Finite Group Databasebeta

← C67×#20041 #30699292

Subgroups — C67×#20041 #30699292

Order 17152.

602 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3384C211yesmaximal, characteristic, complemented
2C67×#3357C211yesmaximal, characteristic, complemented
3C67×#3303C211yesmaximal, characteristic
4C67×#3298C211yesmaximal, characteristic, complemented
5C67×#1718C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1699C211yesmaximal, characteristic, complemented
8C67×#1675C211yesmaximal, characteristic, complemented
9C67×#1671C211yesmaximal, characteristic, complemented
10C67×#1646C211yesmaximal, characteristic, complemented
11C67×#1643C211yesmaximal, characteristic, complemented
12C67×#1637C211yesmaximal, characteristic, complemented
13C67×#1635C211yesmaximal, characteristic, complemented
14C67×#1586C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16#555×C67C2211yescharacteristic
17#554×C67C2211yescharacteristic
18D4×Q8×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#501×C67—21no—
21#500×C67C2211yescharacteristic
22#498×C67C2211yescharacteristic
23#487×C67C2211yescharacteristic
24#487×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#485×C67—21no—
28#485×C67—21no—
29#483×C67—21no—
30#480×C67—21no—
31#478×C67—21no—
32#477×C67C2211yescharacteristic
33#476×C67—21no—
34#476×C67C2211yescharacteristic
35#476×C67C2211yescharacteristic
36#476×C67C2211yescharacteristic
37#475×C67C2211yescharacteristic
38#474×C67C2211yescharacteristic
39#468×C67—21no—
40#465×C67—21no—
41#465×C67—21no—
42#461×C67C2211yescharacteristic
43#461×C67C2211yescharacteristic
44#449×C67—21no—
45#448×C67—21no—
46#446×C67C2211yescharacteristic
47#442×C67—21no—
48C4×SD16×C67C2211yescharacteristic
49C4×D8×C67C2211yescharacteristic
50#436×C67C2211yescharacteristic
51C8×D4×C67C2211yescharacteristic
52C8×D4×C67C2211yescharacteristic
53#428×C67—21no—
54#416×C67—21no—
55C2×#103×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63D4⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C22×Q8×C67—21no—
69C2×SD16×C67—21no—
70C2×SD16×C67—21no—
71C2×SD16×C67D411yescharacteristic
72C2×D8×C67D411yescharacteristic
73C2×M16×C67—21no—
74C22×C8×C67—21no—
75C22×C8×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67C2311yescharacteristic
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81#125×C67—21no—
82#124×C67—41no—
83#123×C67—41no—
84C22⋊Q8×C67—21no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—41no—
88C22⋊Q8×C67—41no—
89#121×C67—21no—
90#121×C67—21no—
91#121×C67—21no—
92C4×Q8×C67C2311yescharacteristic
93C4×D4×C67—21no—
94C4×D4×C67—21no—
95C4×D4×C67C2311yescharacteristic
96C4×D4×C67C2311yescharacteristic
97C4×D4×C67C2311yescharacteristic
98#117×C67—21no—
99#117×C67—21no—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67—21no—
102#107×C67—21no—
103#107×C67—21no—
104#107×C67—21no—
105#107×C67—21no—
106#107×C67D411yescharacteristic
107#106×C67—21no—
108#106×C67—21no—
109#106×C67—21no—
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—21no—
113#106×C67D411yescharacteristic
114C4⋊C8×C67—21no—
115C4⋊C8×C67C2311yescharacteristic
116C4⋊C8×C67C2311yescharacteristic
117C4⋊C8×C67C2311yescharacteristic
118C4⋊C8×C67C2311yescharacteristic
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—41no—
128#103×C67—41no—
129#103×C67D411yescharacteristic
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—81no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67—21no—
