Finite Group Databasebeta

← C67×#20147 #30699398

Subgroups — C67×#20147 #30699398

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3397C211yesmaximal, characteristic
2C67×#3360C211yesmaximal, characteristic, complemented
3C67×#3356C211yesmaximal, characteristic, complemented
4C67×#3304C211yesmaximal, characteristic, complemented
5C67×#1714C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1701C211yesmaximal, characteristic, complemented
8C67×#1671C211yesmaximal, characteristic, complemented
9C67×#1666C211yesmaximal, characteristic, complemented
10C67×#1650C211yesmaximal, characteristic, complemented
11C67×#1650C211yesmaximal, characteristic, complemented
12C67×#1631C211yesmaximal, characteristic, complemented
13C67×#1622C211yesmaximal, characteristic, complemented
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1586C211yesmaximal, characteristic, complemented
16#556×C67C2211yescharacteristic
17#556×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19#550×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#496×C67—21no—
23#492×C67C2211yescharacteristic
24#492×C67C2211yescharacteristic
25#488×C67—21no—
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#485×C67—21no—
29#483×C67—21no—
30#482×C67—21no—
31#479×C67—21no—
32#477×C67C2211yescharacteristic
33#476×C67C2211yescharacteristic
34#475×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#470×C67—21no—
37#465×C67—21no—
38#463×C67—21no—
39#463×C67—21no—
40#461×C67C2211yescharacteristic
41#461×C67C2211yescharacteristic
42#460×C67—21no—
43#460×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#452×C67—21no—
46#446×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48#440×C67—21no—
49C4×Q16×C67C2211yescharacteristic
50C4×SD16×C67—21no—
51C4×D8×C67C2211yescharacteristic
52#436×C67C2211yescharacteristic
53C8×D4×C67C2211yescharacteristic
54#428×C67—21no—
55#416×C67—21no—
56#334×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#331×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×Q16×C67D411yescharacteristic
69C2×SD16×C67—21no—
70C2×SD16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—41no—
75C2×D8×C67—21no—
76C2×D8×C67D411yescharacteristic
77C2×M16×C67—21no—
78C22×C8×C67—21no—
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82#127×C67—21no—
83#127×C67C2311yescharacteristic
84#126×C67—41no—
85#126×C67—41no—
86#125×C67—21no—
87#124×C67—21no—
88#124×C67—21no—
89C22⋊Q8×C67—21no—
90C22⋊Q8×C67—21no—
91C22⋊Q8×C67—21no—
92C22⋊Q8×C67—41no—
93#121×C67—21no—
94#121×C67—41no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67C2311yescharacteristic
98C4×Q8×C67C2311yescharacteristic
99C4×D4×C67—21no—
100C4×D4×C67—21no—
101C4×D4×C67C2311yescharacteristic
102C4×D4×C67C2311yescharacteristic
103#117×C67—21no—
104#117×C67—21no—
105C2×C4⋊C4×C67—21no—
106#107×C67—21no—
107#107×C67—41no—
108#107×C67D411yescharacteristic
109#107×C67D411yescharacteristic
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—21no—
113#106×C67—21no—
114#106×C67—21no—
115C4⋊C8×C67—21no—
116C4⋊C8×C67—21no—
117C4⋊C8×C67C2311yescharacteristic
118C4⋊C8×C67C2311yescharacteristic
119C4⋊C8×C67C2311yescharacteristic
120C4⋊C8×C67C2311yescharacteristic
121#103×C67—21no—
122#103×C67—21no—
123#103×C67—21no—
124#103×C67—21no—
125#103×C67—21no—
126#103×C67—21no—
127#103×C67—21no—
128#103×C67D411yescharacteristic
129#103×C67D411yescharacteristic
130#102×C67—21no—
131#102×C67—21no—
132#102×C67—21no—
133#102×C67—21no—
134#102×C67—21no—
135#102×C67D411yescharacteristic
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138C22⋊C8×C67—21no—
139#97×C67—21no—
140C4×C8×C67—21no—
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#41×C67—42no—
145#41×C67—81no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67—21no—
149C2×Q8×C67C2×D411yescharacteristic
150C2×Q8×C67C2×D411yescharacteristic
151C2×D4×C67—21no—
152C2×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67—41no—
155C2×D4×C67—41no—
156C2×D4×C67C2×D411yescharacteristic
157C2×D4×C67C2×D411yescharacteristic
158C22×C4×C67—21no—
159C22×C4×C67—21no—
160C22×C4×C67—41no—
161C22×C4×C67—41no—
162Q16×C67—41no—
163SD16×C67—42no—
164SD16×C67—42no—
165SD16×C67—81no—
166SD16×C67—82no—
167SD16×C67—81no—
168SD16×C67—81no—
169D8×C67—41no—
170D8×C67—81no—
171M16×C67—42no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67—21no—
176C2×C8×C67—21no—
177C2×C8×C67—81no—
