Finite Group Databasebeta

← C2×D3×D5×D7 #13132946

Subgroups — C2×D3×D5×D7 #13132946

Order 1680.

536 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 132 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×D15⋊D7C211yesmaximal, characteristic, complemented
2C2×D3×D35C211yesmaximal, characteristic, complemented
3C2×D5×D21C211yesmaximal, characteristic, complemented
4C2×D3×C7×D5C211yesmaximal, characteristic, complemented
5C2×D7×D15C211yesmaximal, characteristic, complemented
6C2×C5×D3×D7C211yesmaximal, characteristic, complemented
7C2×C3×D5×D7C211yesmaximal, characteristic, complemented
8D3×D5×D7C218yesmaximal, direct product factor
9C22×D5×D7—31nomaximal, Hall
10C2×D105C2211yescharacteristic, complemented
11C2×C7×D15C2211yescharacteristic, complemented
12C2×C5×D21C2211yescharacteristic, complemented
13C2×C5×D3×C7C2211yescharacteristic, complemented
14C2×C3×D35C2211yescharacteristic, complemented
15C2×C3×C7×D5C2211yescharacteristic, complemented
16C2×C3×C5×D7C2211yescharacteristic, complemented
17D15⋊D7C2214yescomplemented
18D3×D35C2214yescomplemented
19D5×D21C2214yescomplemented
20D3×C7×D5C2214yescomplemented
21D7×D15C2214yescomplemented
22C5×D3×D7C2214yescomplemented
23C3×D5×D7C2214yescomplemented
24C22×D3×D7—51nomaximal, Hall
25C22×D35—31no—
26C22×C7×D5—31no—
27C22×C5×D7—31no—
28C2×D5×D7—31no—
29C2×D5×D7—31no—
30C2×D5×D7—31no—
31C2×D5×D7—38no—
32C2×D5×D7D311yescharacteristic, direct product factor
33C22×D3×D5—71nomaximal, Hall
34C2×C3×C5×C7C2311yescharacteristic, complemented
35D105C2312yescomplemented
36C7×D15C2312yescomplemented
37C5×D21C2312yescomplemented
38C5×D3×C7C2312yescomplemented
39C3×D35C2312yescomplemented
40C3×C7×D5C2312yescomplemented
41C3×C5×D7C2312yescomplemented
42C22×D21—51no—
43C22×D3×C7—51no—
44C22×C3×D7—51no—
45C2×D3×D7—51no—
46C2×D3×D7—51no—
47C2×D3×D7—51no—
48C2×D3×D7—58no—
49C2×D3×D7D511yescharacteristic, direct product factor
50C22×C5×C7—31no—
51C2×D35—31no—
52C2×D35—34no—
53C2×D35C2×D311yescharacteristic, complemented
54C2×C7×D5—31no—
55C2×C7×D5—34no—
56C2×C7×D5C2×D311yescharacteristic, complemented
57C2×C5×D7—31no—
58C2×C5×D7—34no—
59C2×C5×D7C2×D311yescharacteristic, complemented
60D5×D7—34no—
61D5×D7—34no—
62D5×D7—34no—
63D5×D7C2×D314yesdirect product factor
64C22×D15—71no—
65C22×C5×D3—71no—
66C22×C3×D5—71no—
67C2×D3×D5—71no—
68C2×D3×D5—71no—
69C2×D3×D5—71no—
70C2×D3×D5—78no—
71C2×D3×D5D711yescharacteristic, direct product factor
72C23×D7—151noHall
73C3×C5×C7C2411yescharacteristic, complemented, Hall
74C22×C3×C7—51no—
75C2×D21—51no—
76C2×D21—54no—
77C2×D21C2×D511yescharacteristic, complemented
78C2×D3×C7—51no—
79C2×D3×C7—54no—
80C2×D3×C7C2×D511yescharacteristic, complemented
81C2×C3×D7—51no—
82C2×C3×D7—54no—
83C2×C3×D7C2×D511yescharacteristic, complemented
84D3×D7—54no—
85D3×D7—54no—
86D3×D7—54no—
87D3×D7C2×D514yesdirect product factor
88C23×D5—211noHall
89C2×C5×C7—32no—
90C2×C5×C7C22×D311yescharacteristic, complemented
