Finite Group Databasebeta

← C67×#19879 #30699130

Subgroups — C67×#19879 #30699130

Order 17152.

604 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3394C211yesmaximal, characteristic, complemented
2C67×#3394C211yesmaximal, characteristic, complemented
3C67×#3353C211yesmaximal, characteristic, complemented
4C67×#3302C211yesmaximal, characteristic, complemented
5C67×#1714C211yesmaximal, characteristic, complemented
6C67×#1685C211yesmaximal, characteristic, complemented
7C67×#1680C211yesmaximal, characteristic, complemented
8C67×#1674C211yesmaximal, characteristic, complemented
9C67×#1665C211yesmaximal, characteristic, complemented
10C67×#1652C211yesmaximal, characteristic, complemented
11C67×#1652C211yesmaximal, characteristic, complemented
12C67×#1640C211yesmaximal, characteristic, complemented
13C67×#1625C211yesmaximal, characteristic
14C67×#1583C211yesmaximal, characteristic, complemented
15C67×SD16⋊C8C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17#556×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#495×C67—21no—
23#495×C67—21no—
24#492×C67C2211yescharacteristic
25#492×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#485×C67—21no—
29#484×C67—21no—
30#482×C67—21no—
31#475×C67C2211yescharacteristic
32#474×C67C2211yescharacteristic
33#474×C67C2211yescharacteristic
34#468×C67—21no—
35#467×C67—21no—
36#464×C67—21no—
37#464×C67—21no—
38#463×C67—21no—
39#463×C67—21no—
40#462×C67C2211yescharacteristic
41#462×C67C2211yescharacteristic
42#462×C67C2211yescharacteristic
43#459×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#452×C67—21no—
46C8×Q8×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48C4×SD16×C67—21no—
49C4×SD16×C67—21no—
50C4×SD16×C67C2211yescharacteristic
51C4×SD16×C67C2211yescharacteristic
52#436×C67C2211yescharacteristic
53C8×D4×C67C2211yescharacteristic
54#428×C67—21no—
55#416×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#332×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×C67—21no—
69C2×Q16×C67—21no—
70C2×Q16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—21no—
75C2×SD16×C67D411yescharacteristic
76C2×SD16×C67D411yescharacteristic
77C2×D8×C67—21no—
78C2×D8×C67—21no—
79C2×M16×C67—21no—
80C22×C8×C67—21no—
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84#127×C67C2311yescharacteristic
85#126×C67—41no—
86#126×C67—41no—
87#124×C67—21no—
88#124×C67—21no—
89#124×C67—21no—
90#124×C67—21no—
91#123×C67—41no—
92#123×C67—41no—
93C22⋊Q8×C67—21no—
94C22⋊Q8×C67—21no—
95C22⋊Q8×C67—21no—
96#121×C67—21no—
97C4×Q8×C67—21no—
98C4×Q8×C67—21no—
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×D4×C67—21no—
102C4×D4×C67—21no—
103C4×D4×C67C2311yescharacteristic
104C4×D4×C67C2311yescharacteristic
105#117×C67—21no—
106#117×C67—21no—
107C2×C4⋊C4×C67—21no—
108#107×C67—21no—
109#107×C67—21no—
110#107×C67—21no—
111#107×C67—21no—
112#106×C67—21no—
113#106×C67—41no—
114#106×C67D411yescharacteristic
115#106×C67D411yescharacteristic
116C4⋊C8×C67—21no—
117C4⋊C8×C67—21no—
118C4⋊C8×C67C2311yescharacteristic
119C4⋊C8×C67C2311yescharacteristic
120C4⋊C8×C67C2311yescharacteristic
121C4⋊C8×C67C2311yescharacteristic
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—41no—
135#102×C67D411yescharacteristic
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138C22⋊C8×C67—21no—
139C4×C8×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×C67C2×D411yescharacteristic
156C2×D4×C67C2×D411yescharacteristic
157C22×C4×C67—21no—
158C22×C4×C67—21no—
159C22×C4×C67—41no—
160C22×C4×C67—41no—
161Q16×C67—81no—
162Q16×C67—81no—
163Q16×C67—81no—
164SD16×C67—22no—
165SD16×C67—41no—
166SD16×C67—42no—
167SD16×C67—42no—
168SD16×C67—81no—
169SD16×C67—81no—
170D8×C67—81no—
171D8×C67—81no—
172M16×C67—42no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67—21no—
176C2×C8×C67—21no—
177C2×C8×C67—21no—
178C2×C8×C67—81no—
179C2×C8×C67C2×D411yescharacteristic
180C2×C8×C67C2×D411yescharacteristic
181C2×C8×C67C2×D411yescharacteristic
182C2×C8×C67C2×D411yescharacteristic
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—21no—
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—
206C22⋊C4×C67—41no—
207C22⋊C4×C67—41no—
208C42×C67—21no—
