Finite Group Databasebeta

← C67×#20132 #30699383

Subgroups — C67×#20132 #30699383

Order 17152.

624 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3397C211yesmaximal, characteristic, complemented
2C67×#3340C211yesmaximal, characteristic, complemented
3C67×#3332C211yesmaximal, characteristic, complemented
4C67×#3303C211yesmaximal, characteristic, complemented
5C67×#1714C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1691C211yesmaximal, characteristic, complemented
8C67×#1681C211yesmaximal, characteristic, complemented
9C67×#1665C211yesmaximal, characteristic, complemented
10C67×#1650C211yesmaximal, characteristic, complemented
11C67×#1647C211yesmaximal, characteristic, complemented
12C67×#1631C211yesmaximal, characteristic, complemented
13C67×#1624C211yesmaximal, characteristic, complemented
14C67×#1586C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16#556×C67C2211yescharacteristic
17#555×C67C2211yescharacteristic
18#550×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#496×C67—21no—
23#492×C67C2211yescharacteristic
24#492×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#486×C67C2211yescharacteristic
27#485×C67—21no—
28#485×C67—21no—
29#482×C67—21no—
30#481×C67—21no—
31#477×C67C2211yescharacteristic
32#476×C67C2211yescharacteristic
33#475×C67C2211yescharacteristic
34#474×C67C2211yescharacteristic
35#469×C67—21no—
36#465×C67—21no—
37#465×C67—21no—
38#463×C67—21no—
39#461×C67C2211yescharacteristic
40#461×C67C2211yescharacteristic
41#460×C67—21no—
42#459×C67C2211yescharacteristic
43#459×C67C2211yescharacteristic
44#449×C67—21no—
45#449×C67—21no—
46#446×C67C2211yescharacteristic
47#440×C67—21no—
48C4×SD16×C67C2211yescharacteristic
49C4×D8×C67C2211yescharacteristic
50#436×C67C2211yescharacteristic
51C8×D4×C67C2211yescharacteristic
52C8×D4×C67C2211yescharacteristic
53#427×C67—21no—
54#417×C67—21no—
55#416×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#331×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63D4⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×#41×C67—21no—
69C2×SD16×C67—21no—
70C2×SD16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—41no—
74C2×SD16×C67D411yescharacteristic
75C2×D8×C67—21no—
76C2×D8×C67—21no—
77C2×D8×C67D411yescharacteristic
78C2×M16×C67—21no—
79C22×C8×C67—21no—
80C22×C8×C67—21no—
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84#127×C67—21no—
85#127×C67C2311yescharacteristic
86#126×C67—41no—
87#124×C67—21no—
88#124×C67—41no—
89#123×C67—41no—
90C22⋊Q8×C67—21no—
91C22⋊Q8×C67—21no—
92#121×C67—21no—
93#121×C67—21no—
94#121×C67—21no—
95#121×C67—21no—
96#121×C67—41no—
97C4×Q8×C67—21no—
98C4×Q8×C67C2311yescharacteristic
99C4×D4×C67—21no—
100C4×D4×C67C2311yescharacteristic
101C4×D4×C67C2311yescharacteristic
102C4×D4×C67C2311yescharacteristic
103#117×C67—21no—
104#117×C67—21no—
105#117×C67—21no—
106C2×C4⋊C4×C67—21no—
107#107×C67—21no—
108#107×C67—21no—
109#107×C67—41no—
110#107×C67D411yescharacteristic
111#106×C67—21no—
112#106×C67—21no—
113#106×C67—21no—
114#106×C67—21no—
115#106×C67D411yescharacteristic
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×C67D411yescharacteristic
128#102×C67—21no—
129#102×C67—21no—
130#102×C67—21no—
131#102×C67—21no—
132#102×C67—21no—
133#102×C67—21no—
134#102×C67D411yescharacteristic
135#102×C67D411yescharacteristic
