Finite Group Databasebeta

← C2×C3⋊D3×C22⋊D5 #13089304

Subgroups — C2×C3⋊D3×C22⋊D5 #13089304

Order 1440.

1416 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 240 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×D5×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#67184C211yesmaximal, characteristic, complemented
4C2×C32×C22⋊D5C211yesmaximal, characteristic, complemented
5C3⋊D3×C22⋊D5C218yesmaximal, direct product factor
6C2×#67140C211yesmaximal, characteristic, complemented
7C2×#67138C211yesmaximal, characteristic, complemented
8C2×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
9C2×D3×C22⋊D5—34nomaximal
10C23×C32×C5C2211yescharacteristic, complemented
11C22×C3⋊D15C2211yescharacteristic, complemented
12C22×C5×C3⋊D3—24no—
13C22×C5×C3⋊D3C2211yescharacteristic
14C22×C5×C3⋊D3C2214yescomplemented
15C22×C5×C3⋊D3C2211yescharacteristic, complemented
16C22×C32×D5C2211yescharacteristic, complemented
17C2×D5×C3⋊D3—24no—
18C2×D5×C3⋊D3C2214yescomplemented
19#67184C2214yescomplemented
20C2×C3⋊Dic15C2211yescharacteristic, complemented
21C32×C22⋊D5C2214yescomplemented
22C2×C32×Dic5C2211yescharacteristic, complemented
23#67140C2214yescomplemented
24#67138C2214yescomplemented
25C3⋊D3×Dic5C2214yescomplemented
26C2×D4×C3⋊D3—51nomaximal, Hall
27C23×C5×D3—34no—
28C22×D3×D5—34no—
29C2×#1220—34no—
30C2×C3×C22⋊D5D314yescomplemented
31D3×C22⋊D5—332no—
32C2×#1203—34no—
33C2×#1201—34no—
34C2×D3×Dic5—34no—
35C22×C32×C5—22no—
36C22×C32×C5C2311yescharacteristic
37C22×C32×C5C2312yescomplemented
38C2×C3⋊D15—22no—
39C2×C3⋊D15C2312yescomplemented
40C2×C5×C3⋊D3—22no—
41C2×C5×C3⋊D3—28no—
42C2×C5×C3⋊D3D414yescomplemented
43C2×C5×C3⋊D3C2312yes—
44C2×C5×C3⋊D3C2312yescomplemented
45C2×C32×D5—22no—
46C2×C32×D5C2312yescomplemented
47D5×C3⋊D3—28no—
48C3⋊Dic15C2312yescomplemented
49C32×Dic5C2312yescomplemented
50C22×C22⋊D5—91noHall
51C23×C3⋊D3—51no—
52C23×C3⋊D3D511yescharacteristic, complemented
53C2×C32×D4—51no—
54C2×#638—51no—
55C2×#638—51no—
56D4×C3⋊D3—58no—
57C2×C4⋊(C3⋊D3)—51no—
58C2×C4×C3⋊D3—51no—
59C23×C3×C5C2×D314yescomplemented
60C22×D15—34no—
61C22×C5×D3—34no—
62C22×C5×D3—34no—
63C22×C5×D3—316no—
64C22×C5×D3—616no—
65C22×C3×D5C2×D314yescomplemented
66C2×D3×D5—316no—
67C2×D3×D5—616no—
68#1220—316no—
69C2×Dic15—34no—
70C3×C22⋊D5C2×D3116yescomplemented
71C2×C3×Dic5C2×D314yescomplemented
72#1203—316no—
73#1201—316no—
74D3×Dic5—316no—
75C2×D3×D4—154no—
76C2×C32×C5—22no—
77C2×C32×C5C2×D412yescomplemented
78C2×C32×C5C2411yescharacteristic
79C3⋊D15—22no—
80C5×C3⋊D3—22no—
81C5×C3⋊D3C2×D414yescomplemented
82C32×D5—22no—
83C24×C5—91no—
84C23×D5—91no—
85C2×C22⋊D5—91no—
86C2×C22⋊D5—91no—
87C2×C22⋊D5—98no—
88C2×C22⋊D5—91no—
89C2×C22⋊D5C3⋊D311yescharacteristic, direct product factor
90C22×Dic5—91no—
91C23×C32—51no—
92C23×C32C2×D511yescharacteristic, complemented
93C22×C3⋊D3—24no—
94C22×C3⋊D3—51no—
95C22×C3⋊D3—54no—
96C22×C3⋊D3—104no—
97C22×C3⋊D3C2×D511yescharacteristic
98C22×C3⋊D3C2×D514yescomplemented
99C22×C3⋊D3C2×D511yescharacteristic, complemented
100C32×D4—54no—
101C2×C32×C4—51no—
102#638—54no—
