Finite Group Databasebeta

← C2×D3×C7×D8 #13061470

Subgroups — C2×D3×C7×D8 #13061470

Order 1344.

676 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 222 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×D3×C7×D4C212yesmaximal, complemented
2C2×C3×C7×D8C211yesmaximal, characteristic, complemented
3C2×C7×#213C212yesmaximal, complemented
4D3×C7×D8C218yesmaximal, direct product factor
5C2×C7×D24C211yesmaximal, characteristic, complemented
6C2×D3×C7×C8C211yesmaximal, characteristic, complemented
7C22×C7×D8—31nomaximal, Hall
8C23×D3×C7—22no—
9C2×C3×C7×D4C2212yescomplemented
10C2×C7×#67—22no—
11D3×C7×D4—24no—
12D3×C7×D4C2218yescomplemented
13C2×C7×D12C2212yescomplemented
14C2×C4×D3×C7C2211yescharacteristic
15C3×C7×D8C2214yescomplemented
16C2×C3×C7×C8C2211yescharacteristic, complemented
17C7×#213C2218yescomplemented
18C2×C7×C3⋊C8C2211yescharacteristic, complemented
19C7×D24C2214yescomplemented
20D3×C7×C8C2214yescomplemented
21C22×C7×D4—32no—
22C2×C7×D8—32no—
23C2×C7×D8—38no—
24C2×C7×D8—31no—
25C2×C7×D8D311yescharacteristic, direct product factor
26C22×C7×C8—31no—
27C2×D3×D8C711yesmaximal, characteristic, direct product factor, Hall
28C23×C3×C7—22no—
29C22×D3×C7—22no—
30C22×D3×C7—28no—
31C22×D3×C7—48no—
32C22×D3×C7D411yescharacteristic
33C3×C7×D4—22no—
34C3×C7×D4C2314yescomplemented
35C2×C3×C4×C7C2311yescharacteristic
36C7×#67—28no—
37C2×C7×Dic3D411yescharacteristic
38C7×D12—22no—
39C7×D12C2314yescomplemented
40C4×D3×C7D412yes—
41C4×D3×C7C2312yes—
42C3×C7×C8C2312yescomplemented
43C7×C3⋊C8C2312yescomplemented
44C24×C7—62no—
45C2×C7×D4—38no—
46C2×C7×D4—32no—
47C2×C7×D4—64no—
48C2×C7×D4—62no—
49C2×C7×D4C2×D312yescomplemented
50C22×C4×C7—31no—
51C7×D8—38no—
52C7×D8—34no—
53C7×D8C2×D314yesdirect product factor
54C2×C7×C8—31no—
55C2×C7×C8—34no—
56C2×C7×C8C2×D311yescharacteristic, complemented
57C2×D3×D4C2×C712yescomplemented
58C2×C3×D8C2×C711yescharacteristic, complemented
59C2×#213C2×C712yescomplemented
60D3×D8C2×C718yesdirect product factor
61C2×D24C2×C711yescharacteristic, complemented
62C2×D3×C8C2×C711yescharacteristic, complemented
63C22×C3×C7—24no—
64C22×C3×C7—44no—
65C22×C3×C7C2×D411yescharacteristic
66C2×D3×C7—24no—
67C2×D3×C7—44no—
68C2×D3×C7—416no—
69C2×D3×C7D814yesdirect product factor
70C2×D3×C7C2×D412yes—
71C3×C4×C7C2×D411yescharacteristic
72C3×C4×C7C2411yescharacteristic
73C7×Dic3C2×D412yes—
74C22×D8—31noSylow
75C23×C7—22no—
76C23×C7—31no—
77C23×C7—68no—
78C23×C7—62no—
79C23×C7—128no—
80C7×D4—22no—
81C7×D4—34no—
82C7×D4—68no—
83C7×D4—62no—
84C7×D4C22×D314yescomplemented
85C2×C4×C7—31no—
86C2×C4×C7—32no—
87C2×C4×C7—32no—
88C2×C4×C7C22×D311yescharacteristic
89C7×C8—32no—
90C7×C8C22×D312yescomplemented
91C23×D3—22no—
92C2×C3×D4C22×C712yescomplemented
93C2×#67—22no—
94D3×D4—24no—
95D3×D4C22×C718yescomplemented
96C2×D12C22×C712yescomplemented
97C2×C4×D3C22×C711yescharacteristic
98C3×D8C22×C714yescomplemented