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138C22⋊C8×C67—21no—
139C22⋊C8×C67—21no—
140#97×C67—21no—
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#41×C67—42no—
145#41×C67—42no—
146#41×C67—81no—
147#41×C67—81no—
148C2×Q8×C67—21no—
149C2×Q8×C67—21no—
150C2×Q8×C67—21no—
151C2×Q8×C67C2×D411yescharacteristic
152C2×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67—21no—
155C2×D4×C67—41no—
156C2×D4×C67—41no—
157C2×D4×C67—41no—
158C2×D4×C67—41no—
159C2×D4×C67C2×D411yescharacteristic
160C2×D4×C67C2×D411yescharacteristic
161C2×D4×C67C2×D411yescharacteristic
162C22×C4×C67—21no—
163C22×C4×C67—21no—
164C22×C4×C67—21no—
165C22×C4×C67—41no—
166C22×C4×C67—41no—
167SD16×C67—41no—
168SD16×C67—42no—
169SD16×C67—81no—
170SD16×C67—82no—
171SD16×C67—81no—
172SD16×C67—81no—
173D8×C67—41no—
174D8×C67—81no—
175D8×C67—81no—
176M16×C67—81no—
177C2×C8×C67—21no—
178C2×C8×C67—21no—
179C2×C8×C67—21no—
180C2×C8×C67—21no—
181C2×C8×C67—21no—
182C2×C8×C67—42no—
183C2×C8×C67—81no—
184C2×C8×C67C2×D411yescharacteristic
185C2×C8×C67C2×D411yescharacteristic
186C2×C8×C67C2×D411yescharacteristic
187C2×C8×C67C2×D411yescharacteristic
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—41no—
196C4⋊C4×C67—41no—
197C4⋊C4×C67C2×D411yescharacteristic
198C4⋊C4×C67C2×D411yescharacteristic
199C4⋊C4×C67C2×D411yescharacteristic
200C4⋊C4×C67C2×D411yescharacteristic
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C22⋊C4×C67—21no—
204C22⋊C4×C67—41no—
205C22⋊C4×C67—41no—
206C22⋊C4×C67—41no—
207C22⋊C4×C67—41no—
208C22⋊C4×C67—41no—
209C22⋊C4×C67—41no—
210C42×C67—21no—
211C42×C67—41no—
212C42×C67—41no—
213C42×C67C2411yescharacteristic
214C23×C67—21no—
215C23×C67—21no—
216C23×C67—21no—
217C23×C67—41no—
218Q8×C67—21no—
219Q8×C67—21no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—41no—
223Q8×C67—41no—
224Q8×C67—81no—
225D4×C67—21no—
226D4×C67—21no—
227D4×C67—21no—
228D4×C67—21no—
229D4×C67—21no—
230D4×C67—21no—
231D4×C67—41no—
232D4×C67—41no—
233D4×C67—41no—
234D4×C67—41no—
235D4×C67—81no—
236D4×C67—82no—
237D4×C67—82no—
238D4×C67—161no—
239D4×C67—161no—
240C2×C4×C67—21no—
241C2×C4×C67—21no—
242C2×C4×C67—21no—
243C2×C4×C67—21no—
244C2×C4×C67—21no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—41no—
250C2×C4×C67—41no—
251C2×C4×C67—41no—
252C2×C4×C67—41no—
253C2×C4×C67—41no—
254C2×C4×C67—82no—
255C2×C4×C67—82no—
256C2×C4×C67C22×D411yescharacteristic
257C2×C4×C67C22×D411yescharacteristic
258C2×C4×C67H(5,𝔽2)11yescharacteristic
259C8×C67—22no—
260C8×C67—21no—
261C8×C67—21no—
262C8×C67—21no—
263C8×C67—21no—
264C8×C67—41no—
265C8×C67—41no—
266C8×C67—41no—
267C8×C67—41no—
268C22×C67—41no—
269C22×C67—41no—
270C22×C67—41no—
271C22×C67—41no—
272C22×C67—41no—
273C22×C67—41no—
274C22×C67—41no—
275C22×C67—41no—
276C22×C67—41no—
277C22×C67—81no—
278C22×C67—81no—
279C22×C67—81no—
280C22×C67D4211yescharacteristic
281C4×C67—21no—
282C4×C67—41no—
283C4×C67—41no—
284C4×C67—41no—
285C4×C67—41no—
286C4×C67—41no—
287C4×C67—81no—
288C4×C67—81no—
289C4×C67—81no—
290C4×C67#57511yescharacteristic
291C4×C67#57511yescharacteristic
292C4×C67#57611yescharacteristic
293C4×C67#57711yescharacteristic
294#20132C6711yesmaximal, characteristic, direct product factor, Sylow
295C2×C67—81no—
296C2×C67—81no—
297C2×C67—81no—
298C2×C67—161no—
299C2×C67#326711yescharacteristic
300C2×C67#326811yescharacteristic
301C2×C67#329111yescharacteristic
