Finite Group Databasebeta

← #72246670

Subgroups — #72246670

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
1#11759634C211yesmaximal, characteristic, complemented
2#11759607C211yesmaximal, characteristic, complemented
3#11759602C211yesmaximal, characteristic, complemented
4C7×#3799C211yesmaximal, characteristic, complemented
5#11759568C211yesmaximal, characteristic, complemented
6C3×D3×D28C211yesmaximal, characteristic, complemented
7C3×#66670C211yesmaximal, characteristic, complemented
8#11759439C212yesmaximal, complemented
9#11759428C212yesmaximal, complemented
10#11759425C212yesmaximal, complemented
11D7×#626C212yesmaximal, complemented
12#10621582—31nomaximal
13#10621564—31nomaximal
14C2×(C3⋊D3)⋊D7C2212yescomplemented
15C2×D7×C3⋊D3C2212yescomplemented
16C2×C3×D3×D7C2212yescomplemented
17C4×C3⋊D21C2211yescharacteristic, complemented
18C7×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C3×D84C2211yescharacteristic, complemented
20C3×C4×D21C2211yescharacteristic, complemented
21C3×C4×D3×C7C2211yescharacteristic, complemented
22C3×C7×Dic6C2211yescharacteristic, complemented
23C32×D28C2211yescharacteristic, complemented
24#92694C2212yescomplemented
25#92691C2211yescharacteristic, complemented
26#92690C2211yescharacteristic, complemented
27#92689C2212yescomplemented
28#92689C2211yescharacteristic, complemented
29#92688C2211yescharacteristic, complemented
30Dic3×D21C2212yescomplemented
31C7×#626C2212yescomplemented
32C7×#624C2212yescomplemented
33#92676C2212yescomplemented
34C3×#4246C2212yescomplemented
35C3×#4246C2212yescomplemented
36C3×Dic3×D7C2212yescomplemented
37#66732—31no—
38#66728—31no—
39C7×#239—31no—
40C7×#235—31no—
41C3×#1161—31no—
42C2×C3×D28—31no—
43D7×#67—32no—
44D7×#67—32no—
45#66681—32no—
46#66681—31no—
47#66681—31no—
48#66681—31no—
49#66681—61no—
50D3×D28—32no—
51D3×D28—32no—
52D3×D28D311yescharacteristic, complemented
53D7×D12—32no—
54D7×D12—31no—
55#66675—31no—
56#66675—32no—
57#66670—31no—
58#66670D311yescharacteristic, complemented
59#66669—32no—
60#64314—71nomaximal, Hall
61C2×C3⋊D21C2311yescharacteristic
62C2×C7×C3⋊D3C2312yes—
63C2×C3×D21C2312yes—
64C2×C3×D21C2311yescharacteristic
65C2×C3×D3×C7C2311yescharacteristic
66C2×C32×D7C2312yes—
67(C3⋊D3)⋊D7—42no—
68D7×C3⋊D3—42no—
69C3×D3×D7—42no—
70C32×C4×C7C2311yescharacteristic
71C3⋊Dic21C2311yescharacteristic
72C3×Dic21C2311yescharacteristic
73C3×C7×Dic3C2311yescharacteristic
74C3×C7×Dic3C2312yes—
75#6553—91noHall
76C22×C3×D7—32no—
77C2×D3×D7—32no—
78C2×D3×D7—32no—
79C2×D3×D7—32no—
80C2×D3×D7—32no—
81C2×D3×D7—62no—
82C2×D3×D7C2×D312yescomplemented
83C3×C7×Q8—31no—
84C2×C3×C4×C7—31no—
85#4268—31no—
86#4268—31no—
87D84—32no—
88D84—31no—
89D84C2×D311yescharacteristic, complemented
90C4×D21—32no—
91C4×D21—31no—
92C4×D21—31no—
93C4×D21—61no—
94C4×D21C2×D311yescharacteristic, complemented
95Dic42—31no—
96C7×#67—32no—
97C7×D12—32no—
98C7×D12—31no—
99C7×D12—31no—
100C7×D12—61no—
101C4×D3×C7—32no—
102C4×D3×C7—32no—
103C4×D3×C7C2×D311yescharacteristic, complemented
104C7×Dic6C2×D311yescharacteristic, complemented
