Finite Group Databasebeta

← C7×#64314 #72246607

Subgroups — C7×#64314 #72246607

Order 2016.

718 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×C7×#237C211yesmaximal, characteristic, complemented
3C3×D3×C7×D4C211yesmaximal, characteristic, complemented
4C7×#3811C212yesmaximal, complemented
5D3×C7×#67C212yesmaximal, complemented
6C2×C7×#626C212yesmaximal, complemented
7C7×#3804C212yesmaximal, complemented
8D3×C7×D12C211yesmaximal, characteristic, complemented
9C7×#3799C211yesmaximal, characteristic, complemented
10C7×#3798C211yesmaximal, characteristic, complemented
11C7×#3796C211yesmaximal, characteristic, complemented
12C7×#1009—31nomaximal
13C7×#1004—31nomaximal
14C22×C7×C3⋊D3C2212yescomplemented
15C22×C3×D3×C7C2212yescomplemented
16C2×D32×C7C2212yescomplemented
17C32×C7×D4C2211yescharacteristic, complemented
18C7×#638C2212yescomplemented
19C7×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C4×C7×C3⋊D3C2211yescharacteristic, complemented
21C3×C7×#67C2212yescomplemented
22C3×C7×#67C2212yescomplemented
23C2×C3×C7×Dic3C2212yescomplemented
24C3×C7×D12C2211yescharacteristic, complemented
25C3×C4×D3×C7C2211yescharacteristic, complemented
26C3×C4×D3×C7C2211yescharacteristic, complemented
27C3×C7×Dic6C2211yescharacteristic, complemented
28C7×#627C2211yescharacteristic, complemented
29C7×#626C2212yescomplemented
30C7×#626C2211yescharacteristic, complemented
31C7×#626C2211yescharacteristic, complemented
32C7×#626C2212yescomplemented
33C7×#626C2212yescomplemented
34C7×#625C2211yescharacteristic, complemented
35C7×#624C2212yescomplemented
36D3×C7×Dic3C2212yescomplemented
37C3×C7×#41—31no—
38C2×C3×C7×D4—31no—
39C2×C7×#67—32no—
40C2×C7×#67—32no—
41C7×#239—31no—
42C7×#239—31no—
43C7×#237—32no—
44C7×#237—31no—
45C7×#237D311yescharacteristic, complemented
46D3×C7×D4—32no—
47D3×C7×D4—31no—
48D3×C7×D4—31no—
49D3×C7×D4—32no—
50D3×C7×D4—32no—
51D3×C7×D4—31no—
52D3×C7×D4—61no—
53D3×C7×D4D311yescharacteristic, complemented
54C7×#235—32no—
55C7×#235—31no—
56C7×#235—31no—
57C7×#235—31no—
58C2×C7×D12—32no—
59C2×C7×D12—31no—
60#64314C711yesmaximal, characteristic, direct product factor, Hall
61C22×C32×C7C2312yes—
62C2×C7×C3⋊D3—42no—
63C2×C7×C3⋊D3C2312yes—
64C2×C7×C3⋊D3C2311yescharacteristic
65C2×C3×D3×C7—42no—
66C2×C3×D3×C7C2312yes—
67C2×C3×D3×C7C2311yescharacteristic
68C2×C3×D3×C7C2311yescharacteristic
69D32×C7—42no—
70C32×C4×C7C2311yescharacteristic
71C7×C3⋊Dic3C2311yescharacteristic
72C3×C7×Dic3C2312yes—
73C3×C7×Dic3C2311yescharacteristic
74C3×C7×Dic3C2311yescharacteristic
75C7×H(5,𝔽2)—91noHall
76C23×C3×C7—32no—
77C22×D3×C7—32no—
78C22×D3×C7—32no—
79C22×D3×C7—32no—
80C22×D3×C7—32no—
81C22×D3×C7—62no—
82C22×D3×C7C2×D312yescomplemented
83C3×C7×Q8—31no—
84C3×C7×D4—21no—
85C3×C7×D4—32no—
86C3×C7×D4—32no—
87C3×C7×D4—31no—
88C3×C7×D4C2×D311yescharacteristic, complemented
89C3×C7×D4C2×D311yescharacteristic, complemented
90C2×C3×C4×C7—32no—
91C2×C3×C4×C7—31no—
92C2×C3×C4×C7—31no—
93C7×#67—32no—
