Finite Group Databasebeta

← C67×#19914 #30699165

Subgroups — C67×#19914 #30699165

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3367C211yesmaximal, characteristic, complemented
2C67×#3330C211yesmaximal, characteristic
3C67×#3302C211yesmaximal, characteristic, complemented
4C67×#3298C211yesmaximal, characteristic, complemented
5C67×#1710C211yesmaximal, characteristic, complemented
6C67×#1700C211yesmaximal, characteristic, complemented
7C67×#1699C211yesmaximal, characteristic, complemented
8C67×#1679C211yesmaximal, characteristic, complemented
9C67×#1679C211yesmaximal, characteristic, complemented
10C67×#1650C211yesmaximal, characteristic, complemented
11C67×#1649C211yesmaximal, characteristic, complemented
12C67×#1647C211yesmaximal, characteristic, complemented
13C67×#1647C211yesmaximal, characteristic, complemented
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×D8×C67C211yesmaximal, characteristic, complemented
16#556×C67C2211yescharacteristic
17D4×Q8×C67C2211yescharacteristic
18#547×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#500×C67C2211yescharacteristic
22#489×C67—21no—
23#487×C67C2211yescharacteristic
24#487×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#485×C67—21no—
28#484×C67—21no—
29#483×C67—21no—
30#477×C67—21no—
31#475×C67C2211yescharacteristic
32#474×C67C2211yescharacteristic
33#474×C67C2211yescharacteristic
34#474×C67C2211yescharacteristic
35#468×C67—21no—
36#467×C67—21no—
37#465×C67—21no—
38#463×C67—21no—
39#462×C67C2211yescharacteristic
40#461×C67C2211yescharacteristic
41#461×C67C2211yescharacteristic
42#461×C67C2211yescharacteristic
43#452×C67—21no—
44#449×C67—21no—
45#448×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#417×C67—21no—
54#416×C67—21no—
55C2×#103×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#332×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63D4⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66C82×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×#41×C67—21no—
69C22×Q8×C67—21no—
70C2×Q16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67D411yescharacteristic
75C2×D8×C67D411yescharacteristic
76C2×M16×C67—21no—
77C22×C8×C67—21no—
78C22×C8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84#126×C67—41no—
85#124×C67—21no—
86#123×C67—41no—
87#123×C67—41no—
88C22⋊Q8×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—21no—
95C4×Q8×C67—21no—
96C4×Q8×C67C2311yescharacteristic
97C4×D4×C67—21no—
98C4×D4×C67C2311yescharacteristic
99C4×D4×C67C2311yescharacteristic
100C4×D4×C67C2311yescharacteristic
101#117×C67—21no—
102C2×C4⋊C4×C67—21no—
103C2×C4⋊C4×C67—21no—
104#107×C67—21no—
105#107×C67—21no—
106#107×C67—21no—
107#107×C67—41no—
108#107×C67—41no—
109#107×C67D411yescharacteristic
110#106×C67—21no—
111#106×C67D411yescharacteristic
112C4⋊C8×C67—21no—
113C4⋊C8×C67C2311yescharacteristic
114C4⋊C8×C67C2311yescharacteristic
115C4⋊C8×C67C2311yescharacteristic
116C4⋊C8×C67C2311yescharacteristic
117#103×C67—21no—
118#103×C67—21no—
119#103×C67—21no—
120#103×C67—21no—
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×C67—21no—
129#103×C67—21no—
130#103×C67D411yescharacteristic
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×C67—21no—
152C2×Q8×C67—41no—
153C2×Q8×C67—42no—
154C2×Q8×C67—81no—
155C2×Q8×C67C2×D411yescharacteristic
156C2×D4×C67—41no—
157C2×D4×C67—41no—
158C2×D4×C67C2×D411yescharacteristic
159C2×D4×C67C2×D411yescharacteristic
160C2×D4×C67C2×D411yescharacteristic
161C22×C4×C67—21no—
162C22×C4×C67—21no—
163C22×C4×C67—21no—
164C22×C4×C67—41no—
165C22×C4×C67—41no—
166C22×C4×C67—41no—
167Q16×C67—81no—
168SD16×C67—41no—
169SD16×C67—42no—
170SD16×C67—81no—
171SD16×C67—81no—
172D8×C67—22no—
173M16×C67—81no—
174C2×C8×C67—21no—
175C2×C8×C67—21no—
176C2×C8×C67—21no—
177C2×C8×C67—21no—
178C2×C8×C67—21no—
179C2×C8×C67—81no—
180C2×C8×C67—81no—
181C2×C8×C67C2×D411yescharacteristic
182C2×C8×C67C2×D411yescharacteristic
183C2×C8×C67C2×D411yescharacteristic
184C2×C8×C67C2×D411yescharacteristic
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—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—41no—
195C4⋊C4×C67—41no—
