Finite Group Databasebeta

← C67×#19894 #30699145

Subgroups — C67×#19894 #30699145

Order 17152.

640 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3386C211yesmaximal, characteristic, complemented
2C67×#3359C211yesmaximal, characteristic, complemented
3C67×#3298C211yesmaximal, characteristic
4C67×#3276C211yesmaximal, characteristic
5C67×#1707C211yesmaximal, characteristic, complemented
6C67×#1693C211yesmaximal, characteristic, complemented
7C67×#1690C211yesmaximal, characteristic, complemented
8C67×#1680C211yesmaximal, characteristic, complemented
9C67×#1679C211yesmaximal, characteristic, complemented
10C67×#1654C211yesmaximal, characteristic
11C67×#1653C211yesmaximal, characteristic, complemented
12C67×#1651C211yesmaximal, characteristic, complemented
13C67×#1651C211yesmaximal, characteristic, complemented
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×SD16×C67C211yesmaximal, characteristic, complemented
16C2×#137×C67—21no—
17#557×C67C2211yescharacteristic
18D4×Q8×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#547×C67C2211yescharacteristic
21#501×C67—21no—
22#500×C67C2211yescharacteristic
23#498×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#494×C67C2211yescharacteristic
26#487×C67—21no—
27#487×C67C2211yescharacteristic
28#487×C67C2211yescharacteristic
29#483×C67—21no—
30#478×C67—21no—
31#477×C67—21no—
32#475×C67—21no—
33#475×C67C2211yescharacteristic
34#475×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#474×C67C2211yescharacteristic
37#470×C67—21no—
38#465×C67—21no—
39#462×C67C2211yescharacteristic
40#461×C67—21no—
41#461×C67C2211yescharacteristic
42#461×C67C2211yescharacteristic
43#461×C67C2211yescharacteristic
44#452×C67—21no—
45#451×C67—21no—
46#448×C67—21no—
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#440×C67—21no—
50C4×SD16×C67—21no—
51C4×SD16×C67C2211yescharacteristic
52C4×SD16×C67C2211yescharacteristic
53#436×C67C2211yescharacteristic
54C8×D4×C67C2211yescharacteristic
55C2×#103×C67—21no—
56#334×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#332×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66C82×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C22×Q8×C67—21no—
69C22×Q8×C67—21no—
70#137×C67—44no—
71C2×Q16×C67—21no—
72C2×Q16×C67—21no—
73C2×Q16×C67—21no—
74C2×SD16×C67—21no—
75C2×SD16×C67—21no—
76C2×SD16×C67—21no—
77C2×SD16×C67D411yescharacteristic
78C2×SD16×C67D411yescharacteristic
79C2×M16×C67—21no—
80C22×C8×C67—21no—
81C4⋊Q8×C67—21no—
82C4⋊Q8×C67—21no—
83C4⋊Q8×C67—21no—
84C4⋊Q8×C67C2311yescharacteristic
85C4⋊Q8×C67C2311yescharacteristic
86C4⋊Q8×C67C2311yescharacteristic
87C4⋊Q8×C67C2311yescharacteristic
88#125×C67—21no—
89#125×C67—41no—
90#123×C67—41no—
91C22⋊Q8×C67—21no—
92C22⋊Q8×C67—21no—
93C22⋊Q8×C67—21no—
94C22⋊Q8×C67—41no—
95C22⋊Q8×C67—41no—
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
105C2×C4⋊C4×C67—21no—
106#107×C67—21no—
107#107×C67—21no—
108#107×C67—21no—
109#106×C67—21no—
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—41no—
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×C67—21no—
129#103×C67—21no—
130#103×C67—21no—
131#103×C67—21no—
132#103×C67—21no—
133#103×C67D411yescharacteristic
134#103×C67D411yescharacteristic
135#102×C67—21no—
136#102×C67D411yescharacteristic
137#102×C67D411yescharacteristic
138C22⋊C8×C67—21no—
139C22⋊C8×C67—21no—
140#97×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C4×C8×C67C2311yescharacteristic
145#41×C67—42no—
146#41×C67—81no—
147C2×Q8×C67—21no—
148C2×Q8×C67—21no—
149C2×Q8×C67—21no—
150C2×Q8×C67—21no—
151C2×Q8×C67—41no—
152C2×Q8×C67—41no—
153C2×Q8×C67—42no—
154C2×Q8×C67—42no—
155C2×Q8×C67—81no—
156C2×Q8×C67—81no—
157C2×Q8×C67C2×D411yescharacteristic
158C2×Q8×C67C2×D411yescharacteristic
159C2×D4×C67—41no—
160C2×D4×C67C2×D411yescharacteristic
161C2×D4×C67C2×D411yescharacteristic
162C22×C4×C67—21no—
163C22×C4×C67—21no—
164C22×C4×C67—41no—
165C22×C4×C67—41no—
166C22×C4×C67—41no—
167Q16×C67—42no—
168Q16×C67—42no—
169Q16×C67—81no—
170SD16×C67—41no—
171SD16×C67—42no—
172SD16×C67—42no—
173SD16×C67—81no—
174SD16×C67SD1614yescomplemented
175M16×C67—42no—
176C2×C8×C67—21no—
177C2×C8×C67—21no—
178C2×C8×C67—21no—
179C2×C8×C67—21no—
180C2×C8×C67—21no—
181C2×C8×C67—81no—