105C3×D28—21no—
106C3×D28—32no—
107C3×D28—32no—
108C3×D28—31no—
109C3×D28C2×D311yescharacteristic, complemented
110C3×D28C2×D311yescharacteristic, complemented
111C3×C4×D7—32no—
112C3×C4×D7—31no—
113#4248—31no—
114#4247—32no—
115#4247—31no—
116#4246—32no—
117#4246—31no—
118#4246C2×D312yescomplemented
119#4246C2×D312yescomplemented
120#4245—31no—
121#4245—32no—
122#4245—32no—
123#4245—32no—
124#4245—62no—
125Dic3×D7—32no—
126Dic3×D7C2×D312yescomplemented
127D4×C3⋊D3—71no—
128C3×#237—71no—
129C3×D3×D4—71no—
130#3811—72no—
131D3×#67—72no—
132C2×#626—72no—
133#3804—72no—
134D3×D12—71no—
135#3799D711yescharacteristic, complemented
136#3798—71no—
137#3796—71no—
138C2×C32×C7C2411yescharacteristic
139C3⋊D21—21no—
140C7×C3⋊D3—22no—
141C3×D21—22no—
142C3×D21—21no—
143C3×D3×C7—21no—
144C32×D7—22no—
145C7×#41—91no—
146#1161—31no—
147#1161—91no—
148D4×D7—91no—
149D4×D7—92no—
150D4×D7—92no—
151D4×D7—91no—
152#1157—92no—
153#1157—91no—
154C2×D28—31no—
155C2×D28—92no—
156#1009—211no—
157#1004—211no—
158C22×C3×C7—31no—
159C2×D21—31no—
160C2×D21—31no—
161C2×D21—61no—
162C2×D21C22×D311yescharacteristic
163C2×D21C22×D312yes—
164C2×D3×C7—32no—
165C2×D3×C7—32no—
166C2×D3×C7—62no—
167C2×D3×C7C22×D311yescharacteristic
168C2×C3×D7—22no—
169C2×C3×D7—32no—
170C2×C3×D7—31no—
171C2×C3×D7—122no—
172C2×C3×D7C22×D312yes—
173C2×C3×D7C22×D312yes—
174D3×D7—42no—
175D3×D7—122no—
176D3×D7—122no—
177D3×D7—122no—
178D3×D7—122no—
179D3×D7—122no—
180D3×D7—122no—
181C3×C4×C7—21no—
182C3×C4×C7—32no—
183C3×C4×C7—31no—
184C3×C4×C7C22×D311yescharacteristic
185C3×C4×C7C22×D311yescharacteristic
186Dic21—31no—
187Dic21—31no—
188Dic21—61no—
189Dic21C22×D311yescharacteristic
190C3×Dic7—31no—
191C7×Dic3C22×D312yes—
192C7×Dic3C22×D311yescharacteristic
193C22×C3⋊D3—72no—
194C22×C3×D3—72no—
195C2×D32—72no—
196C32×D4—71no—
197#638—72no—
198C4⋊(C3⋊D3)C2×D711yescharacteristic, complemented
199C4×C3⋊D3—71no—
200C3×#67—72no—
201C3×#67—72no—
202C2×C3×Dic3—72no—
203C3×D12—71no—
204C3×C4×D3—71no—
205C3×C4×D3C2×D711yescharacteristic, complemented
206C3×Dic6C2×D711yescharacteristic, complemented
207#627—71no—
208#626—72no—
209#626—71no—
210#626—72no—
211#626—71no—
212#626C2×D712yescomplemented
213#625—71no—
214#624C2×D712yescomplemented
215D3×Dic3—72no—
216C32×C7H(5,𝔽2)11yescharacteristic, complemented, Hall
217C22×D7—32no—
218C22×D7—92no—
219C22×D7—92no—
220C7×Q8—31no—
221C7×D4—92no—
222C7×D4—91no—
223C2×C4×C7—31no—
224C2×C4×C7—92no—
225C22⋊D7—92no—
226C22⋊D7—91no—
227C22⋊D7—91no—
228C22⋊D7—92no—
229D28—32no—
230D28—32no—
231D28—31no—
232D28—91no—
233D28—92no—
234D28D3211yescharacteristic, complemented
235C4×D7—32no—
236C4×D7—31no—
237C4×D7—92no—
238C4×D7—91no—
239Dic14—91no—
240C3×#41—211no—
241C2×C3×D4—211no—
242C2×#67—212no—
243C2×#67—212no—
244#239—31no—
245#239—211no—
246#237—71no—
247#237—212no—
248#237—211no—
249D3×D4—71no—
250D3×D4—212no—
251D3×D4—212no—
252D3×D4—211no—
253D3×D4—211no—
254D3×D4—211no—
255D3×D4—212no—
256D3×D4—421no—
257#235—31no—
258#235—212no—
259#235—211no—
260#235—211no—
261C2×D12—211no—
262C2×D12—212no—