196C4⋊C4×C67—41no—
197C4⋊C4×C67—41no—
198C4⋊C4×C67C2×D411yescharacteristic
199C4⋊C4×C67C2×D411yescharacteristic
200C4⋊C4×C67C2×D411yescharacteristic
201C4⋊C4×C67C2×D411yescharacteristic
202C22⋊C4×C67—21no—
203C22⋊C4×C67—21no—
204C22⋊C4×C67—21no—
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×C67C2411yescharacteristic
213C23×C67—21no—
214C23×C67—21no—
215C23×C67—21no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—41no—
223Q8×C67—41no—
224Q8×C67—41no—
225Q8×C67—41no—
226Q8×C67—82no—
227D4×C67—21no—
228D4×C67—21no—
229D4×C67—21no—
230D4×C67—21no—
231D4×C67—21no—
232D4×C67—82no—
233D4×C67—82no—
234D4×C67#13512yes—
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—41no—
248C2×C4×C67—41no—
249C2×C4×C67—41no—
250C2×C4×C67—82no—
251C2×C4×C67—82no—
252C2×C4×C67—82no—
253C2×C4×C67C22×D411yescharacteristic
254C2×C4×C67C22×D411yescharacteristic
255C2×C4×C67H(5,𝔽2)11yescharacteristic
256C8×C67—22no—
257C8×C67—21no—
258C8×C67—21no—
259C8×C67—21no—
260C8×C67—21no—
261C8×C67—41no—
262C8×C67—41no—
263C8×C67—41no—
264C8×C67—41no—
265C22×C67—22no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67—41no—
270C22×C67—41no—
271C22×C67—41no—
272C22×C67—41no—
273C22×C67—41no—
274C22×C67D4211yescharacteristic
275C4×C67—21no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67—41no—
281C4×C67—81no—
282C4×C67—81no—
283C4×C67—81no—
284C4×C67—81no—
285C4×C67C2×#13511yescharacteristic
286C4×C67#57511yescharacteristic
287C4×C67#57611yescharacteristic
288C4×C67#57711yescharacteristic
289#19914C6711yesmaximal, characteristic, direct product factor, Sylow
290C2×C67—42no—
291C2×C67—81no—
292C2×C67—81no—
293C2×C67#327311yescharacteristic
294C2×C67#328211yescharacteristic
295C2×C67#329311yescharacteristic
296#3367C2×C6711yescharacteristic, complemented
297#3330C2×C6711yescharacteristic
298#3302C2×C6711yescharacteristic, complemented
299#3298C2×C6711yescharacteristic, complemented
300#1710C2×C6711yescharacteristic, complemented
301#1700C2×C6711yescharacteristic, complemented
302#1699C2×C6711yescharacteristic, complemented
303#1679C2×C6711yescharacteristic, complemented
304#1679C2×C6711yescharacteristic, complemented
305#1650C2×C6711yescharacteristic, complemented
306#1649C2×C6711yescharacteristic, complemented
307#1647C2×C6711yescharacteristic, complemented
308#1647C2×C6711yescharacteristic, complemented
309#1590C2×C6711yescharacteristic, complemented
310C8×D8C2×C6711yescharacteristic, complemented
311C67#1991411yescharacteristic, direct product factor, Sylow
312#556C22×C6711yescharacteristic
313D4×Q8C22×C6711yescharacteristic
314#547C22×C6711yescharacteristic
315#547C22×C6711yescharacteristic
316#500C22×C6711yescharacteristic
317#500C22×C6711yescharacteristic
318#489—21no—
319#487C22×C6711yescharacteristic
320#487C22×C6711yescharacteristic
321#487C22×C6711yescharacteristic
322#487C22×C6711yescharacteristic
323#485—21no—
324#484—21no—
325#483—21no—
326#477—21no—
327#475C22×C6711yescharacteristic
328#474C22×C6711yescharacteristic
329#474C22×C6711yescharacteristic
330#474C22×C6711yescharacteristic
331#468—21no—
332#467—21no—
333#465—21no—
334#463—21no—
335#462C22×C6711yescharacteristic
336#461C22×C6711yescharacteristic
337#461C22×C6711yescharacteristic
338#461C22×C6711yescharacteristic
339#452—21no—
340#449—21no—
341#448—21no—
342#446C22×C6711yescharacteristic
343#440—21no—
344C4×SD16C22×C6711yescharacteristic
345C4×D8C22×C6711yescharacteristic
346#436C22×C6711yescharacteristic
347C8×D4C22×C6711yescharacteristic
348C8×D4C22×C6711yescharacteristic
349#417—21no—
350#416—21no—
351C2×#103—21no—
352#335C22×C6711yescharacteristic
353#334C22×C6711yescharacteristic
354#332C22×C6711yescharacteristic
355#332C22×C6711yescharacteristic
356#332C22×C6711yescharacteristic
357#332C22×C6711yescharacteristic
358Q8⋊C8C22×C6711yescharacteristic
359D4⋊C8C22×C6711yescharacteristic
360D4⋊C8C22×C6711yescharacteristic
361D4⋊C8C22×C6711yescharacteristic
362C82C22×C6711yescharacteristic
363C2×#41—21no—
364C2×#41—21no—
365C22×Q8—21no—
366C2×Q16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16D4×C6711yescharacteristic
371C2×D8D4×C6711yescharacteristic
