Finite Group Databasebeta

← C7×#5773 #13061489

Subgroups — C7×#5773 #13061489

Order 1344.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×C7×#41C211yesmaximal, characteristic, complemented
2C2×C7×#237C211yesmaximal, characteristic, complemented
3C3×C7×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4C7×#951C211yesmaximal, characteristic, complemented
5C7×#932C211yesmaximal, characteristic, complemented
6C7×#916C212yesmaximal, complemented
7C7×#915C212yesmaximal, complemented
8C7×#912C212yesmaximal, complemented
9C7×#911C212yesmaximal, complemented
10C7×#909C211yesmaximal, characteristic, complemented
11D3×C7×M16C211yesmaximal, characteristic, complemented
12C7×#575—31nomaximal, Hall
13C3×C7×#41C2211yescharacteristic
14C2×C3×C7×D4C2211yescharacteristic, complemented
15C2×C7×#67—21no—
16C22×C7×Dic3—21no—
17C7×#239C2211yescharacteristic, complemented
18D3×C7×Q8C2211yescharacteristic, complemented
19C7×#237—21no—
20C7×#237—21no—
21C7×#237—21no—
22C7×#237C2212yes—
23C7×#237C2211yescharacteristic, complemented
24C7×#237C2212yescomplemented
25D3×C7×D4—21no—
26D3×C7×D4C2211yescharacteristic, complemented
27C7×#235—21no—
28C7×#235C2211yescharacteristic
29C2×C4×D3×C7—21no—
30C2×C4×D3×C7C2211yescharacteristic
31C2×C7×Dic6C2211yescharacteristic, complemented
32C3×C7×SD16C2212yescomplemented
33C3×C7×D8C2212yescomplemented
34C3×C7×M16C2211yescharacteristic, complemented
35C7×#216C2212yescomplemented
36C7×#214C2212yescomplemented
37C7×#214C2212yescomplemented
38C7×#213C2212yescomplemented
39C7×#208C2211yescharacteristic, complemented
40C7×Dic12C2212yescomplemented
41C7×#204C2212yescomplemented
42C7×ΓL(1,𝔽25)C2212yescomplemented
43D3×C7×C8C2212yescomplemented
44C2×C7×#41—31no—
45C2×C7×#41—31no—
46C7×#137—31no—
47C7×#137—32no—
48C7×#137—31no—
49C7×AGL(1,ℤ/8)—31no—
50C7×AGL(1,ℤ/8)—32no—
51C7×AGL(1,ℤ/8)D311yescharacteristic, complemented
52C7×#135—32no—
53C7×#135—32no—
54C2×C7×M16—31no—
55#5773C711yesmaximal, characteristic, direct product factor, Hall
56C23×C3×C7—21no—
57C22×D3×C7—21no—
58C22×D3×C7D411yescharacteristic
59C3×C7×Q8C2311yescharacteristic
60C3×C7×D4—21no—
61C3×C7×D4—21no—
62C3×C7×D4C2312yes—
63C3×C7×D4C2311yescharacteristic
64C2×C3×C4×C7—21no—
65C2×C3×C4×C7C2311yescharacteristic
66C7×#67—21no—
67C7×#67—21no—
68C7×#67—22no—
69C7×#67—22no—
70C7×#67—21no—
71C2×C7×Dic3—21no—
72C2×C7×Dic3—21no—
73C2×C7×Dic3—22no—
74C2×C7×Dic3—22no—
75C2×C7×Dic3D411yescharacteristic
76C7×D12—21no—
77C7×D12C2311yescharacteristic
78C4×D3×C7—21no—
79C4×D3×C7—21no—
80C4×D3×C7—21no—
81C4×D3×C7D411yescharacteristic
82C4×D3×C7D411yescharacteristic
83C4×D3×C7C2311yescharacteristic
84C4×D3×C7C2311yescharacteristic
85C7×Dic6—21no—
86C7×Dic6—21no—
87C7×Dic6C2312yes—
88C7×Dic6C2311yescharacteristic
89C3×C7×C8C2312yes—
90C7×C3⋊C8C2312yes—
91C7×#41—31no—
92C7×#41—32no—
93C7×#41—31no—
94C7×#41—32no—
95C7×#41—31no—
96C7×#41—61no—
97C7×#41—61no—
98C7×#41—61no—
99C7×#41—61no—
100C7×#41C2×D311yescharacteristic
101C2×C7×Q8—31no—
102C2×C7×Q8—31no—