209C42×C67—21no—
210C42×C67—41no—
211C42×C67—41no—
212C42×C67C2411yescharacteristic
213C23×C67—21no—
214C23×C67—21no—
215C23×C67—41no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—21no—
219Q8×C67—21no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—41no—
223Q8×C67—41no—
224Q8×C67—41no—
225Q8×C67—41no—
226D4×C67—21no—
227D4×C67—21no—
228D4×C67—21no—
229D4×C67—21no—
230D4×C67—41no—
231D4×C67—41no—
232D4×C67—41no—
233D4×C67—41no—
234D4×C67—81no—
235D4×C67—82no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—21no—
239C2×C4×C67—21no—
240C2×C4×C67—21no—
241C2×C4×C67—21no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—41no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—82no—
250C2×C4×C67—161no—
251C2×C4×C67C22×D411yescharacteristic
252C2×C4×C67C22×D411yescharacteristic
253C2×C4×C67H(5,𝔽2)11yescharacteristic
254C8×C67—21no—
255C8×C67—21no—
256C8×C67—21no—
257C8×C67—41no—
258C8×C67—41no—
259C8×C67—41no—
260C8×C67—41no—
261C8×C67—41no—
262C8×C67#13512yes—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67—81no—
270C22×C67—81no—
271C22×C67—81no—
272C22×C67D4211yescharacteristic
273C4×C67—21no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67—81no—
281C4×C67—81no—
282C4×C67—81no—
283C4×C67C2×#13511yescharacteristic
284C4×C67#57511yescharacteristic
285C4×C67#57611yescharacteristic
286C4×C67#57711yescharacteristic
287#19879C6711yesmaximal, characteristic, direct product factor, Sylow
288C2×C67—81no—
289C2×C67—81no—
290C2×C67—161no—
291C2×C67#326811yescharacteristic
292C2×C67#328411yescharacteristic
293C2×C67#328911yescharacteristic
294#3394C2×C6711yescharacteristic, complemented
295#3394C2×C6711yescharacteristic, complemented
296#3353C2×C6711yescharacteristic, complemented
297#3302C2×C6711yescharacteristic, complemented
298#1714C2×C6711yescharacteristic, complemented
299#1685C2×C6711yescharacteristic, complemented
300#1680C2×C6711yescharacteristic, complemented
301#1674C2×C6711yescharacteristic, complemented
302#1665C2×C6711yescharacteristic, complemented
303#1652C2×C6711yescharacteristic, complemented
304#1652C2×C6711yescharacteristic, complemented
305#1640C2×C6711yescharacteristic, complemented
306#1625C2×C6711yescharacteristic
307#1583C2×C6711yescharacteristic, complemented
308SD16⋊C8C2×C6711yescharacteristic, complemented
309C67#1987911yescharacteristic, direct product factor, Sylow
310#559C22×C6711yescharacteristic
311#556C22×C6711yescharacteristic
312#556C22×C6711yescharacteristic
313#547C22×C6711yescharacteristic
314#500C22×C6711yescharacteristic
315#498C22×C6711yescharacteristic
316#495—21no—
317#495—21no—
318#492C22×C6711yescharacteristic
319#492C22×C6711yescharacteristic
320#487C22×C6711yescharacteristic
321#486C22×C6711yescharacteristic
322#485—21no—
323#484—21no—
324#482—21no—
325#475C22×C6711yescharacteristic
326#474C22×C6711yescharacteristic
327#474C22×C6711yescharacteristic
328#468—21no—
329#467—21no—
330#464—21no—
331#464—21no—
332#463—21no—
333#463—21no—
334#462C22×C6711yescharacteristic
335#462C22×C6711yescharacteristic
336#462C22×C6711yescharacteristic
337#459C22×C6711yescharacteristic
338#459C22×C6711yescharacteristic
339#452—21no—
340C8×Q8C22×C6711yescharacteristic
341C8×Q8C22×C6711yescharacteristic
342C4×SD16—21no—
343C4×SD16—21no—
344C4×SD16C22×C6711yescharacteristic
345C4×SD16C22×C6711yescharacteristic
346#436C22×C6711yescharacteristic
347C8×D4C22×C6711yescharacteristic
348#428—21no—
349#416—21no—
350#335C22×C6711yescharacteristic
351#334C22×C6711yescharacteristic
352#332C22×C6711yescharacteristic
353#332C22×C6711yescharacteristic
354#332C22×C6711yescharacteristic
355#331C22×C6711yescharacteristic
356Q8⋊C8C22×C6711yescharacteristic
357Q8⋊C8C22×C6711yescharacteristic
358D4⋊C8C22×C6711yescharacteristic
359D4⋊C8C22×C6711yescharacteristic
360#322C22×C6711yescharacteristic
361C2×#41—21no—
362C2×Q16—21no—
363C2×Q16—21no—
364C2×Q16—21no—
365C2×SD16—21no—
366C2×SD16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16D4×C6711yescharacteristic
370C2×SD16D4×C6711yescharacteristic
371C2×D8—21no—
372C2×D8—21no—
373C2×M16—21no—
374C22×C8—21no—
375C4⋊Q8C23×C6711yescharacteristic
376C4⋊Q8C23×C6711yescharacteristic
