Finite Group Databasebeta

← C7×#5822 #13061541

Subgroups — C7×#5822 #13061541

Order 1344.

524 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C7×#41C211yesmaximal, characteristic, complemented
2C2×C7×#235C211yesmaximal, characteristic, complemented
3C7×#951C212yesmaximal, complemented
4C7×#950C214yesmaximal, complemented
5C7×#949C212yesmaximal, complemented
6C7×#942C212yesmaximal, complemented
7C7×#932C212yesmaximal, complemented
8C2×C7×#208C211yesmaximal, characteristic, complemented
9C7×#575—31nomaximal, Hall
10C3×C7×#41—21no—
11C3×C7×#41—21no—
12C3×C7×#41C2212yes—
13C3×C7×#41C2212yescomplemented
14C2×C3×C7×Q8C2211yescharacteristic, complemented
15C2×C3×C7×D4—21no—
16C2×C3×C7×D4C2211yescharacteristic, complemented
17C22×C3×C4×C7—21no—
18C22×C3×C4×C7C2211yescharacteristic
19C2×C7×#67—21no—
20C7×#235—21no—
21C7×#235—21no—
22C7×#235C2212yes—
23C7×#235C2212yescomplemented
24C2×C7×D12C2211yescharacteristic, complemented
25C2×C4×D3×C7—21no—
26C2×C7×Dic6C2211yescharacteristic, complemented
27C7×#216C2214yescomplemented
28C7×#215C2214yescomplemented
29C7×#214C2214yescomplemented
30C7×#213C2214yescomplemented
31C7×#208C2212yescomplemented
32C7×#208C2212yescomplemented
33C2×C7×C3⋊C8C2212yescomplemented
34C2×C7×#41—31no—
35C2×C7×#41D311yescharacteristic, complemented
36C7×#137—32no—
37C7×#137—32no—
38C7×AGL(1,ℤ/8)—32no—
39C7×AGL(1,ℤ/8)—32no—
40C7×#135—34no—
41C2×C7×M16—31no—
42#5822C711yesmaximal, characteristic, direct product factor, Hall
43C23×C3×C7—21no—
44C23×C3×C7D411yescharacteristic
45C22×D3×C7—21no—
46C3×C7×Q8—21no—
47C3×C7×Q8C2312yes—
48C3×C7×D4—22no—
49C3×C7×D4—22no—
50C3×C7×D4—21no—
51C3×C7×D4C2312yes—
52C2×C3×C4×C7—21no—
53C2×C3×C4×C7—22no—
54C2×C3×C4×C7—22no—
55C2×C3×C4×C7D411yescharacteristic
56C2×C3×C4×C7D411yescharacteristic
57C2×C3×C4×C7D411yescharacteristic
58C2×C3×C4×C7C2311yescharacteristic
59C2×C3×C4×C7C2311yescharacteristic
60C2×C3×C4×C7C2311yescharacteristic
61C7×#67—22no—
62C7×#67—22no—
63C2×C7×Dic3—21no—
64C7×D12—21no—
65C7×D12C2312yes—
66C4×D3×C7—22no—
67C4×D3×C7—22no—
68C7×Dic6—21no—
69C7×Dic6C2312yes—
70C7×C3⋊C8C2314yes—
71C7×#41—21no—
72C7×#41—21no—
73C7×#41—32no—
74C7×#41—32no—
75C7×#41—61no—
76C7×#41—61no—
77C7×#41C2×D312yes—
78C7×#41C2×D312yescomplemented
79C2×C7×Q8—31no—
80C2×C7×Q8C2×D311yescharacteristic, complemented
81C2×C7×D4—21no—
82C2×C7×D4—31no—
83C2×C7×D4—61no—
84C2×C7×D4C2×D311yescharacteristic, complemented
85C22×C4×C7—21no—
86C22×C4×C7—61no—
87C22×C4×C7C2×D311yescharacteristic
88C7×Q16—34no—
89C7×SD16—34no—
90C7×SD16—34no—
91C7×D8—34no—
92C7×M16—32no—
93C7×M16—32no—
94C2×C7×C8—32no—
95C2×C3×#41C2×C711yescharacteristic, complemented
96C2×#235C2×C711yescharacteristic, complemented
97#951C2×C712yescomplemented
98#950C2×C714yescomplemented
99#949C2×C712yescomplemented
100#942C2×C712yescomplemented
101#932C2×C712yescomplemented
102C2×#208C2×C711yescharacteristic, complemented
103C22×C3×C7—22no—
104C22×C3×C7—41no—
105C22×C3×C7—42no—
106C22×C3×C7C2×D411yescharacteristic
107C22×C3×C7C2×D411yescharacteristic
108C22×C3×C7C2×D411yescharacteristic
109C2×D3×C7—22no—
110C2×D3×C7—42no—
111C3×C4×C7—22no—
112C3×C4×C7C2×D411yescharacteristic
113C3×C4×C7C2×D411yescharacteristic
114C3×C4×C7C2×D411yescharacteristic
115C3×C4×C7C2411yescharacteristic
116C7×Dic3—22no—
117#575—31noSylow
