124D3×D4—1516no—
125C2×D12—152no—
126C2×C4×D3—152no—
127C32×C5C22×D411yescharacteristic, complemented, Hall
128C23×C5—91no—
129C23×C5—91no—
130C23×C5—94no—
131C23×C5—184no—
132C23×C5D3211yescharacteristic, complemented
133C22×D5—32no—
134C22×D5—94no—
135C22×D5—184no—
136C22⋊D5—38no—
137C22⋊D5—98no—
138C2×Dic5—32no—
139C2×Dic5—94no—
140C22×C32—22no—
141C22×C32C22×D511yescharacteristic
142C22×C32C22×D512yescomplemented
143C2×C3⋊D3—22no—
144C2×C3⋊D3—28no—
145C2×C3⋊D3C22⋊D514yescomplemented
146C2×C3⋊D3C22×D512yes—
147C2×C3⋊D3C22×D512yescomplemented
148C2×C3×D3—54no—
149C2×C3×D3—104no—
150D32—108no—
151C3×Dic3—54no—
152C22×D4—451noSylow
153C2×C3×C5—24no—
154C2×C3×C5—24no—
155C2×C3×C5—21no—
156C2×C3×C5—22no—
157C2×C3×C5D3×D414yescomplemented
158C2×C3×C5C23×D312yes—
159D15—24no—
160C3×D5—64no—
161C5×D3—38no—
162C5×D3—64no—
163C5×D3—64no—
164C5×D3—64no—
165C23×C3—21no—
166C23×C3—152no—
167C23×C3D3×D512yescomplemented
168C22×D3—32no—
169C22×D3—32no—
170C22×D3—38no—
171C22×D3—52no—
172C22×D3—61no—
173C22×D3—61no—
174C22×D3—64no—
175C22×D3—68no—
176C22×D3—68no—
177C22×D3—158no—
178C22×D3—308no—
179C3×D4—158no—
180C2×C3×C4—152no—
181#67—58no—
182#67—158no—
183C2×Dic3—52no—
184D12—158no—
185C4×D3—158no—
186C22×C5—22no—
187C22×C5—92no—
188C22×C5—94no—
189C22×C5—92no—
190C22×C5—188no—
191C22×C5—182no—
192C22×C5C2×D3211yescharacteristic
193C22×C5C2×D3212yescomplemented
194C2×D5—34no—
195C2×D5—64no—
196C2×D5—188no—
197Dic5—34no—
198C2×C32—22no—
199C2×C32C2×C22⋊D512yescomplemented
200C2×C32C23×D511yescharacteristic
201C3⋊D3—22no—
202C3⋊D3C2×C22⋊D514yescomplemented
203C3×D3—104no—
204C24—91no—
205C24—451no—
206C2×D4—152no—
207C2×D4—452no—
208C2×D4—458no—
209C22×C4—451no—
210C3×C5—21no—
211C3×C5C2×D3×D412yescomplemented
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—21no—
215C22×C3—22no—
216C22×C3—154no—
217C22×C3—304no—
218C22×C3C2×D3×D512yes—
219C22×C3C2×D3×D514yescomplemented
220C2×D3—38no—
221C2×D3—34no—
222C2×D3—34no—
223C2×D3—54no—
224C2×D3—64no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—616no—
228C2×D3—616no—
229C2×D3—64no—
230C2×D3—64no—
231C2×D3—104no—
232C2×D3—3016no—
233C3×C4—154no—
234Dic3—54no—
235C2×C5—22no—
236C2×C5—94no—
237C2×C5—182no—
238C2×C5#381112yescomplemented
239C2×C5C22×D3211yescharacteristic
240D5—64no—
241C32C22×C22⋊D511yescharacteristic, complemented, Sylow
242C23—91no—
243C23—91no—
244C23—94no—
245C23—152no—
246C23—184no—
247C23—454no—
248C23—904no—
249C23(C3⋊D3)⋊D511yescharacteristic, complemented
250D4—158no—
251D4—458no—
252C2×C4—152no—