91D35—32no—
92D35C22×D312yescomplemented
93C5×D7—32no—
94C5×D7C22×D312yescomplemented
95C7×D5—32no—
96C7×D5C22×D312yescomplemented
97C22×C3×C5—71no—
98C2×D15—71no—
99C2×D15—74no—
100C2×D15C2×D711yescharacteristic, complemented
101C2×C5×D3—71no—
102C2×C5×D3—74no—
103C2×C5×D3C2×D711yescharacteristic, complemented
104C2×C3×D5—71no—
105C2×C3×D5—74no—
106C2×C3×D5C2×D711yescharacteristic, complemented
107D3×D5—74no—
108D3×D5—74no—
109D3×D5—74no—
110D3×D5C2×D714yesdirect product factor
111C23×C7—151no—
112C22×D7—31no—
113C22×D7—51no—
114C22×D7—151no—
115C22×D7—158no—
116C22×D7—151no—
117C22×D7—151no—
118C22×D7—151no—
119C23×D3—351noHall
120C2×C3×C7—52no—
121C2×C3×C7C22×D511yescharacteristic, complemented
122D21—52no—
123D21C22×D512yescomplemented
124C3×D7—52no—
125C3×D7C22×D512yescomplemented
126D3×C7—52no—
127D3×C7C22×D512yescomplemented
128C23×C5—211no—
129C22×D5—31no—
130C22×D5—71no—
131C22×D5—211no—
132C22×D5—218no—
133C22×D5—211no—
134C22×D5—211no—
135C22×D5—211no—
136C5×C7C23×D311yescharacteristic, complemented, Hall
137C2×C3×C5—72no—
138C2×C3×C5C22×D711yescharacteristic, complemented
139D15—72no—
140D15C22×D712yescomplemented
141C3×D5—72no—
142C3×D5C22×D712yescomplemented
143C5×D3—72no—
144C5×D3C22×D712yescomplemented
145C22×C7—31no—
146C22×C7—51no—
147C22×C7—154no—
148C22×C7—151no—
149C2×D7—31no—
150C2×D7—34no—
151C2×D7—51no—
152C2×D7—54no—
153C2×D7—154no—
154C2×D7—154no—
155C2×D7—154no—
156C2×D7—151no—
157C2×D7—154no—
158C2×D7D3×D511yescharacteristic, direct product factor
159C23×C3—351no—
160C22×D3—51no—
161C22×D3—71no—
162C22×D3—351no—
163C22×D3—358no—
164C22×D3—351no—
165C22×D3—351no—
166C22×D3—351no—
167C3×C7C23×D511yescharacteristic, complemented, Hall
168C22×C5—31no—
169C22×C5—71no—
170C22×C5—214no—
171C22×C5—211no—
172C2×D5—31no—
173C2×D5—34no—
174C2×D5—71no—
175C2×D5—74no—
176C2×D5—214no—
177C2×D5—214no—
178C2×D5—214no—
179C2×D5—211no—
180C2×D5—214no—
181C2×D5D3×D711yescharacteristic, direct product factor
182C24—1051noSylow
183C3×C5C23×D711yescharacteristic, complemented, Hall
184C2×C7—32no—
185C2×C7—52no—
186C2×C7—152no—
187C2×C7C2×D3×D511yescharacteristic, complemented
188D7—32no—
189D7—52no—
190D7—152no—
191D7C2×D3×D512yesdirect product factor
192C22×C3—51no—
193C22×C3—71no—
194C22×C3—354no—
195C22×C3—351no—
196C2×D3—51no—
197C2×D3—54no—
198C2×D3—71no—
199C2×D3—74no—
200C2×D3—354no—
201C2×D3—354no—
202C2×D3—354no—
203C2×D3—351no—
204C2×D3—354no—
205C2×D3D5×D711yescharacteristic, direct product factor
206C2×C5—32no—
207C2×C5—72no—
208C2×C5—212no—
209C2×C5C2×D3×D711yescharacteristic, complemented
210D5—32no—
211D5—72no—
212D5—212no—
213D5C2×D3×D712yesdirect product factor
214C23—151no—
215C23—211no—
216C23—351no—
217C23—1051no—
218C23—1051no—
219C23—1051no—
220C23—1058no—
221C23—1051no—
222C7C22×D3×D511yescharacteristic, complemented, Sylow
223C2×C3—52no—
224C2×C3—72no—
225C2×C3—352no—
226C2×C3C2×D5×D711yescharacteristic, complemented