118C23×C7—21no—
119C23×C7—61no—
120C23×C7#6711yescharacteristic
121C7×Q8—21no—
122C7×Q8—32no—
123C7×Q8—61no—
124C7×Q8C22×D312yes—
125C7×D4—22no—
126C7×D4—22no—
127C7×D4—21no—
128C7×D4—32no—
129C7×D4—62no—
130C7×D4—62no—
131C7×D4—61no—
132C7×D4C22×D312yes—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—61no—
137C2×C4×C7—62no—
138C2×C4×C7—62no—
139C2×C4×C7#6711yescharacteristic
140C2×C4×C7#6711yescharacteristic
141C2×C4×C7#6711yescharacteristic
142C2×C4×C7C22×D311yescharacteristic
143C2×C4×C7C22×D311yescharacteristic
144C2×C4×C7C22×D311yescharacteristic
145C7×C8—34no—
146C3×#41—21no—
147C3×#41—21no—
148C3×#41C22×C712yes—
149C3×#41C22×C712yescomplemented
150C2×C3×Q8C22×C711yescharacteristic, complemented
151C2×C3×D4—21no—
152C2×C3×D4C22×C711yescharacteristic, complemented
153C22×C3×C4—21no—
154C22×C3×C4C22×C711yescharacteristic
155C2×#67—21no—
156#235—21no—
157#235—21no—
158#235C22×C712yes—
159#235C22×C712yescomplemented
160C2×D12C22×C711yescharacteristic, complemented
161C2×C4×D3—21no—
162C2×Dic6C22×C711yescharacteristic, complemented
163#216C22×C714yescomplemented
164#215C22×C714yescomplemented
165#214C22×C714yescomplemented
166#213C22×C714yescomplemented
167#208C22×C712yescomplemented
168#208C22×C712yescomplemented
169C2×C3⋊C8C22×C712yescomplemented
170C2×C3×C7—21no—
171C2×C3×C7—21no—
172C2×C3×C7—21no—
173C2×C3×C7—42no—
174C2×C3×C7C22×D411yescharacteristic
175D3×C7—42no—
176C2×#41—31no—
177C2×#41D3×C711yescharacteristic, complemented
178#137—32no—
179#137—32no—
180AGL(1,ℤ/8)—32no—
181AGL(1,ℤ/8)—32no—
182#135—34no—
183C2×M16—31no—
184C22×C7—22no—
185C22×C7—41no—
186C22×C7—42no—
187C22×C7—62no—
188C22×C7—122no—
189C22×C7C2×#6711yescharacteristic
190C22×C7C2×#6711yescharacteristic
191C22×C7C2×#6711yescharacteristic
192C4×C7—22no—
193C4×C7—62no—
194C4×C7C2×#6711yescharacteristic
195C4×C7C2×#6711yescharacteristic
196C4×C7C2×#6711yescharacteristic
197C4×C7C23×D311yescharacteristic
198C23×C3—21no—
199C23×C3C7×D411yescharacteristic
200C22×D3—21no—
201C3×Q8—21no—
202C3×Q8C23×C712yes—
203C3×D4—22no—
204C3×D4—22no—
205C3×D4—21no—
206C3×D4C23×C712yes—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4C7×D411yescharacteristic
211C2×C3×C4C7×D411yescharacteristic
212C2×C3×C4C7×D411yescharacteristic
213C2×C3×C4C23×C711yescharacteristic
214C2×C3×C4C23×C711yescharacteristic
215C2×C3×C4C23×C711yescharacteristic
216#67—22no—
217#67—22no—
218C2×Dic3—21no—
219D12—21no—
220D12C23×C712yes—
221C4×D3—22no—
222C4×D3—22no—
223Dic6—21no—
224Dic6C23×C712yes—
225C3⋊C8C23×C714yes—
226C3×C7#57511yescharacteristic, complemented, Hall
227#41—21no—
228#41—21no—
229#41—32no—
230#41—32no—
231#41—61no—
232#41—61no—
233#41C2×D3×C712yes—
234#41C2×D3×C712yescomplemented
235C2×Q8—31no—
236C2×Q8C2×D3×C711yescharacteristic, complemented
237C2×D4—21no—
238C2×D4—31no—
239C2×D4—61no—
240C2×D4C2×D3×C711yescharacteristic, complemented
241C22×C4—21no—
242C22×C4—61no—
243C22×C4C2×D3×C711yescharacteristic
244Q16—34no—
245SD16—34no—
246SD16—34no—
247D8—34no—
248M16—32no—
249M16—32no—
250C2×C8—32no—
251C2×C7—21no—
252C2×C7—21no—
253C2×C7—21no—
254C2×C7—42no—
255C2×C7—122no—
256C2×C7C22×#6711yescharacteristic
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—42no—
260C22×C3C2×C7×D411yescharacteristic
261C22×C3C2×C7×D411yescharacteristic