253C2×C4—454no—
254C2×C3—24no—
255C2×C3—24no—
256C2×C3—21no—
257C2×C3—22no—
258C2×C3—304no—
259C2×C3D3×C22⋊D514yescomplemented
260C2×C3C22×D3×D512yes—
261D3—38no—
262D3—64no—
263D3—64no—
264D3—64no—
265D3—104no—
266C5C2×#381111yescharacteristic, complemented, Sylow
267C22—22no—
268C22—92no—
269C22—94no—
270C22—92no—
271C22—154no—
272C22—188no—
273C22—182no—
274C22—304no—
275C22—908no—
276C22C2×(C3⋊D3)⋊D511yescharacteristic
277C22C2×(C3⋊D3)⋊D512yescomplemented
278C4—154no—
279C3—21no—
280C3C2×D3×C22⋊D512yescomplemented
281C2—22no—
282C2—94no—
283C2—182no—
284C2—304no—
285C2#1062423612yesdirect product factor
286C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊D5
commutator, cosocleC2×C32×C5C24
FrattiniC2C22×(C3⋊D3)⋊D5
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23(C3⋊D3)⋊D5
hypocenterC32×C5C22×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C5×C3⋊D3C2semidirect
2. C22×(C3⋊D3)⋊D5C2semidirect
3. C2×C3×#1220C2semidirect
4. #10624236C2direct (N × H)
5. C2×#67156C2semidirect
6. C2×#67155C2semidirect
8. C23×C32×C5C22semidirect
11. C22×C5×C3⋊D3C22semidirect
12. C22×C5×C3⋊D3C22semidirect
13. C22×C3×D15C22semidirect
15. C2×(C3⋊D3)⋊D5C22semidirect
16. C3×#1220C22semidirect
17. C2×C3×Dic15C22semidirect
18. #67156C22semidirect
19. #67155C22semidirect
24. C2×#1220D3semidirect
32. C22×C32×C5C23semidirect
35. C2×C5×C3⋊D3D4semidirect
37. C2×C5×C3⋊D3C23semidirect
39. C2×C3×D15C23semidirect
41. C3×Dic15C23semidirect
43. C23×C3⋊D3D5semidirect
50. C23×C3×C5C2×D3semidirect
51. C22×D15C2×D3semidirect
63. #1220C2×D3semidirect
64. C2×Dic15C2×D3semidirect
72. C2×C32×C5C2×D4semidirect
75. C5×C3⋊D3C2×D4semidirect
83. C23×C32C2×D5semidirect
86. C22×C3⋊D3C2×D5semidirect
87. C22×C3⋊D3C2×D5semidirect
100. C22×C3×C5C22×D3semidirect
102. C2×D15C22×D3semidirect
116. Dic15C22×D3semidirect
127. C32×C5C22×D4semidirect
132. C23×C5D32semidirect
142. C22×C32C22×D5semidirect
145. C2×C3⋊D3C22⋊D5semidirect
147. C2×C3⋊D3C22×D5semidirect
157. C2×C3×C5D3×D4semidirect
167. C23×C3D3×D5semidirect
193. C22×C5C2×D32semidirect
199. C2×C32C2×C22⋊D5semidirect
202. C3⋊D3C2×C22⋊D5semidirect
211. C3×C5C2×D3×D4semidirect
219. C22×C3C2×D3×D5semidirect
238. C2×C5#3811semidirect
241. C32C22×C22⋊D5semidirect
249. C23(C3⋊D3)⋊D5semidirect
259. C2×C3D3×C22⋊D5semidirect
266. C5C2×#3811semidirect
277. C22C2×(C3⋊D3)⋊D5semidirect
280. C3C2×D3×C22⋊D5semidirect
285. C2#10624236direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×C3⋊D32
C22×(C3⋊D3)⋊D52
C2×C3×#12202
#106242362
C2×#671562
C2×#671552
C2×D3×C22⋊D53
C2×#38115