227D3—52no—
228D3—72no—
229D3—352no—
230D3C2×D5×D712yesdirect product factor
231C5C22×D3×D711yescharacteristic, complemented, Sylow
232C22—31no—
233C22—51no—
234C22—71no—
235C22—154no—
236C22—151no—
237C22—214no—
238C22—211no—
239C22—354no—
240C22—351no—
241C22—1054no—
242C22—1054no—
243C22—1054no—
244C22—1054no—
245C22—1051no—
246C3C22×D5×D711yescharacteristic, complemented, Sylow
247C2—32no—
248C2—52no—
249C2—72no—
250C2—152no—
251C2—212no—
252C2—352no—
253C2—1052no—
254C2D3×D5×D711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D3×D5×D7
commutator, cosocle, hypocenterC3×C5×C7C24
FrattiniC1G
Fitting, socleC2×C3×C5×C7C23
LCSG ▷ C3×C5×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×D15⋊D7C2semidirect
2. C2×D3×D35C2semidirect
3. C2×D5×D21C2semidirect
4. C2×D3×C7×D5C2semidirect
5. C2×D7×D15C2semidirect
6. C2×C5×D3×D7C2semidirect
7. C2×C3×D5×D7C2semidirect
8. D3×D5×D7C2direct (N × H)
10. C2×D105C22semidirect
11. C2×C7×D15C22semidirect
12. C2×C5×D21C22semidirect
13. C2×C5×D3×C7C22semidirect
14. C2×C3×D35C22semidirect
15. C2×C3×C7×D5C22semidirect
16. C2×C3×C5×D7C22semidirect
17. D15⋊D7C22semidirect
18. D3×D35C22semidirect
19. D5×D21C22semidirect
20. D3×C7×D5C22semidirect
21. D7×D15C22semidirect
22. C5×D3×D7C22semidirect
23. C3×D5×D7C22semidirect
32. C2×D5×D7D3direct (N × H)
34. C2×C3×C5×C7C23semidirect
35. D105C23semidirect
36. C7×D15C23semidirect
37. C5×D21C23semidirect
38. C5×D3×C7C23semidirect
39. C3×D35C23semidirect
40. C3×C7×D5C23semidirect
41. C3×C5×D7C23semidirect
49. C2×D3×D7D5direct (N × H)
53. C2×D35C2×D3semidirect
56. C2×C7×D5C2×D3semidirect
59. C2×C5×D7C2×D3semidirect
63. D5×D7C2×D3direct (N × H)
71. C2×D3×D5D7direct (N × H)
73. C3×C5×C7C24semidirect
77. C2×D21C2×D5semidirect
80. C2×D3×C7C2×D5semidirect
83. C2×C3×D7C2×D5semidirect
87. D3×D7C2×D5direct (N × H)
90. C2×C5×C7C22×D3semidirect
92. D35C22×D3semidirect
94. C5×D7C22×D3semidirect
96. C7×D5C22×D3semidirect
100. C2×D15C2×D7semidirect
103. C2×C5×D3C2×D7semidirect
106. C2×C3×D5C2×D7semidirect
110. D3×D5C2×D7direct (N × H)
121. C2×C3×C7C22×D5semidirect
123. D21C22×D5semidirect
125. C3×D7C22×D5semidirect
127. D3×C7C22×D5semidirect
136. C5×C7C23×D3semidirect
138. C2×C3×C5C22×D7semidirect
140. D15C22×D7semidirect
142. C3×D5C22×D7semidirect
144. C5×D3C22×D7semidirect
158. C2×D7D3×D5direct (N × H)
167. C3×C7C23×D5semidirect
181. C2×D5D3×D7direct (N × H)
183. C3×C5C23×D7semidirect
187. C2×C7C2×D3×D5semidirect
191. D7C2×D3×D5direct (N × H)
205. C2×D3D5×D7direct (N × H)
209. C2×C5C2×D3×D7semidirect
213. D5C2×D3×D7direct (N × H)
222. C7C22×D3×D5semidirect
226. C2×C3C2×D5×D7semidirect
230. D3C2×D5×D7direct (N × H)
231. C5C22×D3×D7semidirect
246. C3C22×D5×D7semidirect
254. C2D3×D5×D7direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D15⋊D72
C2×D3×D352
C2×D5×D212
C2×D3×C7×D52
C2×D7×D152
C2×C5×D3×D72
C2×C3×D5×D72
D3×D5×D72
C22×D5×D73
C22×D3×D75
C22×D3×D57