182C2×C8×C67C2×D411yescharacteristic
183C2×C8×C67C2×D411yescharacteristic
184C2×C8×C67C2×D411yescharacteristic
185C2×C8×C67C2×D411yescharacteristic
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—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—41no—
197C4⋊C4×C67—41no—
198C4⋊C4×C67—41no—
199C4⋊C4×C67—41no—
200C4⋊C4×C67—41no—
201C4⋊C4×C67—41no—
202C4⋊C4×C67C2×D411yescharacteristic
203C4⋊C4×C67C2×D411yescharacteristic
204C4⋊C4×C67C2×D411yescharacteristic
205C4⋊C4×C67C2×D411yescharacteristic
206C22⋊C4×C67—21no—
207C22⋊C4×C67—21no—
208C22⋊C4×C67—41no—
209C22⋊C4×C67—41no—
210C22⋊C4×C67—41no—
211C42×C67—21no—
212C42×C67—21no—
213C42×C67—41no—
214C42×C67C2411yescharacteristic
215C23×C67—21no—
216C23×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—
226Q8×C67—41no—
227Q8×C67—41no—
228Q8×C67—82no—
229Q8×C67—82no—
230Q8×C67C2×SD1612yes—
231D4×C67—21no—
232D4×C67—21no—
233D4×C67—21no—
234D4×C67—82no—
235D4×C67C2×SD1612yes—
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—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—41no—
261C8×C67—41no—
262C8×C67—41no—
263C8×C67—41no—
264C8×C67C2×SD1612yes—
265C22×C67—22no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67—41no—
270C22×C67—41no—
271C22×C67D4211yescharacteristic
272C4×C67—22no—
273C4×C67—21no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—41no—
279C4×C67—81no—
280C4×C67—81no—
281C4×C67—81no—
282C4×C67—81no—
283C4×C67C22×SD1611yescharacteristic
284C4×C67#57511yescharacteristic
285C4×C67#57711yescharacteristic
286C4×C67#57711yescharacteristic
287#19894C6711yesmaximal, characteristic, direct product factor, Sylow
288C2×C67—42no—
289C2×C67—81no—
290C2×C67#327511yescharacteristic
291C2×C67D4×SD1611yescharacteristic
292C2×C67#329311yescharacteristic
293#3386C2×C6711yescharacteristic, complemented
294#3359C2×C6711yescharacteristic, complemented
295#3298C2×C6711yescharacteristic
296#3276C2×C6711yescharacteristic
297#1707C2×C6711yescharacteristic, complemented
298#1693C2×C6711yescharacteristic, complemented
299#1690C2×C6711yescharacteristic, complemented
300#1680C2×C6711yescharacteristic, complemented
301#1679C2×C6711yescharacteristic, complemented
302#1654C2×C6711yescharacteristic
303#1653C2×C6711yescharacteristic, complemented
304#1651C2×C6711yescharacteristic, complemented
305#1651C2×C6711yescharacteristic, complemented
306#1590C2×C6711yescharacteristic, complemented
307C8×SD16C2×C6711yescharacteristic, complemented
308C67#1989411yescharacteristic, direct product factor, Sylow
309C2×#137—21no—
310#557C22×C6711yescharacteristic
311D4×Q8C22×C6711yescharacteristic
312D4×Q8C22×C6711yescharacteristic
313#547C22×C6711yescharacteristic
314#501—21no—
315#500C22×C6711yescharacteristic
316#498C22×C6711yescharacteristic
317#498C22×C6711yescharacteristic
318#494C22×C6711yescharacteristic
319#487—21no—
320#487C22×C6711yescharacteristic
321#487C22×C6711yescharacteristic
322#483—21no—
323#478—21no—
324#477—21no—
325#475—21no—
326#475C22×C6711yescharacteristic
327#475C22×C6711yescharacteristic
328#475C22×C6711yescharacteristic
329#474C22×C6711yescharacteristic
330#470—21no—
331#465—21no—
332#462C22×C6711yescharacteristic
333#461—21no—
334#461C22×C6711yescharacteristic
335#461C22×C6711yescharacteristic
336#461C22×C6711yescharacteristic
337#452—21no—
338#451—21no—
339#448—21no—
340#446C22×C6711yescharacteristic
341C8×Q8C22×C6711yescharacteristic
342#440—21no—
343C4×SD16—21no—
344C4×SD16C22×C6711yescharacteristic
345C4×SD16C22×C6711yescharacteristic
346#436C22×C6711yescharacteristic
347C8×D4C22×C6711yescharacteristic
348C2×#103—21no—
349#334C22×C6711yescharacteristic
350#334C22×C6711yescharacteristic
351#332C22×C6711yescharacteristic
352#332C22×C6711yescharacteristic
353#332C22×C6711yescharacteristic
354#332C22×C6711yescharacteristic
355Q8⋊C8C22×C6711yescharacteristic
356Q8⋊C8C22×C6711yescharacteristic
357D4⋊C8C22×C6711yescharacteristic
358D4⋊C8C22×C6711yescharacteristic
359C82C22×C6711yescharacteristic
360C2×#41—21no—
361C22×Q8—21no—
362C22×Q8—21no—
363#137—44no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×Q16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16D4×C6711yescharacteristic
371C2×SD16D4×C6711yescharacteristic