94C7×#67—32no—
95C7×#67—31no—
96C7×#67—31no—
97C7×#67—31no—
98C7×#67—31no—
99C7×#67—32no—
100C7×#67—32no—
101C7×#67—32no—
102C7×#67—62no—
103C7×#67C2×D312yescomplemented
104C7×#67C2×D312yescomplemented
105C2×C7×Dic3—32no—
106C2×C7×Dic3C2×D312yescomplemented
107C7×D12—32no—
108C7×D12—31no—
109C7×D12—32no—
110C7×D12—32no—
111C7×D12—31no—
112C7×D12—31no—
113C7×D12—31no—
114C7×D12—61no—
115C7×D12C2×D311yescharacteristic, complemented
116C4×D3×C7—32no—
117C4×D3×C7—31no—
118C4×D3×C7—31no—
119C4×D3×C7—32no—
120C4×D3×C7—32no—
121C4×D3×C7—61no—
122C4×D3×C7C2×D311yescharacteristic, complemented
123C4×D3×C7C2×D311yescharacteristic, complemented
124C7×Dic6—31no—
125C7×Dic6—31no—
126C7×Dic6C2×D311yescharacteristic, complemented
127D4×C3⋊D3C2×C711yescharacteristic, complemented
128C3×#237C2×C711yescharacteristic, complemented
129C3×D3×D4C2×C711yescharacteristic, complemented
130#3811C2×C712yescomplemented
131D3×#67C2×C712yescomplemented
132C2×#626C2×C712yescomplemented
133#3804C2×C712yescomplemented
134D3×D12C2×C711yescharacteristic, complemented
135#3799C2×C711yescharacteristic, complemented
136#3798C2×C711yescharacteristic, complemented
137#3796C2×C711yescharacteristic, complemented
138C2×C32×C7—22no—
139C2×C32×C7C2411yescharacteristic
140C7×C3⋊D3—21no—
141C7×C3⋊D3—22no—
142C3×D3×C7—21no—
143C3×D3×C7—22no—
144C3×D3×C7—21no—
145C7×#41—31no—
146C7×#41—91no—
147C7×#41—92no—
148C7×#41—91no—
149C7×#41—91no—
150C2×C7×D4—31no—
151C2×C7×D4—91no—
152C2×C7×D4—92no—
153C2×C7×D4—92no—
154C2×C7×D4—92no—
155C2×C7×D4—91no—
156#1009—31no—
157#1004—31no—
158C22×C3×C7—22no—
159C22×C3×C7—32no—
160C22×C3×C7—31no—
161C22×C3×C7—31no—
162C22×C3×C7—122no—
163C22×C3×C7C22×D312yes—
164C22×C3×C7C22×D312yes—
165C2×D3×C7—32no—
166C2×D3×C7—31no—
167C2×D3×C7—32no—
168C2×D3×C7—31no—
169C2×D3×C7—42no—
170C2×D3×C7—61no—
171C2×D3×C7—62no—
172C2×D3×C7—122no—
173C2×D3×C7—122no—
174C2×D3×C7—122no—
175C2×D3×C7—122no—
176C2×D3×C7—122no—
177C2×D3×C7—122no—
178C2×D3×C7C22×D312yes—
179C2×D3×C7C22×D311yescharacteristic
180C2×D3×C7C22×D311yescharacteristic
181C3×C4×C7—21no—
182C3×C4×C7—31no—
183C3×C4×C7—31no—
184C3×C4×C7—32no—
185C3×C4×C7C22×D311yescharacteristic
186C3×C4×C7C22×D311yescharacteristic
187C7×Dic3—31no—
188C7×Dic3—31no—
189C7×Dic3—61no—
190C7×Dic3C22×D311yescharacteristic
191C7×Dic3C22×D311yescharacteristic
192C7×Dic3C22×D312yes—
193C22×C3⋊D3C22×C712yescomplemented
194C22×C3×D3C22×C712yescomplemented
195C2×D32C22×C712yescomplemented
196C32×D4C22×C711yescharacteristic, complemented
197#638C22×C712yescomplemented
198C4⋊(C3⋊D3)C22×C711yescharacteristic, complemented
199C4×C3⋊D3C22×C711yescharacteristic, complemented
200C3×#67C22×C712yescomplemented
201C3×#67C22×C712yescomplemented
202C2×C3×Dic3C22×C712yescomplemented
203C3×D12C22×C711yescharacteristic, complemented
204C3×C4×D3C22×C711yescharacteristic, complemented
205C3×C4×D3C22×C711yescharacteristic, complemented
206C3×Dic6C22×C711yescharacteristic, complemented
