Finite Group Databasebeta

← C2×#10620867 #13059441

Subgroups — C2×#10620867 #13059441

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
1C22×D84C211yesmaximal, characteristic, complemented
2C2×C7×#235C211yesmaximal, characteristic, complemented
3C2×C3×#1136C211yesmaximal, characteristic, complemented
4C2×#66568C212yesmaximal, complemented
5#10620867C218yesmaximal, direct product factor
6C2×#66562C212yesmaximal, complemented
7C2×#6512—31nomaximal, Hall
8C23×D21—21no—
9C22×C3×C4×C7C2211yescharacteristic
10C2×D84—22no—
11C2×D84—21no—
12C2×D84C2212yes—
13C2×D84C2214yescomplemented
14C2×C7×#67—21no—
15C7×#235—22no—
16C7×#235C2214yescomplemented
17C2×C7×D12C2211yescharacteristic, complemented
18C2×C4×D3×C7—21no—
19C2×C7×Dic6C2211yescharacteristic, complemented
20C3×#1136C2214yescomplemented
21C2×C3×C7⋊C8C2212yescomplemented
22#66568C2218yescomplemented
23#66562C2218yescomplemented
24C2×C7×#41—31no—
25C22×D28—31no—
26#6512—38no—
27C2×#1143—32no—
28C2×#1141—32no—
29C2×#1136D311yescharacteristic, complemented
30C2×#908—71nomaximal, Hall
31C23×C3×C7D411yescharacteristic
32C22×D21—22no—
33C22×D21—24no—
34C22×D21—44no—
35C22×D3×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7D412yes—
38C2×C3×C4×C7C2311yescharacteristic
39C2×C3×C4×C7C2312yes—
40D84—22no—
41D84—24no—
42D84C2314yes—
43C7×#67—24no—
44C2×C7×Dic3—21no—
45C7×D12—21no—
46C7×D12C2312yescomplemented
47C4×D3×C7—24no—
48C7×Dic6—21no—
49C7×Dic6C2312yescomplemented
50C3×C7⋊C8C2314yescomplemented
51C23×D7—61no—
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—32no—
60C2×D28—34no—
61C2×D28—61no—
62C2×D28—62no—
63#1143—38no—
64#1141—38no—
65#1136C2×D314yescomplemented
66C2×C7⋊C8C2×D312yescomplemented
67C2×#235D711yescharacteristic, complemented
68C22×D12—71no—
69C2×C3×M16—71no—
70#908—78no—
71C2×D24—72no—
72C2×#204—72no—
73C22×C3×C7—22no—
74C22×C3×C7C2×D412yes—
75C22×C3×C7C2×D411yescharacteristic
76C2×D21—24no—
77C2×D21—48no—
78C2×D21—44no—
79C2×D3×C7—22no—
80C2×D3×C7—42no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D412yes—
83C3×C4×C7C2411yescharacteristic
84C7×Dic3—22no—
85C2×AGL(1,ℤ/8)—211noSylow
86C23×C7—61no—
87C23×C7D1211yescharacteristic
88C22×D7—64no—
89C22×D7—62no—
90C22×D7—124no—
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×C7D1211yescharacteristic
99C2×C4×C7D1212yes—
100C2×C4×C7C22×D311yescharacteristic
101C2×C4×C7C22×D312yes—
102D28—34no—
103D28—64no—
104D28—62no—
105C7⋊C8C22×D314yescomplemented
106C23×D3—141no—
107C22×C3×C4C2×D711yescharacteristic
108C2×#67—21no—
109#235—22no—
110#235C2×D714yescomplemented
111C2×D12—72no—
112C2×D12—74no—
113C2×D12—141no—
114C2×D12—142no—
115C2×D12C2×D711yescharacteristic, complemented
116C2×C4×D3—21no—
117C2×Dic6C2×D711yescharacteristic, complemented
118C3×M16—74no—
119C2×C3×C8—72no—
120D24—78no—
121#204—78no—
122C2×C3×C7—22no—
123C2×C3×C7AGL(1,ℤ/8)12yescomplemented
124C2×C3×C7C22×D411yescharacteristic
125D21—44no—
126D3×C7—42no—
127C2×#41—31no—
128C22×D4—211no—
129AGL(1,ℤ/8)—218no—
130C2×SD16—212no—
131C2×D8—212no—
132C2×M16—71no—
133C22×C7—22no—
134C22×C7—62no—
135C22×C7—122no—
136C22×C7C2×D1212yes—
137C22×C7C2×D1211yescharacteristic