377C4⋊Q8C23×C6711yescharacteristic
378#127C23×C6711yescharacteristic
379#126—41no—
380#126—41no—
381#124—21no—
382#124—21no—
383#124—21no—
384#124—21no—
385#123—41no—
386#123—41no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390#121—21no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8C23×C6711yescharacteristic
394C4×Q8C23×C6711yescharacteristic
395C4×D4—21no—
396C4×D4—21no—
397C4×D4C23×C6711yescharacteristic
398C4×D4C23×C6711yescharacteristic
399#117—21no—
400#117—21no—
401C2×C4⋊C4—21no—
402#107—21no—
403#107—21no—
404#107—21no—
405#107—21no—
406#106—21no—
407#106—41no—
408#106D4×C6711yescharacteristic
409#106D4×C6711yescharacteristic
410C4⋊C8—21no—
411C4⋊C8—21no—
412C4⋊C8C23×C6711yescharacteristic
413C4⋊C8C23×C6711yescharacteristic
414C4⋊C8C23×C6711yescharacteristic
415C4⋊C8C23×C6711yescharacteristic
416#103—21no—
417#103—21no—
418#103—21no—
419#103—21no—
420#103—21no—
421#103—21no—
422#103D4×C6711yescharacteristic
423#103D4×C6711yescharacteristic
424#102—21no—
425#102—21no—
426#102—21no—
427#102—21no—
428#102—41no—
429#102D4×C6711yescharacteristic
430#102D4×C6711yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—21no—
433C4×C8—21no—
434C4×C8—21no—
435C4×C8C23×C6711yescharacteristic
436C4×C8C23×C6711yescharacteristic
437C4×C8C23×C6711yescharacteristic
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×D4×C6711yescharacteristic
444C2×Q8C2×D4×C6711yescharacteristic
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—41no—
449C2×D4C2×D4×C6711yescharacteristic
450C2×D4C2×D4×C6711yescharacteristic
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455Q16—81no—
456Q16—81no—
457Q16—81no—
458SD16—22no—
459SD16—41no—
460SD16—42no—
461SD16—42no—
462SD16—81no—
463SD16—81no—
464D8—81no—
465D8—81no—
466M16—42no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—81no—
473C2×C8C2×D4×C6711yescharacteristic
474C2×C8C2×D4×C6711yescharacteristic
475C2×C8C2×D4×C6711yescharacteristic
476C2×C8C2×D4×C6711yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4C2×D4×C6711yescharacteristic
489C4⋊C4C2×D4×C6711yescharacteristic
490C4⋊C4C2×D4×C6711yescharacteristic
491C4⋊C4C2×D4×C6711yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42—41no—
506C42C24×C6711yescharacteristic
507C23—21no—
508C23—21no—
509C23—41no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520D4—21no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—81no—
529D4—82no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—82no—
544C2×C4—161no—
545C2×C4C22×D4×C6711yescharacteristic
546C2×C4C22×D4×C6711yescharacteristic
547C2×C4H(5,𝔽2)×C6711yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8#135×C6712yes—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22D42×C6711yescharacteristic
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4C2×#135×C6711yescharacteristic
578C4#575×C6711yescharacteristic
579C4#576×C6711yescharacteristic
580C4#577×C6711yescharacteristic
581C2—81no—
582C2—81no—
583C2—161no—
584C2C67×#326811yescharacteristic
585C2C67×#328411yescharacteristic
586C2C67×#328911yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#1625C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3394C2semidirect
2. C67×#3394C2semidirect
3. C67×#3353C2semidirect
4. C67×#3302C2semidirect
5. C67×#1714C2semidirect
6. C67×#1685C2semidirect
7. C67×#1680C2semidirect
8. C67×#1674C2semidirect
9. C67×#1665C2semidirect
10. C67×#1652C2semidirect
11. C67×#1652C2semidirect
12. C67×#1640C2semidirect
14. C67×#1583C2semidirect
15. C67×SD16⋊C8C2semidirect
287. #19879C67direct (N × H)
294. #3394C2×C67semidirect
295. #3394C2×C67semidirect
296. #3353C2×C67semidirect
297. #3302C2×C67semidirect
298. #1714C2×C67semidirect
299. #1685C2×C67semidirect
300. #1680C2×C67semidirect
301. #1674C2×C67semidirect
302. #1665C2×C67semidirect
303. #1652C2×C67semidirect
304. #1652C2×C67semidirect
305. #1640C2×C67semidirect
307. #1583C2×C67semidirect
308. SD16⋊C8C2×C67semidirect
309. C67#19879direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33942
C67×#33532
C67×#33022
C67×#17142
C67×#16852
C67×#16802
C67×#16742
C67×#16652
C67×#16522
C67×#16402
C67×#16252
C67×#15832
C67×SD16⋊C82
#1987967