149C2×Q8×C67—21no—
150C2×Q8×C67—41no—
151C2×Q8×C67—42no—
152C2×Q8×C67—81no—
153C2×Q8×C67C2×D411yescharacteristic
154C2×D4×C67—41no—
155C2×D4×C67—41no—
156C2×D4×C67C2×D411yescharacteristic
157C2×D4×C67C2×D411yescharacteristic
158C2×D4×C67C2×D411yescharacteristic
159C22×C4×C67—21no—
160C22×C4×C67—21no—
161C22×C4×C67—21no—
162C22×C4×C67—41no—
163C22×C4×C67—41no—
164C22×C4×C67—41no—
165SD16×C67—41no—
166SD16×C67—81no—
167SD16×C67—81no—
168D8×C67—22no—
169M16×C67—42no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67—81no—
176C2×C8×C67—81no—
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
179C2×C8×C67C2×D411yescharacteristic
180C2×C8×C67C2×D411yescharacteristic
181C4⋊C4×C67—21no—
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—21no—
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—41no—
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67—21no—
211C23×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—41no—
215Q8×C67—41no—
216Q8×C67—41no—
217Q8×C67—41no—
218Q8×C67—81no—
219Q8×C67—81no—
220Q8×C67—82no—
221D4×C67—21no—
222D4×C67—21no—
223D4×C67—21no—
224D4×C67—21no—
225D4×C67—21no—
226D4×C67—82no—
227D4×C67—161no—
228D4×C67#13712yes—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—21no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—82no—
245C2×C4×C67—82no—
246C2×C4×C67—161no—
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67C22×D411yescharacteristic
249C2×C4×C67H(5,𝔽2)11yescharacteristic
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—21no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C8×C67—41no—
258C8×C67—41no—
259C22×C67—22no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67D4211yescharacteristic
269C4×C67—21no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67—81no—
276C4×C67—81no—
277C4×C67—81no—
278C4×C67—81no—
279C4×C67C2×#13711yescharacteristic
280C4×C67#57511yescharacteristic
281C4×C67#57711yescharacteristic
282C4×C67#57711yescharacteristic
283#20041C6711yesmaximal, characteristic, direct product factor, Sylow
284C2×C67—42no—
285C2×C67—81no—
286C2×C67—81no—
287C2×C67#326611yescharacteristic
288C2×C67#326711yescharacteristic
289C2×C67#329011yescharacteristic
290#3384C2×C6711yescharacteristic, complemented
291#3357C2×C6711yescharacteristic, complemented
292#3303C2×C6711yescharacteristic
293#3298C2×C6711yescharacteristic, complemented
294#1718C2×C6711yescharacteristic, complemented
295#1704C2×C6711yescharacteristic, complemented
296#1699C2×C6711yescharacteristic, complemented
297#1675C2×C6711yescharacteristic, complemented
298#1671C2×C6711yescharacteristic, complemented
299#1646C2×C6711yescharacteristic, complemented
300#1643C2×C6711yescharacteristic, complemented
301#1637C2×C6711yescharacteristic, complemented
302#1635C2×C6711yescharacteristic, complemented
303#1586C2×C6711yescharacteristic, complemented
304#1585C2×C6711yescharacteristic, complemented
305C67#2004111yescharacteristic, direct product factor, Sylow
306#555C22×C6711yescharacteristic
307#554C22×C6711yescharacteristic
308D4×Q8C22×C6711yescharacteristic
309#547C22×C6711yescharacteristic
310#501—21no—
311#500C22×C6711yescharacteristic
312#498C22×C6711yescharacteristic
313#487C22×C6711yescharacteristic
314#487C22×C6711yescharacteristic
315#487C22×C6711yescharacteristic
316#487C22×C6711yescharacteristic
317#485—21no—