372C2×M16—21no—
373C22×C8—21no—
374C22×C8—21no—
375C4⋊Q8—21no—
376C4⋊Q8C23×C6711yescharacteristic
377C4⋊Q8C23×C6711yescharacteristic
378C4⋊Q8C23×C6711yescharacteristic
379C4⋊Q8C23×C6711yescharacteristic
380#126—41no—
381#124—21no—
382#123—41no—
383#123—41no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—41no—
389#121—21no—
390#121—21no—
391C4×Q8—21no—
392C4×Q8C23×C6711yescharacteristic
393C4×D4—21no—
394C4×D4C23×C6711yescharacteristic
395C4×D4C23×C6711yescharacteristic
396C4×D4C23×C6711yescharacteristic
397#117—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400#107—21no—
401#107—21no—
402#107—21no—
403#107—41no—
404#107—41no—
405#107D4×C6711yescharacteristic
406#106—21no—
407#106D4×C6711yescharacteristic
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#103—21no—
421#103—21no—
422#103—21no—
423#103—21no—
424#103—21no—
425#103—21no—
426#103D4×C6711yescharacteristic
427#102—21no—
428#102—21no—
429#102—21no—
430#102D4×C6711yescharacteristic
431#102D4×C6711yescharacteristic
432#102D4×C6711yescharacteristic
433C22⋊C8—21no—
434C22⋊C8—21no—
435C22⋊C8—21no—
436#97—21no—
437C4×C8C23×C6711yescharacteristic
438C4×C8C23×C6711yescharacteristic
439C4×C8C23×C6711yescharacteristic
440#41—42no—
441#41—42no—
442#41—81no—
443#41—81no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—41no—
449C2×Q8—42no—
450C2×Q8—81no—
451C2×Q8C2×D4×C6711yescharacteristic
452C2×D4—41no—
453C2×D4—41no—
454C2×D4C2×D4×C6711yescharacteristic
455C2×D4C2×D4×C6711yescharacteristic
456C2×D4C2×D4×C6711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463Q16—81no—
464SD16—41no—
465SD16—42no—
466SD16—81no—
467SD16—81no—
468D8—22no—
469M16—81no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—81no—
477C2×C8C2×D4×C6711yescharacteristic
478C2×C8C2×D4×C6711yescharacteristic
479C2×C8C2×D4×C6711yescharacteristic
480C2×C8C2×D4×C6711yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4C2×D4×C6711yescharacteristic
495C4⋊C4C2×D4×C6711yescharacteristic
496C4⋊C4C2×D4×C6711yescharacteristic
497C4⋊C4C2×D4×C6711yescharacteristic
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C42—21no—
507C42—41no—
508C42C24×C6711yescharacteristic
509C23—21no—
510C23—21no—
511C23—21no—
512Q8—21no—
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—82no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—21no—
528D4—82no—
529D4—82no—
530D4#135×C6712yes—
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—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4C22×D4×C6711yescharacteristic
550C2×C4C22×D4×C6711yescharacteristic
551C2×C4H(5,𝔽2)×C6711yescharacteristic
552C8—22no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—21no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22D42×C6711yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4C2×#135×C6711yescharacteristic
582C4#575×C6711yescharacteristic
583C4#576×C6711yescharacteristic
584C4#577×C6711yescharacteristic
585C2—42no—
586C2—81no—
587C2—81no—
588C2C67×#327311yescharacteristic
589C2C67×#328211yescharacteristic
590C2C67×#329311yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3330C2
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×#3367C2semidirect
3. C67×#3302C2semidirect
4. C67×#3298C2semidirect
5. C67×#1710C2semidirect
6. C67×#1700C2semidirect
7. C67×#1699C2semidirect
8. C67×#1679C2semidirect
9. C67×#1679C2semidirect
10. C67×#1650C2semidirect
11. C67×#1649C2semidirect
12. C67×#1647C2semidirect
13. C67×#1647C2semidirect
14. C67×#1590C2semidirect
15. C8×D8×C67C2semidirect
289. #19914C67direct (N × H)
296. #3367C2×C67semidirect
298. #3302C2×C67semidirect
299. #3298C2×C67semidirect
300. #1710C2×C67semidirect
301. #1700C2×C67semidirect
302. #1699C2×C67semidirect
303. #1679C2×C67semidirect
304. #1679C2×C67semidirect
305. #1650C2×C67semidirect
306. #1649C2×C67semidirect
307. #1647C2×C67semidirect
308. #1647C2×C67semidirect
309. #1590C2×C67semidirect
310. C8×D8C2×C67semidirect
311. C67#19914direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33672
C67×#33302
C67×#33022
C67×#32982
C67×#17102
C67×#17002
C67×#16992
C67×#16792
C67×#16502
C67×#16492
C67×#16472
C67×#15902
C8×D8×C672
#1991467