99C2×C3×C8C22×C711yescharacteristic, complemented
100#213C22×C718yescomplemented
101C2×C3⋊C8C22×C711yescharacteristic, complemented
102D24C22×C714yescomplemented
103D3×C8C22×C714yescomplemented
104C2×C3×C7—44no—
105C2×C3×C7C2×D812yescomplemented
106C2×C3×C7C22×D411yescharacteristic
107D3×C7—44no—
108D3×C7C2×D814yesdirect product factor
109C22×D4—32no—
110C2×D8—32no—
111C2×D8—38no—
112C2×D8—31no—
113C2×D8D3×C711yescharacteristic, direct product factor
114C22×C8—31no—
115C22×C7—24no—
116C22×C7—34no—
117C22×C7—32no—
118C22×C7—44no—
119C22×C7—64no—
120C22×C7—1216no—
121C22×C7—124no—
122C22×C7D3×D411yescharacteristic
123C4×C7—32no—
124C4×C7D3×D411yescharacteristic
125C4×C7C23×D311yescharacteristic
126C23×C3—22no—
127C22×D3—22no—
128C22×D3—28no—
129C22×D3—48no—
130C22×D3C7×D411yescharacteristic
131C3×D4—22no—
132C3×D4C23×C714yescomplemented
133C2×C3×C4C23×C711yescharacteristic
134#67—28no—
135C2×Dic3C7×D411yescharacteristic
136D12—22no—
137D12C23×C714yescomplemented
138C4×D3C7×D412yes—
139C4×D3C23×C712yes—
140C3×C8C23×C712yescomplemented
141C3⋊C8C23×C712yescomplemented
142C3×C7C22×D811yescharacteristic, complemented, Hall
143C24—62no—
144C2×D4—38no—
145C2×D4—32no—
146C2×D4—64no—
147C2×D4—62no—
148C2×D4C2×D3×C712yescomplemented
149C22×C4—31no—
150D8—38no—
151D8—34no—
152D8C2×D3×C714yesdirect product factor
153C2×C8—31no—
154C2×C8—34no—
155C2×C8C2×D3×C711yescharacteristic, complemented
156C2×C7—34no—
157C2×C7—44no—
158C2×C7—124no—
159C2×C7D3×D812yesdirect product factor
160C2×C7C2×D3×D411yescharacteristic
161C22×C3—24no—
162C22×C3—44no—
163C22×C3C2×C7×D411yescharacteristic
164C2×D3—24no—
165C2×D3—44no—
166C2×D3—416no—
167C2×D3C7×D814yesdirect product factor
168C2×D3C2×C7×D412yes—
169C3×C4C2×C7×D411yescharacteristic
170C3×C4C24×C711yescharacteristic
171Dic3C2×C7×D412yes—
172C23—22no—
173C23—31no—
174C23—68no—
175C23—62no—
176C23—128no—
177D4—22no—
178D4—34no—
179D4—68no—
180D4—62no—
181D4C22×D3×C714yescomplemented
182C2×C4—31no—
183C2×C4—32no—
184C2×C4—32no—
185C2×C4C22×D3×C711yescharacteristic
186C8—32no—
187C8C22×D3×C712yescomplemented
188C7C2×D3×D811yescharacteristic, direct product factor, Sylow
189C2×C3—44no—
190C2×C3C2×C7×D812yescomplemented
191C2×C3C22×C7×D411yescharacteristic
192D3—44no—
193D3C2×C7×D814yesdirect product factor
194C22—24no—
195C22—34no—
196C22—32no—
197C22—44no—
198C22—64no—
199C22—1216no—
200C22—124no—
201C22D3×C7×D411yescharacteristic
202C4—32no—
203C4D3×C7×D411yescharacteristic
204C4C23×D3×C711yescharacteristic
205C3C22×C7×D811yescharacteristic, complemented, Sylow
206C2—34no—
207C2—44no—
208C2—124no—
209C2D3×C7×D812yesdirect product factor
210C2C2×D3×C7×D411yescharacteristic
subgroupG/H
centerC22×C7D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC2×C3×C7×D8C2
socleC22×C3×C7C2×D4
hypercenterC2×C7×D8D3
hypocenterC3C22×C7×D8
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C7 ◁ C2×C4×C7 ◁ C2×C7×D8

