Finite Group Databasebeta

← C2×D7×#213 #13060095

Subgroups — C2×D7×#213 #13060095

Order 1344.

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

normal subgroups only

This group has 676 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×C3×D4×D7C211yesmaximal, characteristic, complemented
2C2×D7×D12C211yesmaximal, characteristic, complemented
3C2×#66632C211yesmaximal, characteristic, complemented
4C2×C7×#213C211yesmaximal, characteristic, complemented
5D7×#213C218yesmaximal, direct product factor
6C2×#66561C211yesmaximal, characteristic, complemented
7C2×#66560C211yesmaximal, characteristic, complemented
8C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×D7×D8—31nomaximal, Hall
10C23×C3×D7—21no—
11C22×D3×D7—21no—
12C2×C3×C7×D4C2211yescharacteristic, complemented
13C2×D84C2211yescharacteristic, complemented
14C2×C7×D12C2211yescharacteristic, complemented
15C2×C3×C22⋊D7—21no—
16C3×D4×D7—22no—
17C3×D4×D7C2214yescomplemented
18C2×C3×D28C2211yescharacteristic, complemented
19C2×C3×C4×D7C2211yescharacteristic
20C2×#4247—21no—
21D7×D12—22no—
22D7×D12C2214yescomplemented
23#66632C2214yescomplemented
24C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
25C7×#213C2214yescomplemented
26C2×C7×C3⋊C8C2211yescharacteristic, complemented
27#66561C2214yescomplemented
28#66560C2214yescomplemented
29D7×C3⋊C8C2214yescomplemented
30C2×D4×D7—31no—
31C2×D4×D7D311yescharacteristic, complemented
32C2×C7×D8—31no—
33C2×#1141—31no—
34C2×#1141—31no—
35D7×D8—38no—
36C2×D56—31no—
37C2×C8×D7—31no—
38C22×#213—71nomaximal, Hall
39C23×C3×C7—21no—
40C22×D21—21no—
41C22×D3×C7—21no—
42C22×C3×D7—21no—
43C22×C3×D7—24no—
44C22×C3×D7—44no—
45C22×C3×D7D411yescharacteristic
46C2×D3×D7—24no—
47C2×D3×D7—44no—
48C3×C7×D4—21no—
49C3×C7×D4C2312yescomplemented
50C2×C3×C4×C7C2311yescharacteristic
51D84—21no—
52D84C2312yescomplemented
53C7×D12—21no—
54C7×D12C2312yescomplemented
55C3×C22⋊D7—24no—
56C2×C3×Dic7D411yescharacteristic
57C3×D28—21no—
58C3×D28C2312yescomplemented
59C3×C4×D7D412yes—
60C3×C4×D7C2312yes—
61#4247—24no—
62C3⋊(C7⋊C8)C2312yescomplemented
63C7×C3⋊C8C2312yescomplemented
64C23×D7—21no—
65C23×D7—61no—
66C2×C7×D4—31no—
67C2×C7×D4C2×D311yescharacteristic, complemented
68C2×C22⋊D7—21no—
69C2×C22⋊D7—61no—
70D4×D7—22no—
71D4×D7—34no—
72D4×D7—62no—
73D4×D7C2×D314yescomplemented
74C2×D28—31no—
75C2×D28C2×D311yescharacteristic, complemented
76C2×C4×D7C2×D311yescharacteristic
77C7×D8—34no—
78C2×C7×C8—31no—
79#1141—34no—
80#1141—34no—
81C2×C7⋊C8—31no—
82D56—34no—
83C8×D7—34no—
84C22×C3×D4—71no—
85C22×D12—71no—
86C2×#213—71no—
87C2×#213—71no—
88C2×#213—78no—
89C2×#213—71no—
90C2×#213D711yescharacteristic, direct product factor
91C22×C3⋊C8—71no—
92C22×C3×C7—22no—
93C22×C3×C7—42no—
94C22×C3×C7C2×D411yescharacteristic
95C2×D21—22no—
96C2×D21—42no—
97C2×D3×C7—22no—
98C2×D3×C7—42no—
99C2×C3×D7—22no—
100C2×C3×D7—42no—
101C2×C3×D7—48no—
102C2×C3×D7D814yescomplemented