178C2×C8×C67C2×D411yescharacteristic
179C2×C8×C67C2×D411yescharacteristic
180C2×C8×C67C2×D411yescharacteristic
181C2×C8×C67C2×D411yescharacteristic
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67—41no—
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C4⋊C4×C67C2×D411yescharacteristic
198C22⋊C4×C67—21no—
199C22⋊C4×C67—21no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C22⋊C4×C67—41no—
204C22⋊C4×C67—41no—
205C22⋊C4×C67—41no—
206C42×C67—21no—
207C42×C67—21no—
208C42×C67—41no—
209C42×C67—41no—
210C42×C67C2411yescharacteristic
211C23×C67—21no—
212C23×C67—21no—
213C23×C67—41no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—81no—
223D4×C67—21no—
224D4×C67—21no—
225D4×C67—21no—
226D4×C67—21no—
227D4×C67—41no—
228D4×C67—41no—
229D4×C67—41no—
230D4×C67—41no—
231D4×C67—81no—
232D4×C67—82no—
233D4×C67—161no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—21no—
239C2×C4×C67—21no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—41no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—82no—
248C2×C4×C67—161no—
249C2×C4×C67C22×D411yescharacteristic
250C2×C4×C67C22×D411yescharacteristic
251C2×C4×C67H(5,𝔽2)11yescharacteristic
252C8×C67—22no—
253C8×C67—21no—
254C8×C67—21no—
255C8×C67—21no—
256C8×C67—21no—
257C8×C67—41no—
258C8×C67—41no—
259C8×C67—41no—
260C8×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—81no—
268C22×C67—81no—
269C22×C67—81no—
270C22×C67D4211yescharacteristic
271C4×C67—21no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—81no—
279C4×C67—81no—
280C4×C67—81no—
281C4×C67#57511yescharacteristic
282C4×C67#57511yescharacteristic
283C4×C67#57711yescharacteristic
284C4×C67#57711yescharacteristic
285#20147C6711yesmaximal, characteristic, direct product factor, Sylow
286C2×C67—81no—
287C2×C67—81no—
288C2×C67—161no—
289C2×C67#326711yescharacteristic
290C2×C67#327111yescharacteristic
291C2×C67#329111yescharacteristic
292#3397C2×C6711yescharacteristic
293#3360C2×C6711yescharacteristic, complemented
294#3356C2×C6711yescharacteristic, complemented
295#3304C2×C6711yescharacteristic, complemented
296#1714C2×C6711yescharacteristic, complemented
297#1704C2×C6711yescharacteristic, complemented
298#1701C2×C6711yescharacteristic, complemented
299#1671C2×C6711yescharacteristic, complemented
300#1666C2×C6711yescharacteristic, complemented
301#1650C2×C6711yescharacteristic, complemented
302#1650C2×C6711yescharacteristic, complemented
303#1631C2×C6711yescharacteristic, complemented
304#1622C2×C6711yescharacteristic, complemented
305#1587C2×C6711yescharacteristic, complemented
306#1586C2×C6711yescharacteristic, complemented
307C67#2014711yescharacteristic, direct product factor, Sylow
308#556C22×C6711yescharacteristic
309#556C22×C6711yescharacteristic
310#554C22×C6711yescharacteristic
311#550C22×C6711yescharacteristic
312#500C22×C6711yescharacteristic
313#498C22×C6711yescharacteristic
314#496—21no—
315#492C22×C6711yescharacteristic
316#492C22×C6711yescharacteristic
317#488—21no—
318#487C22×C6711yescharacteristic
319#486C22×C6711yescharacteristic
320#485—21no—
321#483—21no—
322#482—21no—
323#479—21no—
324#477C22×C6711yescharacteristic
325#476C22×C6711yescharacteristic
326#475C22×C6711yescharacteristic
327#475C22×C6711yescharacteristic
328#470—21no—
329#465—21no—
330#463—21no—
331#463—21no—
332#461C22×C6711yescharacteristic
333#461C22×C6711yescharacteristic
334#460—21no—
335#460C22×C6711yescharacteristic
336#459C22×C6711yescharacteristic
337#452—21no—
338#446C22×C6711yescharacteristic
339C8×Q8C22×C6711yescharacteristic
340#440—21no—
341C4×Q16C22×C6711yescharacteristic
342C4×SD16—21no—
343C4×D8C22×C6711yescharacteristic
344#436C22×C6711yescharacteristic
345C8×D4C22×C6711yescharacteristic
346#428—21no—
347#416—21no—
348#334C22×C6711yescharacteristic
349#334C22×C6711yescharacteristic
350#333C22×C6711yescharacteristic
351#332C22×C6711yescharacteristic
352#332C22×C6711yescharacteristic
353#331C22×C6711yescharacteristic
354Q8⋊C8C22×C6711yescharacteristic
355Q8⋊C8C22×C6711yescharacteristic
356D4⋊C8C22×C6711yescharacteristic
357D4⋊C8C22×C6711yescharacteristic
358#322C22×C6711yescharacteristic
359C2×#41—21no—
360C2×Q16D4×C6711yescharacteristic
361C2×SD16—21no—
362C2×SD16—21no—
363C2×SD16—21no—
364C2×SD16—21no—
365C2×SD16—21no—
366C2×SD16—41no—
367C2×D8—21no—
368C2×D8D4×C6711yescharacteristic
369C2×M16—21no—
370C22×C8—21no—