103#638—54no—
104C2×C3⋊Dic3—51no—
105C4⋊(C3⋊D3)—54no—
106C4×C3⋊D3—54no—
107C22×C3×C5—28no—
108C22×C3×C5C22×D314yes—
109C22×C3×C5C22×D318yescomplemented
110C2×D15—38no—
111C2×D15—68no—
112C2×C5×D3—316no—
113C2×C5×D3—38no—
114C2×C5×D3—38no—
115C2×C5×D3—632no—
116C2×C5×D3—68no—
117C2×C3×D5—28no—
118C2×C3×D5C22×D318yescomplemented
119D3×D5—632no—
120Dic15—38no—
121C3×Dic5C22×D318yescomplemented
122C23×D3—34no—
123C23×D3—154no—
124C2×C3×D4—54no—
125C2×#67—154no—
126C2×#67—154no—
127D3×D4—1532no—
128C2×D12—154no—
129C2×C4×D3—154no—
130C32×C5C22×D411yescharacteristic, complemented, Hall
131C23×C5—91no—
132C23×C5—91no—
133C23×C5—94no—
134C23×C5—184no—
135C23×C5C2×C3⋊D311yescharacteristic, complemented
136C22×D5—91no—
137C22×D5—94no—
138C22×D5—184no—
139C22×D5C2×C3⋊D311yescharacteristic, complemented
140C22⋊D5—94no—
141C22⋊D5—94no—
142C22⋊D5—94no—
143C22⋊D5C2×C3⋊D314yesdirect product factor
144C2×Dic5—91no—
145C2×Dic5—94no—
146C2×Dic5C2×C3⋊D311yescharacteristic, complemented
147C22×C32—22no—
148C22×C32—52no—
149C22×C32—102no—
150C22×C32C22×D511yescharacteristic
151C22×C32C22×D512yescomplemented
152C2×C3⋊D3—22no—
153C2×C3⋊D3—28no—
154C2×C3⋊D3—52no—
155C2×C3⋊D3—102no—
156C2×C3⋊D3—108no—
157C2×C3⋊D3C22⋊D514yesdirect product factor
158C2×C3⋊D3C22×D512yes—
159C2×C3⋊D3C22×D512yescomplemented
160C32×C4—52no—
161C3⋊Dic3—52no—
162C22×D4—451noSylow
163C2×C3×C5—28no—
164C2×C3×C5D3×D418yescomplemented
165C2×C3×C5C23×D314yes—
166D15—68no—
167C3×D5—28no—
168C5×D3—316no—
169C5×D3—68no—
170C23×C3—54no—
171C23×C3D3×D514yescomplemented
172C22×D3—34no—
173C22×D3—34no—
174C22×D3—316no—
175C22×D3—616no—
176C22×D3—1516no—
177C22×D3—154no—
178C22×D3—3016no—
179C3×D4—516no—
180C2×C3×C4—54no—
181#67—1516no—
182#67—1516no—
183C2×Dic3—154no—
184D12—1516no—
185C4×D3—1516no—
186C22×C5—22no—
187C22×C5—92no—
188C22×C5—94no—
189C22×C5—92no—
190C22×C5—188no—
191C22×C5—182no—
192C22×C5C22×C3⋊D311yescharacteristic
193C22×C5C22×C3⋊D312yescomplemented
194C2×D5—22no—
195C2×D5—92no—
196C2×D5—188no—
197C2×D5—182no—
198C2×D5C22×C3⋊D312yescomplemented
199Dic5—92no—
200Dic5C22×C3⋊D312yescomplemented
201C2×C32—22no—
202C2×C32—102no—
203C2×C32C2×C22⋊D512yescomplemented
204C2×C32C23×D511yescharacteristic
205C3⋊D3—22no—
206C3⋊D3—102no—
207C3⋊D3C2×C22⋊D514yesdirect product factor
208C24—91no—
209C24—451no—
210C2×D4—51no—
211C2×D4—451no—
212C2×D4—458no—
213C2×D4—451no—
214C2×D4—451no—
215C22×C4—451no—
216C3×C5C2×D3×D414yescomplemented
217C22×C3—28no—
218C22×C3—58no—
219C22×C3—108no—
220C22×C3C2×D3×D514yes—
221C22×C3C2×D3×D518yescomplemented
222C2×D3—316no—
223C2×D3—38no—
224C2×D3—38no—
225C2×D3—632no—
226C2×D3—68no—
227C2×D3—158no—
228C2×D3—3032no—
229C2×D3—308no—
230C3×C4—58no—
231Dic3—158no—
232C2×C5—22no—
233C2×C5—94no—
234C2×C5—182no—
235C2×C5D4×C3⋊D312yescomplemented
236C2×C5C23×C3⋊D311yescharacteristic
237D5—22no—
238D5—182no—
239C32C22×C22⋊D511yescharacteristic, complemented, Sylow
240C23—51no—
241C23—91no—
242C23—91no—
243C23—94no—
244C23—184no—
245C23—454no—
246C23—451no—
247C23—904no—
248C23D5×C3⋊D311yescharacteristic, complemented