372C2×M16—21no—
373C22×C8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8C23×C6711yescharacteristic
378C4⋊Q8C23×C6711yescharacteristic
379C4⋊Q8C23×C6711yescharacteristic
380C4⋊Q8C23×C6711yescharacteristic
381#125—21no—
382#125—41no—
383#123—41no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—41no—
388C22⋊Q8—41no—
389#121—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8C23×C6711yescharacteristic
393C4×Q8C23×C6711yescharacteristic
394C4×D4—21no—
395C4×D4—21no—
396C4×D4C23×C6711yescharacteristic
397C4×D4C23×C6711yescharacteristic
398C2×C4⋊C4—21no—
399#107—21no—
400#107—21no—
401#107—21no—
402#106—21no—
403#106—21no—
404#106—21no—
405#106—41no—
406#106—41no—
407#106D4×C6711yescharacteristic
408#106D4×C6711yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8C23×C6711yescharacteristic
412C4⋊C8C23×C6711yescharacteristic
413C4⋊C8C23×C6711yescharacteristic
414C4⋊C8C23×C6711yescharacteristic
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#103D4×C6711yescharacteristic
428#102—21no—
429#102D4×C6711yescharacteristic
430#102D4×C6711yescharacteristic
431C22⋊C8—21no—
432C22⋊C8—21no—
433#97—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×Q8—21no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—81no—
449C2×Q8—81no—
450C2×Q8C2×D4×C6711yescharacteristic
451C2×Q8C2×D4×C6711yescharacteristic
452C2×D4—41no—
453C2×D4C2×D4×C6711yescharacteristic
454C2×D4C2×D4×C6711yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460Q16—42no—
461Q16—42no—
462Q16—81no—
463SD16—41no—
464SD16—42no—
465SD16—42no—
466SD16—81no—
467SD16SD16×C6714yescomplemented
468M16—42no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—81no—
475C2×C8C2×D4×C6711yescharacteristic
476C2×C8C2×D4×C6711yescharacteristic
477C2×C8C2×D4×C6711yescharacteristic
478C2×C8C2×D4×C6711yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—21no—
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—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4C2×D4×C6711yescharacteristic
496C4⋊C4C2×D4×C6711yescharacteristic
497C4⋊C4C2×D4×C6711yescharacteristic
498C4⋊C4C2×D4×C6711yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C42—21no—
505C42—21no—
506C42—41no—
507C42C24×C6711yescharacteristic
508C23—21no—
509C23—21no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—82no—
522Q8—82no—
523Q8C2×SD16×C6712yes—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—82no—
528D4C2×SD16×C6712yes—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
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—82no—
545C2×C4—82no—
546C2×C4C22×D4×C6711yescharacteristic
547C2×C4C22×D4×C6711yescharacteristic
548C2×C4H(5,𝔽2)×C6711yescharacteristic
549C8—22no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8C2×SD16×C6712yes—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22D42×C6711yescharacteristic
565C4—22no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C22×SD16×C6711yescharacteristic
577C4#575×C6711yescharacteristic
578C4#577×C6711yescharacteristic
579C4#577×C6711yescharacteristic
580C2—42no—
581C2—81no—
582C2C67×#327511yescharacteristic
583C2D4×SD16×C6711yescharacteristic
584C2C67×#329311yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#547×C67C22
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×#3386C2semidirect
2. C67×#3359C2semidirect
5. C67×#1707C2semidirect
6. C67×#1693C2semidirect
7. C67×#1690C2semidirect
8. C67×#1680C2semidirect
9. C67×#1679C2semidirect
11. C67×#1653C2semidirect
12. C67×#1651C2semidirect
13. C67×#1651C2semidirect
14. C67×#1590C2semidirect
15. C8×SD16×C67C2semidirect
174. SD16×C67SD16semidirect
287. #19894C67direct (N × H)
293. #3386C2×C67semidirect
294. #3359C2×C67semidirect
297. #1707C2×C67semidirect
298. #1693C2×C67semidirect
299. #1690C2×C67semidirect
300. #1680C2×C67semidirect
301. #1679C2×C67semidirect
303. #1653C2×C67semidirect
304. #1651C2×C67semidirect
305. #1651C2×C67semidirect
306. #1590C2×C67semidirect
307. C8×SD16C2×C67semidirect
308. C67#19894direct (N × H)
467. SD16SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#33862
C67×#33592
C67×#32982
C67×#32762
C67×#17072
C67×#16932
C67×#16902
C67×#16802
C67×#16792
C67×#16542
C67×#16532
C67×#16512
C67×#15902
C8×SD16×C672
#1989467