Finite Group Databasebeta

← C2×#10620860 #13059433

Subgroups — C2×#10620860 #13059433

Order 1344.

596 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 160 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×C7×#235C211yesmaximal, characteristic, complemented
2C22×C3×D28C211yesmaximal, characteristic, complemented
3C2×#66627C211yesmaximal, characteristic, complemented
4C2×#66563C212yesmaximal, complemented
5#10620860C218yesmaximal, direct product factor
6C2×#66560C212yesmaximal, complemented
7C2×#6512—31nomaximal, Hall
8C23×C3×D7—21no—
9C22×C3×C4×C7C2211yescharacteristic
10C2×C7×#67—21no—
11C7×#235—22no—
12C7×#235C2214yescomplemented
13C2×C7×D12C2211yescharacteristic, complemented
14C2×C4×D3×C7—21no—
15C2×C7×Dic6C2211yescharacteristic, complemented
16C2×C3×D28—22no—
17C2×C3×D28—21no—
18C2×C3×D28C2212yes—
19C2×C3×D28C2214yescomplemented
20#66627C2214yescomplemented
21C2×C3⋊(C7⋊C8)C2212yescomplemented
22#66563C2218yescomplemented
23#66560C2218yescomplemented
24C2×C7×#41—31no—
25C22×D28D311yescharacteristic, complemented
26#6512—38no—
27C2×#1143—32no—
28C2×#1141—32no—
29C2×#1136—31no—
30C2×#932—71nomaximal, Hall
31C23×C3×C7D411yescharacteristic
32C22×D3×C7—21no—
33C22×C3×D7—22no—
34C22×C3×D7—24no—
35C22×C3×D7—44no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7D412yes—
38C2×C3×C4×C7C2311yescharacteristic
39C2×C3×C4×C7C2312yes—
40C7×#67—24no—
41C2×C7×Dic3—21no—
42C7×D12—21no—
43C7×D12C2312yescomplemented
44C4×D3×C7—24no—
45C7×Dic6—21no—
46C7×Dic6C2312yescomplemented
47C3×D28—22no—
48C3×D28—24no—
49C3×D28C2314yes—
50C3⋊(C7⋊C8)C2314yescomplemented
51C23×D7—21no—
52C7×#41—34no—
53C7×#41—62no—
54C2×C7×Q8—31no—
55C2×C7×D4—31no—
56C2×C7×D4—61no—
57C22×C4×C7—61no—
58C22×C4×C7C2×D311yescharacteristic
59C2×D28—22no—
60C2×D28—21no—
61C2×D28C2×D312yes—
62C2×D28C2×D314yescomplemented
63#1143—38no—
64#1141—38no—
65#1136—34no—
66C2×C7⋊C8—32no—
67C22×C3×D4—71no—
68C2×#235D711yescharacteristic, complemented
69C2×#214—72no—
70#932—78no—
71C2×#213—72no—
72C2×#208—71no—
73C22×C3×C7—22no—
74C22×C3×C7C2×D412yes—
75C22×C3×C7C2×D411yescharacteristic
76C2×D3×C7—22no—
77C2×D3×C7—42no—
78C2×C3×D7—24no—
79C2×C3×D7—48no—
80C2×C3×D7—44no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D412yes—
83C3×C4×C7C2411yescharacteristic
84C7×Dic3—22no—
85C2×AGL(1,ℤ/8)—211noSylow
86C23×C7—61no—
87C23×C7#6711yescharacteristic
88C22×D7—22no—
89C22×D7—24no—
90C22×D7—44no—
91C7×Q8—32no—
92C7×Q8—61no—
93C7×D4—32no—
94C7×D4—64no—
95C7×D4—61no—
96C2×C4×C7—61no—
97C2×C4×C7—64no—
98C2×C4×C7#6711yescharacteristic
99C2×C4×C7#6712yes—
100C2×C4×C7C22×D311yescharacteristic
101C2×C4×C7C22×D312yes—
102D28—22no—
103D28—24no—
104D28C22×D314yes—
105C7⋊C8—34no—
106C24×C3—141no—
107C2×C3×D4—72no—
108C2×C3×D4—74no—
109C2×C3×D4—141no—
110C2×C3×D4—142no—
111C22×C3×C4C2×D711yescharacteristic
112C2×#67—21no—
113#235—22no—
114#235C2×D714yescomplemented
115C2×D12C2×D711yescharacteristic, complemented
116C2×C4×D3—21no—
117C2×Dic6C2×D711yescharacteristic, complemented
118#214—78no—
119#213—78no—
120#208—74no—
121C2×C3⋊C8—72no—
122C2×C3×C7—22no—
123C2×C3×C7AGL(1,ℤ/8)12yescomplemented
124C2×C3×C7C22×D411yescharacteristic
125C3×D7—44no—
126D3×C7—42no—
127C2×#41—31no—
128C22×D4—71no—
129AGL(1,ℤ/8)—218no—
130C2×SD16—212no—
131C2×D8—212no—
132C2×M16—211no—
133C22×C7—22no—
134C22×C7—62no—
135C22×C7—122no—
136C22×C7C2×#6712yes—