103C2×C7×D4—31no—
104C2×C7×D4—61no—
105C2×C7×D4—61no—
106C2×C7×D4C2×D311yescharacteristic, complemented
107C22×C4×C7—31no—
108C22×C4×C7—61no—
109C22×C4×C7—61no—
110C7×Q16—32no—
111C7×Q16—32no—
112C7×SD16—32no—
113C7×SD16—32no—
114C7×SD16—32no—
115C7×SD16C2×D312yescomplemented
116C7×D8—32no—
117C7×D8C2×D312yescomplemented
118C7×M16—31no—
119C7×M16—32no—
120C7×M16C2×D311yescharacteristic, complemented
121C2×C7×C8—32no—
122D3×#41C2×C711yescharacteristic, complemented
123C2×#237C2×C711yescharacteristic, complemented
124C3×AGL(1,ℤ/8)C2×C711yescharacteristic, complemented
125#951C2×C711yescharacteristic, complemented
126#932C2×C711yescharacteristic, complemented
127#916C2×C712yescomplemented
128#915C2×C712yescomplemented
129#912C2×C712yescomplemented
130#911C2×C712yescomplemented
131#909C2×C711yescharacteristic, complemented
132D3×M16C2×C711yescharacteristic, complemented
133C22×C3×C7—22no—
134C22×C3×C7—21no—
135C22×C3×C7—42no—
136C22×C3×C7C2×D411yescharacteristic
137C2×D3×C7—21no—
138C2×D3×C7—41no—
139C2×D3×C7—42no—
140C2×D3×C7C2×D411yescharacteristic
141C2×D3×C7C2×D411yescharacteristic
142C3×C4×C7—21no—
143C3×C4×C7C2×D411yescharacteristic
144C3×C4×C7C2411yescharacteristic
145C7×Dic3—21no—
146C7×Dic3—22no—
147C7×Dic3C2×D411yescharacteristic
148C7×Dic3C2×D411yescharacteristic
149#575—31noSylow
150C23×C7—21no—
151C23×C7—31no—
152C23×C7—61no—
153C7×Q8—31no—
154C7×Q8—32no—
155C7×Q8—61no—
156C7×Q8—61no—
157C7×Q8C22×D311yescharacteristic
158C7×D4—21no—
159C7×D4—21no—
160C7×D4—31no—
161C7×D4—61no—
162C7×D4—61no—
163C7×D4—61no—
164C7×D4—62no—
165C7×D4—62no—
166C7×D4—61no—
167C7×D4C22×D312yes—
168C7×D4C22×D311yescharacteristic
169C2×C4×C7—21no—
170C2×C4×C7—31no—
171C2×C4×C7—31no—
172C2×C4×C7—31no—
173C2×C4×C7—31no—
174C2×C4×C7—31no—
175C2×C4×C7—61no—
176C2×C4×C7—61no—
177C2×C4×C7—61no—
178C2×C4×C7—62no—
179C2×C4×C7—62no—
180C2×C4×C7—61no—
181C2×C4×C7—61no—
182C2×C4×C7C22×D311yescharacteristic
183C7×C8—32no—
184C7×C8C22×D312yes—
185C3×#41C22×C711yescharacteristic
186C2×C3×D4C22×C711yescharacteristic, complemented
187C2×#67—21no—
188C22×Dic3—21no—
189#239C22×C711yescharacteristic, complemented
190D3×Q8C22×C711yescharacteristic, complemented
191#237—21no—
192#237—21no—
193#237—21no—
194#237C22×C712yes—
195#237C22×C711yescharacteristic, complemented
196#237C22×C712yescomplemented
197D3×D4—21no—
198D3×D4C22×C711yescharacteristic, complemented
199#235—21no—
200#235C22×C711yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3C22×C711yescharacteristic
203C2×Dic6C22×C711yescharacteristic, complemented
204C3×SD16C22×C712yescomplemented
205C3×D8C22×C712yescomplemented
206C3×M16C22×C711yescharacteristic, complemented
207#216C22×C712yescomplemented
208#214C22×C712yescomplemented
209#214C22×C712yescomplemented
210#213C22×C712yescomplemented
211#208C22×C711yescharacteristic, complemented
212Dic12C22×C712yescomplemented
213#204C22×C712yescomplemented
214ΓL(1,𝔽25)C22×C712yescomplemented
215D3×C8C22×C712yescomplemented
216C2×C3×C7—21no—