302#3397C2×C6711yescharacteristic, complemented
303#3340C2×C6711yescharacteristic, complemented
304#3332C2×C6711yescharacteristic, complemented
305#3303C2×C6711yescharacteristic, complemented
306#1714C2×C6711yescharacteristic, complemented
307#1704C2×C6711yescharacteristic, complemented
308#1691C2×C6711yescharacteristic, complemented
309#1681C2×C6711yescharacteristic, complemented
310#1665C2×C6711yescharacteristic, complemented
311#1650C2×C6711yescharacteristic, complemented
312#1647C2×C6711yescharacteristic, complemented
313#1631C2×C6711yescharacteristic, complemented
314#1624C2×C6711yescharacteristic, complemented
315#1586C2×C6711yescharacteristic, complemented
316#1585C2×C6711yescharacteristic, complemented
317C67#2013211yescharacteristic, direct product factor, Sylow
318#556C22×C6711yescharacteristic
319#555C22×C6711yescharacteristic
320#550C22×C6711yescharacteristic
321#547C22×C6711yescharacteristic
322#500C22×C6711yescharacteristic
323#498C22×C6711yescharacteristic
324#496—21no—
325#492C22×C6711yescharacteristic
326#492C22×C6711yescharacteristic
327#487C22×C6711yescharacteristic
328#486C22×C6711yescharacteristic
329#485—21no—
330#485—21no—
331#482—21no—
332#481—21no—
333#477C22×C6711yescharacteristic
334#476C22×C6711yescharacteristic
335#475C22×C6711yescharacteristic
336#474C22×C6711yescharacteristic
337#469—21no—
338#465—21no—
339#465—21no—
340#463—21no—
341#461C22×C6711yescharacteristic
342#461C22×C6711yescharacteristic
343#460—21no—
344#459C22×C6711yescharacteristic
345#459C22×C6711yescharacteristic
346#449—21no—
347#449—21no—
348#446C22×C6711yescharacteristic
349#440—21no—
350C4×SD16C22×C6711yescharacteristic
351C4×D8C22×C6711yescharacteristic
352#436C22×C6711yescharacteristic
353C8×D4C22×C6711yescharacteristic
354C8×D4C22×C6711yescharacteristic
355#427—21no—
356#417—21no—
357#416—21no—
358#335C22×C6711yescharacteristic
359#334C22×C6711yescharacteristic
360#333C22×C6711yescharacteristic
361#332C22×C6711yescharacteristic
362#332C22×C6711yescharacteristic
363#331C22×C6711yescharacteristic
364Q8⋊C8C22×C6711yescharacteristic
365D4⋊C8C22×C6711yescharacteristic
366D4⋊C8C22×C6711yescharacteristic
367D4⋊C8C22×C6711yescharacteristic
368#322C22×C6711yescharacteristic
369C2×#41—21no—
370C2×#41—21no—
371C2×SD16—21no—
372C2×SD16—21no—
373C2×SD16—21no—
374C2×SD16—21no—
375C2×SD16—41no—
376C2×SD16D4×C6711yescharacteristic
377C2×D8—21no—
378C2×D8—21no—
379C2×D8D4×C6711yescharacteristic
380C2×M16—21no—
381C22×C8—21no—
382C22×C8—21no—
383C4⋊Q8C23×C6711yescharacteristic
384C4⋊Q8C23×C6711yescharacteristic
385C4⋊Q8C23×C6711yescharacteristic
386#127—21no—
387#127C23×C6711yescharacteristic
388#126—41no—
389#124—21no—
390#124—41no—
391#123—41no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#121—41no—
399C4×Q8—21no—
400C4×Q8C23×C6711yescharacteristic
401C4×D4—21no—
402C4×D4C23×C6711yescharacteristic
403C4×D4C23×C6711yescharacteristic
404C4×D4C23×C6711yescharacteristic
405#117—21no—
406#117—21no—
407#117—21no—
408C2×C4⋊C4—21no—
409#107—21no—
410#107—21no—
411#107—41no—
412#107D4×C6711yescharacteristic
413#106—21no—
414#106—21no—
415#106—21no—
416#106—21no—
417#106D4×C6711yescharacteristic
418C4⋊C8—21no—
419C4⋊C8C23×C6711yescharacteristic
420C4⋊C8C23×C6711yescharacteristic
421C4⋊C8C23×C6711yescharacteristic
422C4⋊C8C23×C6711yescharacteristic
423#103—21no—