137C22×C7C2×#6711yescharacteristic
138C2×D7—24no—
139C2×D7—48no—
140C2×D7—44no—
141C4×C7—62no—
142C4×C7C2×#6712yes—
143C4×C7C2×#6711yescharacteristic
144C4×C7C23×D311yescharacteristic
145C23×C3—144no—
146C23×C3—142no—
147C23×C3—284no—
148C23×C3C22⋊D711yescharacteristic
149C22×D3—21no—
150C3×D4—74no—
151C3×D4—142no—
152C3×D4—144no—
153C2×C3×C4C22⋊D711yescharacteristic
154C2×C3×C4C22⋊D712yes—
155C2×C3×C4C22×D711yescharacteristic
156C2×C3×C4C22×D712yes—
157#67—24no—
158C2×Dic3—21no—
159D12—21no—
160D12C22×D712yescomplemented
161C4×D3—24no—
162Dic6—21no—
163Dic6C22×D712yescomplemented
164C3⋊C8—74no—
165C3×C7C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
166C24—141no—
167#41—34no—
168#41—62no—
169C2×Q8—31no—
170C2×D4—31no—
171C2×D4—61no—
172C2×D4—72no—
173C2×D4—74no—
174C2×D4—142no—
175C2×D4—141no—
176C22×C4—61no—
177C22×C4D3×D711yescharacteristic
178SD16—218no—
179D8—218no—
180M16—214no—
181C2×C8—212no—
182C2×C7—22no—
183C2×C7—122no—
184C2×C7#93212yescomplemented
185C2×C7C22×#6711yescharacteristic
186D7—44no—
187C22×C3—22no—
188C22×C3—144no—
189C22×C3—288no—
190C22×C3—284no—
191C22×C3C2×C22⋊D712yes—
192C22×C3C2×C22⋊D711yescharacteristic
193C2×D3—22no—
194C2×D3—42no—
195C3×C4C2×C22⋊D711yescharacteristic
196C3×C4C2×C22⋊D712yes—
197C3×C4C23×D711yescharacteristic
198Dic3—22no—
199C23—61no—
200C23—144no—
201C23—142no—
202C23—284no—
203C23#424511yescharacteristic
204Q8—32no—
205Q8—61no—
206D4—32no—
207D4—64no—
208D4—61no—
209D4—74no—
210D4—144no—
211D4—142no—
212C2×C4—61no—
213C2×C4—64no—
214C2×C4#424512yes—
215C2×C4#424511yescharacteristic
216C2×C4C2×D3×D712yes—
217C2×C4C2×D3×D711yescharacteristic
218C8—214no—
219C7C2×#93211yescharacteristic, complemented, Sylow
220C2×C3—22no—
221C2×C3—284no—
222C2×C3#651212yescomplemented
223C2×C3C22×C22⋊D711yescharacteristic
224D3—42no—
225C22—22no—
226C22—62no—
227C22—122no—
228C22—144no—
229C22—288no—
230C22—284no—
231C22C2×#424512yes—
232C22C2×#424511yescharacteristic
233C4—62no—
234C4C2×#424512yes—
235C4C2×#424511yescharacteristic
236C4C22×D3×D711yescharacteristic
237C3C2×#651211yescharacteristic, complemented, Sylow
238C2—22no—
239C2—122no—
240C2—284no—
241C2#1062086012yesdirect product factor
242C2C22×#424511yescharacteristic
subgroupG/H
centerC22C2×#4245
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
hypercenterC22×C4D3×D7
hypocenterC3×C7C2×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C7×#235C2semidirect
2. C22×C3×D28C2semidirect
3. C2×#66627C2semidirect
4. C2×#66563C2semidirect
5. #10620860C2direct (N × H)
6. C2×#66560C2semidirect
12. C7×#235C22semidirect
13. C2×C7×D12C22semidirect
15. C2×C7×Dic6C22semidirect
19. C2×C3×D28C22semidirect
20. #66627C22semidirect
21. C2×C3⋊(C7⋊C8)C22semidirect
22. #66563C22semidirect
23. #66560C22semidirect
25. C22×D28D3semidirect
43. C7×D12C23semidirect
46. C7×Dic6C23semidirect
50. C3⋊(C7⋊C8)C23semidirect
62. C2×D28C2×D3semidirect
68. C2×#235D7semidirect
114. #235C2×D7semidirect
115. C2×D12C2×D7semidirect
117. C2×Dic6C2×D7semidirect
123. C2×C3×C7AGL(1,ℤ/8)semidirect
160. D12C22×D7semidirect
163. Dic6C22×D7semidirect
165. C3×C7C2×AGL(1,ℤ/8)semidirect
184. C2×C7#932semidirect
219. C7C2×#932semidirect
222. C2×C3#6512semidirect
237. C3C2×#6512semidirect
241. C2#10620860direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#2352
C22×C3×D282
C2×#666272
C2×#665632
#106208602
C2×#665602
C2×#65123
C2×#9327