207#627C22×C711yescharacteristic, complemented
208#626C22×C712yescomplemented
209#626C22×C711yescharacteristic, complemented
210#626C22×C711yescharacteristic, complemented
211#626C22×C712yescomplemented
212#626C22×C712yescomplemented
213#625C22×C711yescharacteristic, complemented
214#624C22×C712yescomplemented
215D3×Dic3C22×C712yescomplemented
216C32×C7H(5,𝔽2)11yescharacteristic, complemented, Hall
217C23×C7—32no—
218C23×C7—92no—
219C23×C7—92no—
220C7×Q8—31no—
221C7×Q8—91no—
222C7×D4—32no—
223C7×D4—32no—
224C7×D4—31no—
225C7×D4—91no—
226C7×D4—92no—
227C7×D4—91no—
228C7×D4—92no—
229C7×D4—92no—
230C7×D4—91no—
231C7×D4—92no—
232C7×D4—91no—
233C7×D4D3211yescharacteristic, complemented
234C2×C4×C7—32no—
235C2×C4×C7—31no—
236C2×C4×C7—31no—
237C2×C4×C7—92no—
238C2×C4×C7—92no—
239C2×C4×C7—91no—
240C3×#41—31no—
241C2×C3×D4—31no—
242C2×#67—32no—
243C2×#67—32no—
244#239—31no—
245#239—31no—
246#237—32no—
247#237—31no—
248#237D3×C711yescharacteristic, complemented
249D3×D4—32no—
250D3×D4—31no—
251D3×D4—31no—
252D3×D4—32no—
253D3×D4—32no—
254D3×D4—31no—
255D3×D4—61no—
256D3×D4D3×C711yescharacteristic, complemented
257#235—32no—
258#235—31no—
259#235—31no—
260#235—31no—
261C2×D12—32no—
262C2×D12—31no—
263C2×C3×C7—22no—
264C2×C3×C7—22no—
265C2×C3×C7—21no—
266C2×C3×C7—42no—
267C2×C3×C7—61no—
268C2×C3×C7—62no—
269C2×C3×C7—61no—
270C2×C3×C7C23×D311yescharacteristic
271C2×C3×C7C23×D311yescharacteristic
272D3×C7—21no—
273D3×C7—22no—
274D3×C7—21no—
275D3×C7—61no—
276D3×C7—61no—
277D3×C7—61no—
278D3×C7—62no—
279D3×C7—61no—
280D3×C7—62no—
281D3×C7—122no—
282C22×C32C23×C712yes—
283C2×C3⋊D3—42no—
284C2×C3⋊D3C23×C712yes—
285C2×C3⋊D3C23×C711yescharacteristic
286C2×C3×D3—42no—
287C2×C3×D3C23×C712yes—
288C2×C3×D3C23×C711yescharacteristic
289C2×C3×D3C23×C711yescharacteristic
290D32—42no—
291C32×C4C23×C711yescharacteristic
292C3⋊Dic3C23×C711yescharacteristic
293C3×Dic3C23×C712yes—
294C3×Dic3C23×C711yescharacteristic
295C3×Dic3C23×C711yescharacteristic
296H(5,𝔽2)—91noSylow
297C22×C7—31no—
298C22×C7—31no—
299C22×C7—32no—
300C22×C7—91no—
301C22×C7—92no—
302C22×C7—122no—
303C22×C7—362no—
304C22×C7—362no—
305C22×C7C2×D3212yes—
306C4×C7—31no—
307C4×C7—31no—
308C4×C7—32no—
309C4×C7—91no—
310C4×C7C2×D3211yescharacteristic
311C23×C3—32no—
312C22×D3—32no—
313C22×D3—32no—
314C22×D3—32no—
315C22×D3—32no—
316C22×D3—62no—
317C22×D3C2×D3×C712yescomplemented
318C3×Q8—31no—
319C3×D4—21no—
320C3×D4—32no—
321C3×D4—32no—
322C3×D4—31no—
323C3×D4C2×D3×C711yescharacteristic, complemented
324C3×D4C2×D3×C711yescharacteristic, complemented
325C2×C3×C4—32no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328#67—32no—
329#67—32no—
330#67—31no—
331#67—31no—
332#67—31no—
333#67—31no—
334#67—32no—
335#67—32no—
336#67—32no—
337#67—62no—
338#67C2×D3×C712yescomplemented
339#67C2×D3×C712yescomplemented
340C2×Dic3—32no—
341C2×Dic3C2×D3×C712yescomplemented
342D12—32no—
343D12—31no—
344D12—32no—
345D12—32no—
346D12—31no—
347D12—31no—
348D12—31no—
349D12—61no—