424#103—21no—
425#103—21no—
426#103—21no—
427#103—21no—
428#103—21no—
429#103D4×C6711yescharacteristic
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102D4×C6711yescharacteristic
437#102D4×C6711yescharacteristic
438#102D4×C6711yescharacteristic
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442#97—21no—
443C4×C8C23×C6711yescharacteristic
444C4×C8C23×C6711yescharacteristic
445C4×C8C23×C6711yescharacteristic
446#41—42no—
447#41—42no—
448#41—81no—
449#41—81no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8C2×D4×C6711yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4C2×D4×C6711yescharacteristic
462C2×D4C2×D4×C6711yescharacteristic
463C2×D4C2×D4×C6711yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—41no—
469SD16—41no—
470SD16—42no—
471SD16—81no—
472SD16—82no—
473SD16—81no—
474SD16—81no—
475D8—41no—
476D8—81no—
477D8—81no—
478M16—81no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—81no—
486C2×C8C2×D4×C6711yescharacteristic
487C2×C8C2×D4×C6711yescharacteristic
488C2×C8C2×D4×C6711yescharacteristic
489C2×C8C2×D4×C6711yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C2×D4×C6711yescharacteristic
500C4⋊C4C2×D4×C6711yescharacteristic
501C4⋊C4C2×D4×C6711yescharacteristic
502C4⋊C4C2×D4×C6711yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C42—21no—
513C42—41no—
514C42—41no—
515C42C24×C6711yescharacteristic
516C23—21no—
517C23—21no—
518C23—21no—
519C23—41no—
520Q8—21no—
521Q8—21no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—81no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—81no—
538D4—82no—
539D4—82no—
540D4—161no—
541D4—161no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4C22×D4×C6711yescharacteristic
559C2×C4C22×D4×C6711yescharacteristic
560C2×C4H(5,𝔽2)×C6711yescharacteristic
561C8—22no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22D42×C6711yescharacteristic
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4#575×C6711yescharacteristic
593C4#575×C6711yescharacteristic
594C4#576×C6711yescharacteristic
595C4#577×C6711yescharacteristic
596C2—81no—
597C2—81no—
598C2—81no—
599C2—161no—
600C2C67×#326711yescharacteristic
601C2C67×#326811yescharacteristic
602C2C67×#329111yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
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×#3397C2semidirect
2. C67×#3340C2semidirect
3. C67×#3332C2semidirect
4. C67×#3303C2semidirect
5. C67×#1714C2semidirect
6. C67×#1704C2semidirect
7. C67×#1691C2semidirect
8. C67×#1681C2semidirect
9. C67×#1665C2semidirect
10. C67×#1650C2semidirect
11. C67×#1647C2semidirect
12. C67×#1631C2semidirect
13. C67×#1624C2semidirect
14. C67×#1586C2semidirect
15. C67×#1585C2semidirect
294. #20132C67direct (N × H)
302. #3397C2×C67semidirect
303. #3340C2×C67semidirect
304. #3332C2×C67semidirect
305. #3303C2×C67semidirect
306. #1714C2×C67semidirect
307. #1704C2×C67semidirect
308. #1691C2×C67semidirect
309. #1681C2×C67semidirect
310. #1665C2×C67semidirect
311. #1650C2×C67semidirect
312. #1647C2×C67semidirect
313. #1631C2×C67semidirect
314. #1624C2×C67semidirect
315. #1586C2×C67semidirect
316. #1585C2×C67semidirect
317. C67#20132direct (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×#33402
C67×#33322
C67×#33032
C67×#17142
C67×#17042
C67×#16912
C67×#16812
C67×#16652
C67×#16502
C67×#16472
C67×#16312
C67×#16242
C67×#15862
C67×#15852
#2013267