Complements

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

NH ≅ G/Ntype
1. C2×D3×C7×D4C2semidirect
2. C2×C3×C7×D8C2semidirect
3. C2×C7×#213C2semidirect
4. D3×C7×D8C2direct (N × H)
5. C2×C7×D24C2semidirect
6. C2×D3×C7×C8C2semidirect
9. C2×C3×C7×D4C22semidirect
12. D3×C7×D4C22semidirect
13. C2×C7×D12C22semidirect
15. C3×C7×D8C22semidirect
16. C2×C3×C7×C8C22semidirect
17. C7×#213C22semidirect
18. C2×C7×C3⋊C8C22semidirect
19. C7×D24C22semidirect
20. D3×C7×C8C22semidirect
25. C2×C7×D8D3direct (N × H)
27. C2×D3×D8C7direct (N × H)
34. C3×C7×D4C23semidirect
39. C7×D12C23semidirect
42. C3×C7×C8C23semidirect
43. C7×C3⋊C8C23semidirect
49. C2×C7×D4C2×D3semidirect
53. C7×D8C2×D3direct (N × H)
56. C2×C7×C8C2×D3semidirect
57. C2×D3×D4C2×C7semidirect
58. C2×C3×D8C2×C7semidirect
59. C2×#213C2×C7semidirect
60. D3×D8C2×C7direct (N × H)
61. C2×D24C2×C7semidirect
62. C2×D3×C8C2×C7semidirect
69. C2×D3×C7D8direct (N × H)
84. C7×D4C22×D3semidirect
90. C7×C8C22×D3semidirect
92. C2×C3×D4C22×C7semidirect
95. D3×D4C22×C7semidirect
96. C2×D12C22×C7semidirect
98. C3×D8C22×C7semidirect
99. C2×C3×C8C22×C7semidirect
100. #213C22×C7semidirect
101. C2×C3⋊C8C22×C7semidirect
102. D24C22×C7semidirect
103. D3×C8C22×C7semidirect
105. C2×C3×C7C2×D8semidirect
108. D3×C7C2×D8direct (N × H)
113. C2×D8D3×C7direct (N × H)
132. C3×D4C23×C7semidirect
137. D12C23×C7semidirect
140. C3×C8C23×C7semidirect
141. C3⋊C8C23×C7semidirect
142. C3×C7C22×D8semidirect
148. C2×D4C2×D3×C7semidirect
152. D8C2×D3×C7direct (N × H)
155. C2×C8C2×D3×C7semidirect
159. C2×C7D3×D8direct (N × H)
167. C2×D3C7×D8direct (N × H)
181. D4C22×D3×C7semidirect
187. C8C22×D3×C7semidirect
188. C7C2×D3×D8direct (N × H)
190. C2×C3C2×C7×D8semidirect
193. D3C2×C7×D8direct (N × H)
205. C3C22×C7×D8semidirect
209. C2D3×C7×D8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C7×D42
C2×C3×C7×D82
C2×C7×#2132
D3×C7×D82
C2×C7×D242
C2×D3×C7×C82
C22×C7×D83
C2×D3×D87