263C2×C3×C7—21no—
264C2×C3×C7—61no—
265C2×C3×C7C23×D311yescharacteristic
266C2×C3×C7C23×D311yescharacteristic
267D21—22no—
268D21—21no—
269D21—61no—
270D21—61no—
271D21—61no—
272D21—61no—
273C3×D7—22no—
274C3×D7—22no—
275C3×D7—42no—
276C3×D7—61no—
277C3×D7—62no—
278D3×C7—21no—
279D3×C7—62no—
280D3×C7—62no—
281D3×C7—122no—
282C22×C32—72no—
283C2×C3⋊D3—71no—
284C2×C3⋊D3—282no—
285C2×C3⋊D3C22×D712yes—
286C2×C3×D3—72no—
287C2×C3×D3—71no—
288C2×C3×D3—282no—
289C2×C3×D3C22×D711yescharacteristic
290D32—282no—
291C32×C4C22×D711yescharacteristic
292C3⋊Dic3—71no—
293C3×Dic3—71no—
294C3×Dic3C22×D712yes—
295C3×Dic3C22×D711yescharacteristic
296H(5,𝔽2)—631noSylow
297C22×C7—31no—
298C22×C7—92no—
299C2×D7—31no—
300C2×D7—32no—
301C2×D7—91no—
302C2×D7—122no—
303C2×D7—362no—
304C2×D7—362no—
305C2×D7C2×D3212yes—
306C4×C7—31no—
307C4×C7—32no—
308C4×C7C2×D3211yescharacteristic
309Dic7—31no—
310Dic7—91no—
311C23×C3—212no—
312C22×D3—72no—
313C22×D3—212no—
314C22×D3—212no—
315C22×D3—212no—
316C22×D3—212no—
317C22×D3—422no—
318C3×Q8—31no—
319C3×D4—71no—
320C3×D4—71no—
321C3×D4—141no—
322C3×D4—212no—
323C3×D4—212no—
324C3×D4—211no—
325C2×C3×C4—31no—
326C2×C3×C4—212no—
327C2×C3×C4—211no—
328#67—32no—
329#67—72no—
330#67—72no—
331#67—212no—
332#67—211no—
333#67—212no—
334#67—212no—
335#67—211no—
336#67—211no—
337#67—211no—
338#67—212no—
339#67—422no—
340C2×Dic3—72no—
341C2×Dic3—212no—
342D12—32no—
343D12—31no—
344D12—31no—
345D12—61no—
346D12—71no—
347D12—212no—
348D12—211no—
349D12—212no—
350D12—211no—
351C4×D3—32no—
352C4×D3—32no—
353C4×D3—71no—
354C4×D3—212no—
355C4×D3—211no—
356C4×D3—211no—
357C4×D3—421no—
358C4×D3D3×D711yescharacteristic, complemented
359Dic6—211no—
360Dic6—211no—
361Dic6D3×D711yescharacteristic, complemented
362C3×C7—21no—
363C3×C7#100411yescharacteristic, complemented
364C3×C7#100911yescharacteristic, complemented
365C2×C32—142no—
366C2×C32C23×D711yescharacteristic
367C3⋊D3—22no—
368C3⋊D3—141no—
369C3×D3—21no—
370C3×D3—141no—
371C3×D3—142no—
372#41—91no—
373#41—211no—
374#41—631no—
375#41—632no—
376#41—631no—
377C2×D4—211no—
378C2×D4—632no—
379C2×D4—631no—
380C2×D4—632no—
381C2×D4—631no—
382C2×D4—632no—
383C2×C7—61no—
384C2×C7—182no—
385C2×C7C22×D3211yescharacteristic
386D7—22no—
387D7—62no—
388D7—61no—
389D7—181no—
390C22×C3—31no—
391C22×C3—72no—
392C22×C3—72no—
393C22×C3—142no—
394C22×C3—212no—
395C22×C3—211no—
396C22×C3—842no—
397C2×D3—32no—
398C2×D3—32no—
399C2×D3—62no—
400C2×D3—71no—
401C2×D3—72no—
402C2×D3—211no—
403C2×D3—211no—
404C2×D3—282no—
405C2×D3—421no—
406C2×D3—842no—
407C2×D3—842no—
408C2×D3—842no—
409C2×D3—842no—
410C2×D3—842no—
411C2×D3—842no—
412C2×D3C2×D3×D711yescharacteristic
413C3×C4—21no—
414C3×C4—31no—
415C3×C4—32no—
416C3×C4—211no—
417C3×C4C2×D3×D711yescharacteristic
418C3×C4C2×D3×D711yescharacteristic
419Dic3—71no—
420Dic3—211no—
421Dic3—211no—
422Dic3—421no—
423Dic3C2×D3×D711yescharacteristic
424Dic3C2×D3×D712yes—
425C32#655311yescharacteristic, complemented, Sylow
426C23—212no—
427C23—632no—
428C23—632no—
429Q8—31no—
430Q8—631no—
431D4—71no—
432D4—92no—
433D4—91no—