217C2×C3×C7—41no—
218C2×C3×C7—42no—
219C2×C3×C7C22×D411yescharacteristic
220D3×C7—21no—
221D3×C7—21no—
222D3×C7—41no—
223C2×#41—31no—
224C2×#41—31no—
225#137—31no—
226#137—32no—
227#137—31no—
228AGL(1,ℤ/8)—31no—
229AGL(1,ℤ/8)—32no—
230AGL(1,ℤ/8)D3×C711yescharacteristic, complemented
231#135—32no—
232#135—32no—
233C2×M16—31no—
234C22×C7—21no—
235C22×C7—22no—
236C22×C7—31no—
237C22×C7—31no—
238C22×C7—42no—
239C22×C7—61no—
240C22×C7—121no—
241C22×C7—122no—
242C22×C7D3×D411yescharacteristic
243C4×C7—21no—
244C4×C7—31no—
245C4×C7—31no—
246C4×C7—61no—
247C4×C7—62no—
248C4×C7D3×D411yescharacteristic
249C4×C7C23×D311yescharacteristic
250C23×C3—21no—
251C22×D3—21no—
252C22×D3C7×D411yescharacteristic
253C3×Q8C23×C711yescharacteristic
254C3×D4—21no—
255C3×D4—21no—
256C3×D4C23×C712yes—
257C3×D4C23×C711yescharacteristic
258C2×C3×C4—21no—
259C2×C3×C4C23×C711yescharacteristic
260#67—21no—
261#67—21no—
262#67—22no—
263#67—22no—
264#67—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3C7×D411yescharacteristic
270D12—21no—
271D12C23×C711yescharacteristic
272C4×D3—21no—
273C4×D3—21no—
274C4×D3—21no—
275C4×D3C7×D411yescharacteristic
276C4×D3C7×D411yescharacteristic
277C4×D3C23×C711yescharacteristic
278C4×D3C23×C711yescharacteristic
279Dic6—21no—
280Dic6—21no—
281Dic6C23×C712yes—
282Dic6C23×C711yescharacteristic
283C3×C8C23×C712yes—
284C3⋊C8C23×C712yes—
285C3×C7#57511yescharacteristic, complemented, Hall
286#41—31no—
287#41—32no—
288#41—31no—
289#41—32no—
290#41—31no—
291#41—61no—
292#41—61no—
293#41—61no—
294#41—61no—
295#41C2×D3×C711yescharacteristic
296C2×Q8—31no—
297C2×Q8—31no—
298C2×D4—31no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4C2×D3×C711yescharacteristic, complemented
302C22×C4—31no—
303C22×C4—61no—
304C22×C4—61no—
305Q16—32no—
306Q16—32no—
307SD16—32no—
308SD16—32no—
309SD16—32no—
310SD16C2×D3×C712yescomplemented
311D8—32no—
312D8C2×D3×C712yescomplemented
313M16—31no—
314M16—32no—
315M16C2×D3×C711yescharacteristic, complemented
316C2×C8—32no—
317C2×C7—21no—
318C2×C7—42no—
319C2×C7—41no—
320C2×C7—61no—
321C2×C7—61no—
322C2×C7—121no—
323C2×C7C2×D3×D411yescharacteristic
324C22×C3—22no—
325C22×C3—21no—
326C22×C3—42no—
327C22×C3C2×C7×D411yescharacteristic
328C2×D3—21no—
329C2×D3—41no—
330C2×D3—42no—
331C2×D3C2×C7×D411yescharacteristic
332C2×D3C2×C7×D411yescharacteristic
333C3×C4—21no—
334C3×C4C2×C7×D411yescharacteristic
335C3×C4C24×C711yescharacteristic
336Dic3—21no—
337Dic3—22no—
338Dic3C2×C7×D411yescharacteristic
339Dic3C2×C7×D411yescharacteristic
340C23—21no—
341C23—31no—
342C23—61no—
343Q8—31no—
344Q8—32no—
345Q8—61no—
346Q8—61no—
347Q8C22×D3×C711yescharacteristic
348D4—21no—
349D4—21no—
350D4—31no—
351D4—61no—
352D4—61no—
353D4—61no—
354D4—62no—
355D4—62no—
356D4—61no—
357D4C22×D3×C712yes—
358D4C22×D3×C711yescharacteristic
359C2×C4—21no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4C22×D3×C711yescharacteristic
373C8—32no—
374C8C22×D3×C712yes—
375C7#577311yescharacteristic, direct product factor, Sylow