371C4⋊Q8C23×C6711yescharacteristic
372C4⋊Q8C23×C6711yescharacteristic
373C4⋊Q8C23×C6711yescharacteristic
374#127—21no—
375#127C23×C6711yescharacteristic
376#126—41no—
377#126—41no—
378#125—21no—
379#124—21no—
380#124—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—41no—
385#121—21no—
386#121—41no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8C23×C6711yescharacteristic
390C4×Q8C23×C6711yescharacteristic
391C4×D4—21no—
392C4×D4—21no—
393C4×D4C23×C6711yescharacteristic
394C4×D4C23×C6711yescharacteristic
395#117—21no—
396#117—21no—
397C2×C4⋊C4—21no—
398#107—21no—
399#107—41no—
400#107D4×C6711yescharacteristic
401#107D4×C6711yescharacteristic
402#106—21no—
403#106—21no—
404#106—21no—
405#106—21no—
406#106—21no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8C23×C6711yescharacteristic
410C4⋊C8C23×C6711yescharacteristic
411C4⋊C8C23×C6711yescharacteristic
412C4⋊C8C23×C6711yescharacteristic
413#103—21no—
414#103—21no—
415#103—21no—
416#103—21no—
417#103—21no—
418#103—21no—
419#103—21no—
420#103D4×C6711yescharacteristic
421#103D4×C6711yescharacteristic
422#102—21no—
423#102—21no—
424#102—21no—
425#102—21no—
426#102—21no—
427#102D4×C6711yescharacteristic
428#102D4×C6711yescharacteristic
429C22⋊C8—21no—
430C22⋊C8—21no—
431#97—21no—
432C4×C8—21no—
433C4×C8C23×C6711yescharacteristic
434C4×C8C23×C6711yescharacteristic
435C4×C8C23×C6711yescharacteristic
436#41—42no—
437#41—81no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×D4×C6711yescharacteristic
442C2×Q8C2×D4×C6711yescharacteristic
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4C2×D4×C6711yescharacteristic
449C2×D4C2×D4×C6711yescharacteristic
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454Q16—41no—
455SD16—42no—
456SD16—42no—
457SD16—81no—
458SD16—82no—
459SD16—81no—
460SD16—81no—
461D8—41no—
462D8—81no—
463M16—42no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—81no—
470C2×C8C2×D4×C6711yescharacteristic
471C2×C8C2×D4×C6711yescharacteristic
472C2×C8C2×D4×C6711yescharacteristic
473C2×C8C2×D4×C6711yescharacteristic
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4C2×D4×C6711yescharacteristic
487C4⋊C4C2×D4×C6711yescharacteristic
488C4⋊C4C2×D4×C6711yescharacteristic
489C4⋊C4C2×D4×C6711yescharacteristic
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C42—21no—
499C42—21no—
500C42—41no—
501C42—41no—
502C42C24×C6711yescharacteristic
503C23—21no—
504C23—21no—
505C23—41no—
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—21no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—81no—
515D4—21no—
516D4—21no—
517D4—21no—
518D4—21no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—81no—
524D4—82no—
525D4—161no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—161no—
541C2×C4C22×D4×C6711yescharacteristic
542C2×C4C22×D4×C6711yescharacteristic
543C2×C4H(5,𝔽2)×C6711yescharacteristic
544C8—22no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22D42×C6711yescharacteristic
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4#575×C6711yescharacteristic
574C4#575×C6711yescharacteristic
575C4#577×C6711yescharacteristic
576C4#577×C6711yescharacteristic
577C2—81no—
578C2—81no—
579C2—161no—
580C2C67×#326711yescharacteristic
581C2C67×#327111yescharacteristic
582C2C67×#329111yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3397C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3360C2semidirect
3. C67×#3356C2semidirect
4. C67×#3304C2semidirect
5. C67×#1714C2semidirect
6. C67×#1704C2semidirect
7. C67×#1701C2semidirect
8. C67×#1671C2semidirect
9. C67×#1666C2semidirect
10. C67×#1650C2semidirect
11. C67×#1650C2semidirect
12. C67×#1631C2semidirect
13. C67×#1622C2semidirect
14. C67×#1587C2semidirect
15. C67×#1586C2semidirect
285. #20147C67direct (N × H)
293. #3360C2×C67semidirect
294. #3356C2×C67semidirect
295. #3304C2×C67semidirect
296. #1714C2×C67semidirect
297. #1704C2×C67semidirect
298. #1701C2×C67semidirect
299. #1671C2×C67semidirect
300. #1666C2×C67semidirect
301. #1650C2×C67semidirect
302. #1650C2×C67semidirect
303. #1631C2×C67semidirect
304. #1622C2×C67semidirect
305. #1587C2×C67semidirect
306. #1586C2×C67semidirect
307. C67#20147direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33972
C67×#33602
C67×#33562
C67×#33042
C67×#17142
C67×#17042
C67×#17012
C67×#16712
C67×#16662
C67×#16502
C67×#16312
C67×#16222
C67×#15872
C67×#15862
#2014767