350D12C2×D3×C711yescharacteristic, complemented
351C4×D3—32no—
352C4×D3—31no—
353C4×D3—31no—
354C4×D3—32no—
355C4×D3—32no—
356C4×D3—61no—
357C4×D3C2×D3×C711yescharacteristic, complemented
358C4×D3C2×D3×C711yescharacteristic, complemented
359Dic6—31no—
360Dic6—31no—
361Dic6C2×D3×C711yescharacteristic, complemented
362C3×C7—21no—
363C3×C7#100411yescharacteristic, complemented
364C3×C7#100911yescharacteristic, complemented
365C2×C32—22no—
366C2×C32C24×C711yescharacteristic
367C3⋊D3—21no—
368C3⋊D3—22no—
369C3×D3—21no—
370C3×D3—22no—
371C3×D3—21no—
372#41—31no—
373#41—91no—
374#41—92no—
375#41—91no—
376#41—91no—
377C2×D4—31no—
378C2×D4—91no—
379C2×D4—92no—
380C2×D4—92no—
381C2×D4—92no—
382C2×D4—91no—
383C2×C7—22no—
384C2×C7—61no—
385C2×C7—62no—
386C2×C7—61no—
387C2×C7—181no—
388C2×C7—182no—
389C2×C7C22×D3211yescharacteristic
390C22×C3—22no—
391C22×C3—32no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—122no—
395C22×C3C22×D3×C712yes—
396C22×C3C22×D3×C712yes—
397C2×D3—31no—
398C2×D3—32no—
399C2×D3—31no—
400C2×D3—32no—
401C2×D3—42no—
402C2×D3—61no—
403C2×D3—62no—
404C2×D3—122no—
405C2×D3—122no—
406C2×D3—122no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3—122no—
410C2×D3C22×D3×C712yes—
411C2×D3C22×D3×C711yescharacteristic
412C2×D3C22×D3×C711yescharacteristic
413C3×C4—21no—
414C3×C4—31no—
415C3×C4—31no—
416C3×C4—32no—
417C3×C4C22×D3×C711yescharacteristic
418C3×C4C22×D3×C711yescharacteristic
419Dic3—31no—
420Dic3—31no—
421Dic3—61no—
422Dic3C22×D3×C711yescharacteristic
423Dic3C22×D3×C711yescharacteristic
424Dic3C22×D3×C712yes—
425C32C7×H(5,𝔽2)11yescharacteristic, complemented, Sylow
426C23—32no—
427C23—92no—
428C23—92no—
429Q8—31no—
430Q8—91no—
431D4—32no—
432D4—32no—
433D4—31no—
434D4—91no—
435D4—92no—
436D4—91no—
437D4—92no—
438D4—92no—
439D4—91no—
440D4—92no—
441D4—91no—
442D4D32×C711yescharacteristic, complemented
443C2×C4—32no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—91no—
449C7#6431411yescharacteristic, direct product factor, Sylow
450C2×C3—22no—
451C2×C3—22no—
452C2×C3—21no—
453C2×C3—42no—
454C2×C3—61no—
455C2×C3—62no—
456C2×C3—61no—
457C2×C3C23×D3×C711yescharacteristic
458C2×C3C23×D3×C711yescharacteristic
459D3—21no—
460D3—22no—
461D3—21no—
462D3—61no—
463D3—61no—
464D3—61no—
465D3—62no—
466D3—61no—
467D3—62no—
468D3—122no—
469C22—31no—
470C22—31no—
471C22—32no—
472C22—91no—
473C22—92no—
474C22—122no—
475C22—362no—
476C22—362no—
477C22C2×D32×C712yes—
478C4—31no—
479C4—31no—
480C4—32no—
481C4—91no—
482C4C2×D32×C711yescharacteristic
483C3—21no—
484C3C7×#100411yescharacteristic, complemented
485C3C7×#100911yescharacteristic, complemented
486C2—22no—
487C2—61no—
488C2—62no—
489C2—61no—
490C2—181no—
491C2—182no—
492C2C22×D32×C711yescharacteristic
subgroupG/H
centerC2×C7C22×D32
commutator, cosocleC2×C32C24×C7
FrattiniC2C22×D32×C7
FittingC32×C7×D4C22
socleC2×C32×C7C24
hypercenterC7×D4D32
hypocenterC32C7×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C7 ◁ C7×D4