103C2×C3×D7C2×D412yes—
104D3×D7—48no—
105C3×C4×C7C2×D411yescharacteristic
106C3×C4×C7C2411yescharacteristic
107C3×Dic7C2×D412yes—
108C22×D8—211noSylow
109C23×C7—21no—
110C23×C7—61no—
111C22×D7—21no—
112C22×D7—24no—
113C22×D7—44no—
114C22×D7—61no—
115C22×D7—64no—
116C22×D7—124no—
117C22×D7#6711yescharacteristic
118C7×D4—21no—
119C7×D4—32no—
120C7×D4—61no—
121C7×D4C22×D312yescomplemented
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—24no—
124C22⋊D7—64no—
125C2×Dic7#6711yescharacteristic
126D28—21no—
127D28—32no—
128D28—61no—
129D28C22×D312yescomplemented
130C4×D7#6712yes—
131C4×D7C22×D312yes—
132C7×C8—32no—
133C7⋊C8—32no—
134C24×C3—141no—
135C23×D3—141no—
136C2×C3×D4—74no—
137C2×C3×D4—71no—
138C2×C3×D4—142no—
139C2×C3×D4—141no—
140C2×C3×D4C2×D711yescharacteristic, complemented
141C22×C3×C4—71no—
142C2×D12—71no—
143C2×D12—74no—
144C2×D12—142no—
145C2×D12—141no—
146C2×D12C2×D711yescharacteristic, complemented
147#213—74no—
148#213—74no—
149#213—74no—
150#213C2×D714yesdirect product factor
151C2×C3⋊C8—71no—
152C2×C3⋊C8—74no—
153C2×C3⋊C8C2×D711yescharacteristic, complemented
154C2×C3×C7—42no—
155C2×C3×C7C2×D812yescomplemented
156C2×C3×C7C22×D411yescharacteristic
157D21—42no—
158C3×D7—42no—
159C3×D7C2×D814yescomplemented
160D3×C7—42no—
161C22×D4—71no—
162C22×D4—211no—
163C2×D8—31no—
164C2×D8—211no—
165C2×D8—218no—
166C2×D8—211no—
167C2×D8—211no—
168C22×C8—211no—
169C22×C7—22no—
170C22×C7—42no—
171C22×C7—62no—
172C22×C7—122no—
173C22×C7C2×#6711yescharacteristic
174C2×D7—22no—
175C2×D7—42no—
176C2×D7—48no—
177C2×D7—62no—
178C2×D7—122no—
179C2×D7—128no—
180C2×D7#21314yesdirect product factor
181C2×D7C2×#6712yes—
182C4×C7C2×#6711yescharacteristic
183C4×C7C23×D311yescharacteristic
184Dic7C2×#6712yes—
185C23×C3—21no—
186C23×C3—71no—
187C23×C3—144no—
188C23×C3—141no—
189C23×C3—284no—
190C22×D3—21no—
191C22×D3—144no—
192C22×D3—141no—
193C22×D3—284no—
194C3×D4—21no—
195C3×D4—72no—
196C3×D4—144no—
197C3×D4—141no—
198C3×D4C22×D712yescomplemented
199C2×C3×C4—71no—
200C2×C3×C4—72no—
201C2×C3×C4—72no—
202C2×C3×C4C22×D711yescharacteristic
203D12—21no—
204D12—72no—
205D12—144no—
206D12—141no—
207D12C22×D712yescomplemented
208C3⋊C8—72no—
209C3⋊C8C22×D712yescomplemented
210C3×C7C22×D811yescharacteristic, complemented, Hall
211C24—141no—
212C24—421no—
213C2×D4—31no—
214C2×D4—74no—
215C2×D4—71no—
216C2×D4—142no—
217C2×D4—141no—
218C2×D4—214no—
219C2×D4—211no—
220C2×D4—422no—
221C2×D4—421no—
222C2×D4D3×D711yescharacteristic, complemented
223C22×C4—71no—
224D8—34no—
225D8—214no—
226D8—214no—
227D8—214no—
228C2×C8—31no—
229C2×C8—214no—
230C2×C8—211no—
231C2×C7—42no—
232C2×C7—122no—
233C2×C7C2×#21312yescomplemented
234C2×C7C22×#6711yescharacteristic
235D7—42no—
236D7—122no—
237D7C2×#21314yesdirect product factor
238C22×C3—22no—
239C22×C3—42no—
240C22×C3—74no—
241C22×C3—72no—
242C22×C3—142no—
243C22×C3—288no—
244C22×C3—282no—