262C22×C3C2×C7×D411yescharacteristic
263C2×D3—22no—
264C2×D3—42no—
265C3×C4—22no—
266C3×C4C2×C7×D411yescharacteristic
267C3×C4C2×C7×D411yescharacteristic
268C3×C4C2×C7×D411yescharacteristic
269C3×C4C24×C711yescharacteristic
270Dic3—22no—
271C23—21no—
272C23—61no—
273C23C7×#6711yescharacteristic
274Q8—21no—
275Q8—32no—
276Q8—61no—
277Q8C22×D3×C712yes—
278D4—22no—
279D4—22no—
280D4—21no—
281D4—32no—
282D4—62no—
283D4—62no—
284D4—61no—
285D4C22×D3×C712yes—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—61no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4C7×#6711yescharacteristic
293C2×C4C7×#6711yescharacteristic
294C2×C4C7×#6711yescharacteristic
295C2×C4C22×D3×C711yescharacteristic
296C2×C4C22×D3×C711yescharacteristic
297C2×C4C22×D3×C711yescharacteristic
298C8—34no—
299C7#582211yescharacteristic, direct product factor, Sylow
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—42no—
304C2×C3C22×C7×D411yescharacteristic
305D3—42no—
306C22—22no—
307C22—41no—
308C22—42no—
309C22—62no—
310C22—122no—
311C22C2×C7×#6711yescharacteristic
312C22C2×C7×#6711yescharacteristic
313C22C2×C7×#6711yescharacteristic
314C4—22no—
315C4—62no—
316C4C2×C7×#6711yescharacteristic
317C4C2×C7×#6711yescharacteristic
318C4C2×C7×#6711yescharacteristic
319C4C23×D3×C711yescharacteristic
320C3C7×#57511yescharacteristic, complemented, Sylow
321C2—21no—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—122no—
326C2C22×C7×#6711yescharacteristic
subgroupG/H
centerC4×C7C2×#67
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC2×C3×C7×#41C2
socleC2×C3×C7C22×D4
hypercenterC2×C7×#41D3
hypocenterC3C7×#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C7×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C7×#41C2semidirect
2. C2×C7×#235C2semidirect
3. C7×#951C2semidirect
4. C7×#950C2semidirect
5. C7×#949C2semidirect
6. C7×#942C2semidirect
7. C7×#932C2semidirect
8. C2×C7×#208C2semidirect
13. C3×C7×#41C22semidirect
14. C2×C3×C7×Q8C22semidirect
16. C2×C3×C7×D4C22semidirect
23. C7×#235C22semidirect
24. C2×C7×D12C22semidirect
26. C2×C7×Dic6C22semidirect
27. C7×#216C22semidirect
28. C7×#215C22semidirect
29. C7×#214C22semidirect
30. C7×#213C22semidirect
31. C7×#208C22semidirect
32. C7×#208C22semidirect
33. C2×C7×C3⋊C8C22semidirect
35. C2×C7×#41D3semidirect
42. #5822C7direct (N × H)
78. C7×#41C2×D3semidirect
80. C2×C7×Q8C2×D3semidirect
84. C2×C7×D4C2×D3semidirect
95. C2×C3×#41C2×C7semidirect
96. C2×#235C2×C7semidirect
97. #951C2×C7semidirect
98. #950C2×C7semidirect
99. #949C2×C7semidirect
100. #942C2×C7semidirect
101. #932C2×C7semidirect
102. C2×#208C2×C7semidirect
149. C3×#41C22×C7semidirect
150. C2×C3×Q8C22×C7semidirect
152. C2×C3×D4C22×C7semidirect
159. #235C22×C7semidirect
160. C2×D12C22×C7semidirect
162. C2×Dic6C22×C7semidirect
163. #216C22×C7semidirect
164. #215C22×C7semidirect
165. #214C22×C7semidirect
166. #213C22×C7semidirect
167. #208C22×C7semidirect
168. #208C22×C7semidirect
169. C2×C3⋊C8C22×C7semidirect
177. C2×#41D3×C7semidirect
226. C3×C7#575semidirect
234. #41C2×D3×C7semidirect
236. C2×Q8C2×D3×C7semidirect
240. C2×D4C2×D3×C7semidirect
299. C7#5822direct (N × H)
320. C3C7×#575semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×#412
C2×C7×#2352
C7×#9512
C7×#9502
C7×#9492
C7×#9422
C7×#9322
C2×C7×#2082
C7×#5753
#58227