Finite Group Databasebeta

← C2×#72245569 #72245705

Subgroups — C2×#72245569 #72245705

Order 4704.

2360 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 2360 conjugacy classes of subgroups — too many to draw. Only its 340 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
1C2×#11924241C211yesmaximal, characteristic, complemented
2C23×C3⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
3C2×C72×#67C211yesmaximal, characteristic, complemented
4C2×#11924242C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊(C7⋊D7)C211yesmaximal, characteristic, complemented
6#72245569C218yesmaximal, direct product factor
7C2×#11924239C211yesmaximal, characteristic, complemented
8C22×D3×C7⋊D7C211yesmaximal, characteristic, complemented
9C2×D4×C7⋊D7—31nomaximal, Hall
10C23×C3×C72C2211yescharacteristic, complemented
11C22×C3⋊(C7⋊D7)—24no—
12C22×C3⋊(C7⋊D7)C2211yescharacteristic
13C22×C3⋊(C7⋊D7)C2214yescomplemented
14C22×C3⋊(C7⋊D7)C2211yescharacteristic, complemented
15C22×D3×C72C2211yescharacteristic, complemented
16C22×C3×C7⋊D7C2211yescharacteristic, complemented
17C2×D3×C7⋊D7—24no—
18C2×D3×C7⋊D7C2214yescomplemented
19C72×#67C2214yescomplemented
20C2×C72×Dic3C2211yescharacteristic, complemented
21C3×C22⋊(C7⋊D7)C2214yescomplemented
22C2×C3×C7⋊Dic7C2211yescharacteristic, complemented
23#11924242C2214yescomplemented
24#11924241C2214yescomplemented
25#11924239C2214yescomplemented
26C23×C7⋊D7—31no—
27C23×C7⋊D7—31no—
28C2×C72×D4—31no—
29C2×C22⋊(C7⋊D7)—31no—
30C2×C22⋊(C7⋊D7)D311yescharacteristic, complemented
31D4×C7⋊D7—38no—
32C2×C4⋊(C7⋊D7)—31no—
33C2×C4×C7⋊D7—31no—
34C2×#66694—78nomaximal
35C22×C3×C72—22no—
36C22×C3×C72C2311yescharacteristic
37C22×C3×C72C2312yescomplemented
38C2×C3⋊(C7⋊D7)—22no—
39C2×C3⋊(C7⋊D7)—28no—
40C2×C3⋊(C7⋊D7)D414yescomplemented
41C2×C3⋊(C7⋊D7)C2312yes—
42C2×C3⋊(C7⋊D7)C2312yescomplemented
43C2×D3×C72—22no—
44C2×D3×C72C2312yescomplemented
45C2×C3×C7⋊D7—22no—
46C2×C3×C7⋊D7C2312yescomplemented
47D3×C7⋊D7—28no—
48C72×Dic3C2312yescomplemented
49C3×C7⋊Dic7C2312yescomplemented
50C23×C72—31no—
51C23×C72C2×D311yescharacteristic, complemented
52C22×C7⋊D7—31no—
53C22×C7⋊D7—31no—
54C22×C7⋊D7—34no—
55C22×C7⋊D7—34no—
56C22×C7⋊D7—64no—
57C22×C7⋊D7—64no—
58C22×C7⋊D7C2×D311yescharacteristic, complemented
59C72×D4—34no—
60C2×C4×C72—31no—
61C22⋊(C7⋊D7)—34no—
62C22⋊(C7⋊D7)C2×D314yescomplemented
63C2×C7⋊Dic7C2×D311yescharacteristic, complemented
64C4⋊(C7⋊D7)—34no—
65C4×C7⋊D7—34no—
66C23×D21—78no—
67C22×D3×D7—78no—
68C2×C7×#67D718yescomplemented
69C2×C3×C22⋊D7—78no—
70#66694—764no—
71C2×#4247—78no—
72C2×#4246—78no—
73C2×D21⋊C4—78no—
74C2×C3×C72—22no—
75C2×C3×C72C2×D412yescomplemented
76C2×C3×C72C2411yescharacteristic
77C3⋊(C7⋊D7)—22no—
78C3⋊(C7⋊D7)C2×D414yescomplemented
79D3×C72—22no—
80C3×C7⋊D7—22no—
81C2×D4×D7—218no—
82C22×C72—22no—
83C22×C72—32no—
84C22×C72—62no—
85C22×C72C22×D311yescharacteristic
86C22×C72C22×D312yescomplemented
87C2×C7⋊D7—22no—
88C2×C7⋊D7—34no—
89C2×C7⋊D7—32no—
90C2×C7⋊D7—32no—
91C2×C7⋊D7—68no—
92C2×C7⋊D7—68no—
93C2×C7⋊D7—62no—
94C2×C7⋊D7C22×D312yescomplemented
95C4×C72—32no—
96C7⋊Dic7C22×D312yescomplemented
97C23×C3×C7C2×D718yescomplemented