376C2×C3—21no—
377C2×C3—41no—
378C2×C3—42no—
379C2×C3C22×C7×D411yescharacteristic
380D3—21no—
381D3—21no—
382D3—41no—
383C22—21no—
384C22—22no—
385C22—31no—
386C22—31no—
387C22—42no—
388C22—61no—
389C22—121no—
390C22—122no—
391C22D3×C7×D411yescharacteristic
392C4—21no—
393C4—31no—
394C4—31no—
395C4—61no—
396C4—62no—
397C4D3×C7×D411yescharacteristic
398C4C23×D3×C711yescharacteristic
399C3C7×#57511yescharacteristic, complemented, Sylow
400C2—21no—
401C2—42no—
402C2—41no—
403C2—61no—
404C2—61no—
405C2—121no—
406C2C2×D3×C7×D411yescharacteristic
subgroupG/H
centerC2×C7C2×D3×D4
commutator, cosocleC3×C4C24×C7
FrattiniC4C23×D3×C7
FittingC3×C7×AGL(1,ℤ/8)C2
socleC2×C3×C7C22×D4
hypercenterC7×AGL(1,ℤ/8)D3
hypocenterC3C7×#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. D3×C7×#41C2semidirect
2. C2×C7×#237C2semidirect
3. C3×C7×AGL(1,ℤ/8)C2semidirect
4. C7×#951C2semidirect
5. C7×#932C2semidirect
6. C7×#916C2semidirect
7. C7×#915C2semidirect
8. C7×#912C2semidirect
9. C7×#911C2semidirect
10. C7×#909C2semidirect
11. D3×C7×M16C2semidirect
14. C2×C3×C7×D4C22semidirect
17. C7×#239C22semidirect
18. D3×C7×Q8C22semidirect
23. C7×#237C22semidirect
24. C7×#237C22semidirect
26. D3×C7×D4C22semidirect
31. C2×C7×Dic6C22semidirect
32. C3×C7×SD16C22semidirect
33. C3×C7×D8C22semidirect
34. C3×C7×M16C22semidirect
35. C7×#216C22semidirect
36. C7×#214C22semidirect
37. C7×#214C22semidirect
38. C7×#213C22semidirect
39. C7×#208C22semidirect
40. C7×Dic12C22semidirect
41. C7×#204C22semidirect
42. C7×ΓL(1,𝔽25)C22semidirect
43. D3×C7×C8C22semidirect
51. C7×AGL(1,ℤ/8)D3semidirect
55. #5773C7direct (N × H)
106. C2×C7×D4C2×D3semidirect
115. C7×SD16C2×D3semidirect
117. C7×D8C2×D3semidirect
120. C7×M16C2×D3semidirect
122. D3×#41C2×C7semidirect
123. C2×#237C2×C7semidirect
124. C3×AGL(1,ℤ/8)C2×C7semidirect
125. #951C2×C7semidirect
126. #932C2×C7semidirect
127. #916C2×C7semidirect
128. #915C2×C7semidirect
129. #912C2×C7semidirect
130. #911C2×C7semidirect
131. #909C2×C7semidirect
132. D3×M16C2×C7semidirect
186. C2×C3×D4C22×C7semidirect
189. #239C22×C7semidirect
190. D3×Q8C22×C7semidirect
195. #237C22×C7semidirect
196. #237C22×C7semidirect
198. D3×D4C22×C7semidirect
203. C2×Dic6C22×C7semidirect
204. C3×SD16C22×C7semidirect
205. C3×D8C22×C7semidirect
206. C3×M16C22×C7semidirect
207. #216C22×C7semidirect
208. #214C22×C7semidirect
209. #214C22×C7semidirect
210. #213C22×C7semidirect
211. #208C22×C7semidirect
212. Dic12C22×C7semidirect
213. #204C22×C7semidirect
214. ΓL(1,𝔽25)C22×C7semidirect
215. D3×C8C22×C7semidirect
230. AGL(1,ℤ/8)D3×C7semidirect
285. C3×C7#575semidirect
301. C2×D4C2×D3×C7semidirect
310. SD16C2×D3×C7semidirect
312. D8C2×D3×C7semidirect
315. M16C2×D3×C7semidirect
375. C7#5773direct (N × H)
399. C3C7×#575semidirect

Maximal subgroups

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

subgroupindex
D3×C7×#412
C2×C7×#2372
C3×C7×AGL(1,ℤ/8)2
C7×#9512
C7×#9322
C7×#9162
C7×#9152
C7×#9122
C7×#9112
C7×#9092
D3×C7×M162
C7×#5753
#57737