138C2×D7—64no—
139C2×D7—128no—
140C2×D7—124no—
141C4×C7—62no—
142C4×C7C2×D1212yes—
143C4×C7C2×D1211yescharacteristic
144C4×C7C23×D311yescharacteristic
145C23×C3C22⋊D711yescharacteristic
146C22×D3—21no—
147C22×D3—144no—
148C22×D3—142no—
149C22×D3—284no—
150C2×C3×C4C22⋊D711yescharacteristic
151C2×C3×C4C22⋊D712yes—
152C2×C3×C4C22×D711yescharacteristic
153C2×C3×C4C22×D712yes—
154#67—24no—
155C2×Dic3—21no—
156D12—21no—
157D12—74no—
158D12—142no—
159D12—144no—
160D12C22×D712yescomplemented
161C4×D3—24no—
162Dic6—21no—
163Dic6C22×D712yescomplemented
164C3×C8—74no—
165C3×C7C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
166C24—421no—
167#41—34no—
168#41—62no—
169C2×Q8—31no—
170C2×D4—31no—
171C2×D4—61no—
172C2×D4—212no—
173C2×D4—214no—
174C2×D4—422no—
175C2×D4—421no—
176C22×C4—61no—
177C22×C4D3×D711yescharacteristic
178SD16—218no—
179D8—218no—
180M16—74no—
181C2×C8—72no—
182C2×C7—22no—
183C2×C7—122no—
184C2×C7#90812yescomplemented
185C2×C7C22×D1211yescharacteristic
186D7—124no—
187C22×C3—22no—
188C22×C3C2×C22⋊D712yes—
189C22×C3C2×C22⋊D711yescharacteristic
190C2×D3—22no—
191C2×D3—42no—
192C2×D3—144no—
193C2×D3—288no—
194C2×D3—284no—
195C3×C4C2×C22⋊D711yescharacteristic
196C3×C4C2×C22⋊D712yes—
197C3×C4C23×D711yescharacteristic
198Dic3—22no—
199C23—61no—
200C23—424no—
201C23—422no—
202C23—844no—
203C23#424711yescharacteristic
204Q8—32no—
205Q8—61no—
206D4—32no—
207D4—64no—
208D4—61no—
209D4—214no—
210D4—424no—
211D4—422no—
212C2×C4—61no—
213C2×C4—64no—
214C2×C4#424712yes—
215C2×C4#424711yescharacteristic
216C2×C4C2×D3×D712yes—
217C2×C4C2×D3×D711yescharacteristic
218C8—74no—
219C7C2×#90811yescharacteristic, complemented, Sylow
220C2×C3—22no—
221C2×C3#651212yescomplemented
222C2×C3C22×C22⋊D711yescharacteristic
223D3—42no—
224D3—284no—
225C22—22no—
226C22—62no—
227C22—122no—
228C22—424no—
229C22—848no—
230C22—844no—
231C22C2×#424712yes—
232C22C2×#424711yescharacteristic
233C4—62no—
234C4C2×#424712yes—
235C4C2×#424711yescharacteristic
236C4C22×D3×D711yescharacteristic
237C3C2×#651211yescharacteristic, complemented, Sylow
238C2—22no—
239C2—122no—
240C2—844no—
241C2#1062086712yesdirect product factor
242C2C22×#424711yescharacteristic
subgroupG/H
centerC22C2×#4247
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. C22×D84C2semidirect
2. C2×C7×#235C2semidirect
3. C2×C3×#1136C2semidirect
4. C2×#66568C2semidirect
5. #10620867C2direct (N × H)
6. C2×#66562C2semidirect
13. C2×D84C22semidirect
16. C7×#235C22semidirect
17. C2×C7×D12C22semidirect
19. C2×C7×Dic6C22semidirect
20. C3×#1136C22semidirect
21. C2×C3×C7⋊C8C22semidirect
22. #66568C22semidirect
23. #66562C22semidirect
29. C2×#1136D3semidirect
46. C7×D12C23semidirect
49. C7×Dic6C23semidirect
50. C3×C7⋊C8C23semidirect
65. #1136C2×D3semidirect
66. C2×C7⋊C8C2×D3semidirect
67. C2×#235D7semidirect
105. C7⋊C8C22×D3semidirect
110. #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#908semidirect
219. C7C2×#908semidirect
221. C2×C3#6512semidirect
237. C3C2×#6512semidirect
241. C2#10620867direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D842
C2×C7×#2352
C2×C3×#11362
C2×#665682
#106208672
C2×#665622
C2×#65123
C2×#9087