245C22×C3D4×D711yescharacteristic
246C2×D3—22no—
247C2×D3—42no—
248C2×D3—142no—
249C2×D3—288no—
250C2×D3—282no—
251C3×C4—72no—
252C3×C4D4×D711yescharacteristic
253C3×C4C23×D711yescharacteristic
254C23—21no—
255C23—61no—
256C23—71no—
257C23—144no—
258C23—141no—
259C23—284no—
260C23—424no—
261C23—421no—
262C23—844no—
263D4—21no—
264D4—32no—
265D4—61no—
266D4—72no—
267D4—144no—
268D4—141no—
269D4—212no—
270D4—424no—
271D4—421no—
272D4C2×D3×D712yescomplemented
273C2×C4—71no—
274C2×C4—72no—
275C2×C4—72no—
276C2×C4C2×D3×D711yescharacteristic
277C8—32no—
278C8—212no—
279C7C22×#21311yescharacteristic, complemented, Sylow
280C2×C3—42no—
281C2×C3—74no—
282C2×C3—282no—
283C2×C3D7×D812yescomplemented
284C2×C3C2×D4×D711yescharacteristic
285D3—42no—
286D3—282no—
287C22—22no—
288C22—42no—
289C22—62no—
290C22—74no—
291C22—72no—
292C22—122no—
293C22—142no—
294C22—288no—
295C22—282no—
296C22—422no—
297C22—848no—
298C22—842no—
299C22D7×#6711yescharacteristic
300C4—72no—
301C4D7×#6711yescharacteristic
302C4C22×D3×D711yescharacteristic
303C3C2×D7×D811yescharacteristic, complemented, Sylow
304C2—42no—
305C2—74no—
306C2—122no—
307C2—282no—
308C2—842no—
309C2D7×#21312yesdirect product factor
310C2C2×D7×#6711yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7C22×D8
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D7C2semidirect
2. C2×D7×D12C2semidirect
3. C2×#66632C2semidirect
4. C2×C7×#213C2semidirect
5. D7×#213C2direct (N × H)
6. C2×#66561C2semidirect
7. C2×#66560C2semidirect
8. C2×D7×C3⋊C8C2semidirect
12. C2×C3×C7×D4C22semidirect
13. C2×D84C22semidirect
14. C2×C7×D12C22semidirect
17. C3×D4×D7C22semidirect
18. C2×C3×D28C22semidirect
22. D7×D12C22semidirect
23. #66632C22semidirect
24. C2×C3⋊(C7⋊C8)C22semidirect
25. C7×#213C22semidirect
26. C2×C7×C3⋊C8C22semidirect
27. #66561C22semidirect
28. #66560C22semidirect
29. D7×C3⋊C8C22semidirect
31. C2×D4×D7D3semidirect
49. C3×C7×D4C23semidirect
52. D84C23semidirect
54. C7×D12C23semidirect
58. C3×D28C23semidirect
62. C3⋊(C7⋊C8)C23semidirect
63. C7×C3⋊C8C23semidirect
67. C2×C7×D4C2×D3semidirect
73. D4×D7C2×D3semidirect
75. C2×D28C2×D3semidirect
90. C2×#213D7direct (N × H)
102. C2×C3×D7D8semidirect
121. C7×D4C22×D3semidirect
129. D28C22×D3semidirect
140. C2×C3×D4C2×D7semidirect
146. C2×D12C2×D7semidirect
150. #213C2×D7direct (N × H)
153. C2×C3⋊C8C2×D7semidirect
155. C2×C3×C7C2×D8semidirect
159. C3×D7C2×D8semidirect
180. C2×D7#213direct (N × H)
198. C3×D4C22×D7semidirect
207. D12C22×D7semidirect
209. C3⋊C8C22×D7semidirect
210. C3×C7C22×D8semidirect
222. C2×D4D3×D7semidirect
233. C2×C7C2×#213semidirect
237. D7C2×#213direct (N × H)
272. D4C2×D3×D7semidirect
279. C7C22×#213semidirect
283. C2×C3D7×D8semidirect
303. C3C2×D7×D8semidirect
309. C2D7×#213direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D72
C2×D7×D122
C2×#666322
C2×C7×#2132
D7×#2132
C2×#665612
C2×#665602
C2×D7×C3⋊C82
C2×D7×D83
C22×#2137