Complements

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

NH ≅ G/Ntype
1. C7×D4×C3⋊D3C2semidirect
2. C3×C7×#237C2semidirect
3. C3×D3×C7×D4C2semidirect
4. C7×#3811C2semidirect
5. D3×C7×#67C2semidirect
6. C2×C7×#626C2semidirect
7. C7×#3804C2semidirect
8. D3×C7×D12C2semidirect
9. C7×#3799C2semidirect
10. C7×#3798C2semidirect
11. C7×#3796C2semidirect
14. C22×C7×C3⋊D3C22semidirect
15. C22×C3×D3×C7C22semidirect
16. C2×D32×C7C22semidirect
17. C32×C7×D4C22semidirect
18. C7×#638C22semidirect
19. C7×C4⋊(C3⋊D3)C22semidirect
20. C4×C7×C3⋊D3C22semidirect
21. C3×C7×#67C22semidirect
22. C3×C7×#67C22semidirect
23. C2×C3×C7×Dic3C22semidirect
24. C3×C7×D12C22semidirect
25. C3×C4×D3×C7C22semidirect
26. C3×C4×D3×C7C22semidirect
27. C3×C7×Dic6C22semidirect
28. C7×#627C22semidirect
29. C7×#626C22semidirect
30. C7×#626C22semidirect
31. C7×#626C22semidirect
32. C7×#626C22semidirect
33. C7×#626C22semidirect
34. C7×#625C22semidirect
35. C7×#624C22semidirect
36. D3×C7×Dic3C22semidirect
45. C7×#237D3semidirect
53. D3×C7×D4D3semidirect
60. #64314C7direct (N × H)
82. C22×D3×C7C2×D3semidirect
88. C3×C7×D4C2×D3semidirect
89. C3×C7×D4C2×D3semidirect
103. C7×#67C2×D3semidirect
104. C7×#67C2×D3semidirect
106. C2×C7×Dic3C2×D3semidirect
115. C7×D12C2×D3semidirect
122. C4×D3×C7C2×D3semidirect
123. C4×D3×C7C2×D3semidirect
126. C7×Dic6C2×D3semidirect
127. D4×C3⋊D3C2×C7semidirect
128. C3×#237C2×C7semidirect
129. C3×D3×D4C2×C7semidirect
130. #3811C2×C7semidirect
131. D3×#67C2×C7semidirect
132. C2×#626C2×C7semidirect
133. #3804C2×C7semidirect
134. D3×D12C2×C7semidirect
135. #3799C2×C7semidirect
136. #3798C2×C7semidirect
137. #3796C2×C7semidirect
193. C22×C3⋊D3C22×C7semidirect
194. C22×C3×D3C22×C7semidirect
195. C2×D32C22×C7semidirect
196. C32×D4C22×C7semidirect
197. #638C22×C7semidirect
198. C4⋊(C3⋊D3)C22×C7semidirect
199. C4×C3⋊D3C22×C7semidirect
200. C3×#67C22×C7semidirect
201. C3×#67C22×C7semidirect
202. C2×C3×Dic3C22×C7semidirect
203. C3×D12C22×C7semidirect
204. C3×C4×D3C22×C7semidirect
205. C3×C4×D3C22×C7semidirect
206. C3×Dic6C22×C7semidirect
207. #627C22×C7semidirect
208. #626C22×C7semidirect
209. #626C22×C7semidirect
210. #626C22×C7semidirect
211. #626C22×C7semidirect
212. #626C22×C7semidirect
213. #625C22×C7semidirect
214. #624C22×C7semidirect
215. D3×Dic3C22×C7semidirect
216. C32×C7H(5,𝔽2)semidirect
233. C7×D4D32semidirect
248. #237D3×C7semidirect
256. D3×D4D3×C7semidirect
317. C22×D3C2×D3×C7semidirect
323. C3×D4C2×D3×C7semidirect
324. C3×D4C2×D3×C7semidirect
338. #67C2×D3×C7semidirect
339. #67C2×D3×C7semidirect
341. C2×Dic3C2×D3×C7semidirect
350. D12C2×D3×C7semidirect
357. C4×D3C2×D3×C7semidirect
358. C4×D3C2×D3×C7semidirect
361. Dic6C2×D3×C7semidirect
363. C3×C7#1004semidirect
364. C3×C7#1009semidirect
425. C32C7×H(5,𝔽2)semidirect
442. D4D32×C7semidirect
449. C7#64314direct (N × H)
484. C3C7×#1004semidirect
485. C3C7×#1009semidirect

Maximal subgroups

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

subgroupindex
C7×D4×C3⋊D32
C3×C7×#2372
C3×D3×C7×D42
C7×#38112
D3×C7×#672
C2×C7×#6262
C7×#38042
D3×C7×D122
C7×#37992
C7×#37982
C7×#37962
C7×#10093
C7×#10043
#643147