434D4—212no—
435D4—212no—
436D4—211no—
437D4—631no—
438D4—632no—
439D4—631no—
440D4—632no—
441D4—631no—
442D4—632no—
443C2×C4—31no—
444C2×C4—92no—
445C2×C4—212no—
446C2×C4—211no—
447C2×C4—632no—
448C2×C4—631no—
449C7#6431411yescharacteristic, complemented, Sylow
450C2×C3—21no—
451C2×C3—61no—
452C2×C3—142no—
453C2×C3—142no—
454C2×C3—282no—
455C2×C3—421no—
456C2×C3—422no—
457C2×C3C22×D3×D711yescharacteristic
458C2×C3C22×D3×D711yescharacteristic
459D3—21no—
460D3—62no—
461D3—62no—
462D3—122no—
463D3—142no—
464D3—141no—
465D3—421no—
466D3—421no—
467D3—421no—
468D3—421no—
469C22—31no—
470C22—72no—
471C22—92no—
472C22—211no—
473C22—212no—
474C22—631no—
475C22—842no—
476C22—2522no—
477C22—2522no—
478C4—31no—
479C4—32no—
480C4—211no—
481C4—631no—
482C4D32×D711yescharacteristic
483C3—21no—
484C3#1062156411yescharacteristic, complemented
485C3#1062158211yescharacteristic, complemented
486C2—61no—
487C2—142no—
488C2—182no—
489C2—422no—
490C2—421no—
491C2—1261no—
492C2C2×D32×D711yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D7
commutator, socle, cosocleC2×C32×C7C24
FittingC32×C4×C7C23
hypercenterC4D32×D7
hypocenterC32×C7H(5,𝔽2)
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #11759634C2semidirect
2. #11759607C2semidirect
3. #11759602C2semidirect
4. C7×#3799C2semidirect
5. #11759568C2semidirect
6. C3×D3×D28C2semidirect
7. C3×#66670C2semidirect
8. #11759439C2semidirect
9. #11759428C2semidirect
10. #11759425C2semidirect
11. D7×#626C2semidirect
14. C2×(C3⋊D3)⋊D7C22semidirect
15. C2×D7×C3⋊D3C22semidirect
16. C2×C3×D3×D7C22semidirect
17. C4×C3⋊D21C22semidirect
18. C7×C4⋊(C3⋊D3)C22semidirect
19. C3×D84C22semidirect
20. C3×C4×D21C22semidirect
21. C3×C4×D3×C7C22semidirect
22. C3×C7×Dic6C22semidirect
23. C32×D28C22semidirect
24. #92694C22semidirect
25. #92691C22semidirect
26. #92690C22semidirect
27. #92689C22semidirect
28. #92689C22semidirect
29. #92688C22semidirect
30. Dic3×D21C22semidirect
31. C7×#626C22semidirect
32. C7×#624C22semidirect
33. #92676C22semidirect
34. C3×#4246C22semidirect
35. C3×#4246C22semidirect
36. C3×Dic3×D7C22semidirect
52. D3×D28D3semidirect
58. #66670D3semidirect
82. C2×D3×D7C2×D3semidirect
89. D84C2×D3semidirect
94. C4×D21C2×D3semidirect
103. C4×D3×C7C2×D3semidirect
104. C7×Dic6C2×D3semidirect
109. C3×D28C2×D3semidirect
110. C3×D28C2×D3semidirect
118. #4246C2×D3semidirect
119. #4246C2×D3semidirect
126. Dic3×D7C2×D3semidirect
135. #3799D7semidirect
198. C4⋊(C3⋊D3)C2×D7semidirect
205. C3×C4×D3C2×D7semidirect
206. C3×Dic6C2×D7semidirect
212. #626C2×D7semidirect
214. #624C2×D7semidirect
216. C32×C7H(5,𝔽2)semidirect
234. D28D32semidirect
358. C4×D3D3×D7semidirect
361. Dic6D3×D7semidirect
363. C3×C7#1004semidirect
364. C3×C7#1009semidirect
425. C32#6553semidirect
449. C7#64314semidirect
484. C3#10621564semidirect
485. C3#10621582semidirect

Maximal subgroups

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

subgroupindex
#117596342
#117596072
#117596022
C7×#37992
#117595682
C3×D3×D282
C3×#666702
#117594392
#117594282
#117594252
D7×#6262
#106215823
#106215643
#643147