249D4—54no—
250D4—454no—
251D4—454no—
252D4—454no—
253C2×C4—51no—
254C2×C4—454no—
255C2×C4—451no—
256C2×C3—28no—
257C2×C3—108no—
258C2×C3D3×C22⋊D518yescomplemented
259C2×C3C22×D3×D514yes—
260D3—316no—
261D3—68no—
262D3—308no—
263C5C2×D4×C3⋊D311yescharacteristic, complemented, Sylow
264C22—22no—
265C22—52no—
266C22—92no—
267C22—94no—
268C22—92no—
269C22—102no—
270C22—188no—
271C22—182no—
272C22—452no—
273C22—908no—
274C22—902no—
275C22C2×D5×C3⋊D311yescharacteristic
276C22C2×D5×C3⋊D312yescomplemented
277C4—52no—
278C4—452no—
279C3C2×D3×C22⋊D514yescomplemented
280C2—22no—
281C2—94no—
282C2—102no—
283C2—182no—
284C2—902no—
285C2C3⋊D3×C22⋊D512yesdirect product factor
286C2C22×D5×C3⋊D311yescharacteristic
subgroupG/H
centerC22C2×D5×C3⋊D3
commutator, cosocleC2×C32×C5C24
FrattiniC2C22×D5×C3⋊D3
FittingC23×C32×C5C22
socleC22×C32×C5C23
hypercenterC23D5×C3⋊D3
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×D5×C3⋊D3C2semidirect
3. C2×#67184C2semidirect
4. C2×C32×C22⋊D5C2semidirect
5. C3⋊D3×C22⋊D5C2direct (N × H)
6. C2×#67140C2semidirect
7. C2×#67138C2semidirect
8. C2×C3⋊D3×Dic5C2semidirect
10. C23×C32×C5C22semidirect
11. C22×C3⋊D15C22semidirect
14. C22×C5×C3⋊D3C22semidirect
15. C22×C5×C3⋊D3C22semidirect
16. C22×C32×D5C22semidirect
18. C2×D5×C3⋊D3C22semidirect
19. #67184C22semidirect
20. C2×C3⋊Dic15C22semidirect
21. C32×C22⋊D5C22semidirect
22. C2×C32×Dic5C22semidirect
23. #67140C22semidirect
24. #67138C22semidirect
25. C3⋊D3×Dic5C22semidirect
30. C2×C3×C22⋊D5D3semidirect
37. C22×C32×C5C23semidirect
39. C2×C3⋊D15C23semidirect
42. C2×C5×C3⋊D3D4semidirect
44. C2×C5×C3⋊D3C23semidirect
46. C2×C32×D5C23semidirect
48. C3⋊Dic15C23semidirect
49. C32×Dic5C23semidirect
52. C23×C3⋊D3D5semidirect
59. C23×C3×C5C2×D3semidirect
65. C22×C3×D5C2×D3semidirect
70. C3×C22⋊D5C2×D3semidirect
71. C2×C3×Dic5C2×D3semidirect
77. C2×C32×C5C2×D4semidirect
81. C5×C3⋊D3C2×D4semidirect
89. C2×C22⋊D5C3⋊D3direct (N × H)
92. C23×C32C2×D5semidirect
98. C22×C3⋊D3C2×D5semidirect
99. C22×C3⋊D3C2×D5semidirect
109. C22×C3×C5C22×D3semidirect
118. C2×C3×D5C22×D3semidirect
121. C3×Dic5C22×D3semidirect
130. C32×C5C22×D4semidirect
135. C23×C5C2×C3⋊D3semidirect
139. C22×D5C2×C3⋊D3semidirect
143. C22⋊D5C2×C3⋊D3direct (N × H)
146. C2×Dic5C2×C3⋊D3semidirect
151. C22×C32C22×D5semidirect
157. C2×C3⋊D3C22⋊D5direct (N × H)
159. C2×C3⋊D3C22×D5semidirect
164. C2×C3×C5D3×D4semidirect
171. C23×C3D3×D5semidirect
193. C22×C5C22×C3⋊D3semidirect
198. C2×D5C22×C3⋊D3semidirect
200. Dic5C22×C3⋊D3semidirect
203. C2×C32C2×C22⋊D5semidirect
207. C3⋊D3C2×C22⋊D5direct (N × H)
216. C3×C5C2×D3×D4semidirect
221. C22×C3C2×D3×D5semidirect
235. C2×C5D4×C3⋊D3semidirect
239. C32C22×C22⋊D5semidirect
248. C23D5×C3⋊D3semidirect
258. C2×C3D3×C22⋊D5semidirect
263. C5C2×D4×C3⋊D3semidirect
276. C22C2×D5×C3⋊D3semidirect
279. C3C2×D3×C22⋊D5semidirect
285. C2C3⋊D3×C22⋊D5direct (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×D5×C3⋊D32
C2×#671842
C2×C32×C22⋊D52
C3⋊D3×C22⋊D52
C2×#671402
C2×#671382
C2×C3⋊D3×Dic52
C2×D3×C22⋊D53
C2×D4×C3⋊D35