98C22×D21—78no—
99C22×D21—78no—
100C22×D21—732no—
101C22×D21—1432no—
102C22×D3×C7C2×D718yescomplemented
103C22×C3×D7—78no—
104C2×D3×D7—732no—
105C2×D3×D7—1432no—
106C7×#67C2×D7132yescomplemented
107C2×C7×Dic3C2×D718yescomplemented
108C3×C22⋊D7—732no—
109C2×C3×Dic7—78no—
110#4247—732no—
111#4246—732no—
112D21⋊C4—732no—
113C3×C72C22×D411yescharacteristic, complemented, Hall
114C23×D7—218no—
115C23×D7—218no—
116C2×C7×D4—38no—
117C2×C22⋊D7—78no—
118C2×C22⋊D7—218no—
119D4×D7—2164no—
120C2×D28—218no—
121C2×C4×D7—218no—
122C2×C72—22no—
123C2×C72—62no—
124C2×C72D3×D412yescomplemented
125C2×C72C23×D311yescharacteristic
126C7⋊D7—22no—
127C7⋊D7—34no—
128C7⋊D7—62no—
129C2×D3×D4—491noHall
130C22×C3×C7—216no—
131C22×C3×C7C22×D718yes—
132C22×C3×C7C22×D7116yescomplemented
133C2×D21—732no—
134C2×D21—716no—
135C2×D21—716no—
136C2×D21—1464no—
137C2×D21—1416no—
138C2×D3×C7—216no—
139C2×D3×C7C22×D7116yescomplemented
140C2×C3×D7—716no—
141C2×C3×D7—1416no—
142D3×D7—1464no—
143C3×Dic7—716no—
144C7×Dic3C22×D7116yescomplemented
145C23×C7—38no—
146C23×C7D3×D718yescomplemented
147C22×D7—78no—
148C22×D7—2132no—
149C22×D7—218no—
150C22×D7—218no—
151C22×D7—2132no—
152C22×D7—4232no—
153C22×D7—4232no—
154C7×D4—332no—
155C2×C4×C7—38no—
156C22⋊D7—732no—
157C22⋊D7—2132no—
158C2×Dic7—78no—
159D28—2132no—
160C4×D7—2132no—
161C72C2×D3×D411yescharacteristic, complemented, Sylow
162C23×D3—491no—
163C23×D3—491no—
164C2×C3×D4—491no—
165C2×#67—491no—
166C2×#67C7⋊D711yescharacteristic, complemented
167D3×D4—498no—
168C2×D12—491no—
169C2×C4×D3—491no—
170C2×C3×C7—216no—
171C2×C3×C7D4×D7116yescomplemented
172C2×C3×C7C23×D718yes—
173D21—732no—
174D21—1416no—
175C3×D7—1416no—
176D3×C7—216no—
177C22×D4—1471noSylow
178C22×C7—216no—
179C22×C7—316no—
180C22×C7—616no—
181C22×C7C2×D3×D718yes—
182C22×C7C2×D3×D7116yescomplemented
183C2×D7—716no—
184C2×D7—1416no—
185C2×D7—2132no—
186C2×D7—2116no—
187C2×D7—2116no—
188C2×D7—4264no—
189C2×D7—4264no—
190C2×D7—4216no—
191C4×C7—316no—
192Dic7—716no—
193C23×C3—491no—
194C23×C3C2×C7⋊D711yescharacteristic, complemented
195C22×D3—491no—
196C22×D3—491no—
197C22×D3—494no—
198C22×D3—494no—
199C22×D3—984no—
200C22×D3—984no—
201C22×D3C2×C7⋊D711yescharacteristic, complemented
202C3×D4—494no—
203C2×C3×C4—491no—
204#67—494no—
205#67C2×C7⋊D714yescomplemented
206C2×Dic3C2×C7⋊D711yescharacteristic, complemented
207D12—494no—
208C4×D3—494no—
209C3×C7C2×D4×D718yescomplemented
210C24—1471no—
211C24—1471no—
212C2×D4—31no—
213C2×D4—491no—
214C2×D4—1471no—
215C2×D4—1471no—
216C2×D4—1478no—
217C22×C4—1471no—
218C2×C7—216no—
219C2×C7—616no—
220C2×C7#66694116yescomplemented
221C2×C7C22×D3×D718yes—
222D7—1416no—
223D7—2132no—
224D7—4216no—
225C22×C3—22no—
226C22×C3—492no—
227C22×C3—982no—
228C22×C3C22×C7⋊D711yescharacteristic
229C22×C3C22×C7⋊D712yescomplemented
230C2×D3—22no—
231C2×D3—494no—
232C2×D3—492no—
233C2×D3—492no—
234C2×D3—988no—
235C2×D3—988no—
236C2×D3—982no—
237C2×D3C22×C7⋊D712yescomplemented
238C3×C4—492no—
239Dic3C22×C7⋊D712yescomplemented
240C23—31no—
241C23—2944no—
242C23—2944no—
243C23—491no—
244C23—1474no—
245C23—1471no—
246C23—1474no—
247C23—1471no—
248C23D3×C7⋊D711yescharacteristic, complemented
249D4—34no—
250D4—494no—
251D4—1474no—
252D4—1474no—
253C2×C4—31no—
254C2×C4—491no—
255C2×C4—1474no—
256C7C2×#6669418yescomplemented
257C2×C3—22no—
258C2×C3—982no—
259C2×C3D4×C7⋊D712yescomplemented
260C2×C3C23×C7⋊D711yescharacteristic
261D3—22no—
262D3—494no—
263D3—982no—
264C22—22no—
265C22—32no—
266C22—62no—
267C22—2948no—
268C22—2948no—
269C22—2942no—
270C22—492no—
271C22—982no—
272C22—1472no—
273C22—1474no—
274C22—1472no—
275C22C2×D3×C7⋊D711yescharacteristic
276C22C2×D3×C7⋊D712yescomplemented
277C4—32no—
278C4—492no—
279C3C2×D4×C7⋊D711yescharacteristic, complemented, Sylow
280C2—22no—
281C2—62no—
282C2—2942no—
283C2—982no—
284C2—1474no—
285C2C22×D3×C7⋊D711yescharacteristic
286C2#7224556912yesdirect product factor
subgroupG/H
centerC22C2×D3×C7⋊D7
commutator, cosocleC2×C3×C72C24
FrattiniC2C22×D3×C7⋊D7
FittingC23×C3×C72C22
socleC22×C3×C72C23
hypercenterC23D3×C7⋊D7
hypocenterC3×C72C22×D4
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#11924241C2semidirect
2. C23×C3⋊(C7⋊D7)C2semidirect
3. C2×C72×#67C2semidirect
4. C2×#11924242C2semidirect
5. C2×C3×C22⋊(C7⋊D7)C2semidirect
6. #72245569C2direct (N × H)
7. C2×#11924239C2semidirect
8. C22×D3×C7⋊D7C2semidirect
10. C23×C3×C72C22semidirect
13. C22×C3⋊(C7⋊D7)C22semidirect
14. C22×C3⋊(C7⋊D7)C22semidirect
15. C22×D3×C72C22semidirect
16. C22×C3×C7⋊D7C22semidirect
18. C2×D3×C7⋊D7C22semidirect
19. C72×#67C22semidirect
20. C2×C72×Dic3C22semidirect
21. C3×C22⋊(C7⋊D7)C22semidirect
22. C2×C3×C7⋊Dic7C22semidirect
23. #11924242C22semidirect
24. #11924241C22semidirect
25. #11924239C22semidirect
30. C2×C22⋊(C7⋊D7)D3semidirect
37. C22×C3×C72C23semidirect
40. C2×C3⋊(C7⋊D7)D4semidirect
42. C2×C3⋊(C7⋊D7)C23semidirect
44. C2×D3×C72C23semidirect
46. C2×C3×C7⋊D7C23semidirect
48. C72×Dic3C23semidirect
49. C3×C7⋊Dic7C23semidirect
51. C23×C72C2×D3semidirect
58. C22×C7⋊D7C2×D3semidirect
62. C22⋊(C7⋊D7)C2×D3semidirect
63. C2×C7⋊Dic7C2×D3semidirect
68. C2×C7×#67D7semidirect
75. C2×C3×C72C2×D4semidirect
78. C3⋊(C7⋊D7)C2×D4semidirect
86. C22×C72C22×D3semidirect
94. C2×C7⋊D7C22×D3semidirect
96. C7⋊Dic7C22×D3semidirect
97. C23×C3×C7C2×D7semidirect
102. C22×D3×C7C2×D7semidirect
106. C7×#67C2×D7semidirect
107. C2×C7×Dic3C2×D7semidirect
113. C3×C72C22×D4semidirect
124. C2×C72D3×D4semidirect
132. C22×C3×C7C22×D7semidirect
139. C2×D3×C7C22×D7semidirect
144. C7×Dic3C22×D7semidirect
146. C23×C7D3×D7semidirect
161. C72C2×D3×D4semidirect
166. C2×#67C7⋊D7semidirect
171. C2×C3×C7D4×D7semidirect
182. C22×C7C2×D3×D7semidirect
194. C23×C3C2×C7⋊D7semidirect
201. C22×D3C2×C7⋊D7semidirect
205. #67C2×C7⋊D7semidirect
206. C2×Dic3C2×C7⋊D7semidirect
209. C3×C7C2×D4×D7semidirect
220. C2×C7#66694semidirect
229. C22×C3C22×C7⋊D7semidirect
237. C2×D3C22×C7⋊D7semidirect
239. Dic3C22×C7⋊D7semidirect
248. C23D3×C7⋊D7semidirect
256. C7C2×#66694semidirect
259. C2×C3D4×C7⋊D7semidirect
276. C22C2×D3×C7⋊D7semidirect
279. C3C2×D4×C7⋊D7semidirect
286. C2#72245569direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#119242412
C23×C3⋊(C7⋊D7)2
C2×C72×#672
C2×#119242422
C2×C3×C22⋊(C7⋊D7)2
#722455692
C2×#119242392
C22×D3×C7⋊D72
C2×D4×C7⋊D73
C2×#666947