318#485—21no—
319#483—21no—
320#480—21no—
321#478—21no—
322#477C22×C6711yescharacteristic
323#476—21no—
324#476C22×C6711yescharacteristic
325#476C22×C6711yescharacteristic
326#476C22×C6711yescharacteristic
327#475C22×C6711yescharacteristic
328#474C22×C6711yescharacteristic
329#468—21no—
330#465—21no—
331#465—21no—
332#461C22×C6711yescharacteristic
333#461C22×C6711yescharacteristic
334#449—21no—
335#448—21no—
336#446C22×C6711yescharacteristic
337#442—21no—
338C4×SD16C22×C6711yescharacteristic
339C4×D8C22×C6711yescharacteristic
340#436C22×C6711yescharacteristic
341C8×D4C22×C6711yescharacteristic
342C8×D4C22×C6711yescharacteristic
343#428—21no—
344#416—21no—
345C2×#103—21no—
346#335C22×C6711yescharacteristic
347#334C22×C6711yescharacteristic
348#333C22×C6711yescharacteristic
349#333C22×C6711yescharacteristic
350#332C22×C6711yescharacteristic
351#332C22×C6711yescharacteristic
352Q8⋊C8C22×C6711yescharacteristic
353D4⋊C8C22×C6711yescharacteristic
354D4⋊C8C22×C6711yescharacteristic
355D4⋊C8C22×C6711yescharacteristic
356#322C22×C6711yescharacteristic
357C2×#41—21no—
358C22×Q8—21no—
359C2×SD16—21no—
360C2×SD16—21no—
361C2×SD16D4×C6711yescharacteristic
362C2×D8D4×C6711yescharacteristic
363C2×M16—21no—
364C22×C8—21no—
365C22×C8—21no—
366C4⋊Q8—21no—
367C4⋊Q8C23×C6711yescharacteristic
368C4⋊Q8C23×C6711yescharacteristic
369C4⋊Q8C23×C6711yescharacteristic
370C4⋊Q8C23×C6711yescharacteristic
371#125—21no—
372#124—41no—
373#123—41no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—41no—
378C22⋊Q8—41no—
379#121—21no—
380#121—21no—
381#121—21no—
382C4×Q8C23×C6711yescharacteristic
383C4×D4—21no—
384C4×D4—21no—
385C4×D4C23×C6711yescharacteristic
386C4×D4C23×C6711yescharacteristic
387C4×D4C23×C6711yescharacteristic
388#117—21no—
389#117—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392#107—21no—
393#107—21no—
394#107—21no—
395#107—21no—
396#107D4×C6711yescharacteristic
397#106—21no—
398#106—21no—
399#106—21no—
400#106—21no—
401#106—21no—
402#106—21no—
403#106D4×C6711yescharacteristic
404C4⋊C8—21no—
405C4⋊C8C23×C6711yescharacteristic
406C4⋊C8C23×C6711yescharacteristic
407C4⋊C8C23×C6711yescharacteristic
408C4⋊C8C23×C6711yescharacteristic
409#103—21no—
410#103—21no—
411#103—21no—
412#103—21no—
413#103—21no—
414#103—21no—
415#103—21no—
416#103—21no—
417#103—41no—
418#103—41no—
419#103D4×C6711yescharacteristic
420#102—21no—
421#102—21no—
422#102—21no—
423#102—21no—
424#102D4×C6711yescharacteristic
425#102D4×C6711yescharacteristic
426#102D4×C6711yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—21no—
430#97—21no—
431C4×C8C23×C6711yescharacteristic
432C4×C8C23×C6711yescharacteristic
433C4×C8C23×C6711yescharacteristic
434#41—42no—
435#41—81no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—81no—
443C2×Q8C2×D4×C6711yescharacteristic
444C2×D4—41no—
445C2×D4—41no—
446C2×D4C2×D4×C6711yescharacteristic
447C2×D4C2×D4×C6711yescharacteristic
448C2×D4C2×D4×C6711yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455SD16—41no—
456SD16—81no—
457SD16—81no—
458D8—22no—
459M16—42no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—81no—
466C2×C8—81no—
467C2×C8C2×D4×C6711yescharacteristic
468C2×C8C2×D4×C6711yescharacteristic
469C2×C8C2×D4×C6711yescharacteristic
470C2×C8C2×D4×C6711yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4C2×D4×C6711yescharacteristic
485C4⋊C4C2×D4×C6711yescharacteristic
486C4⋊C4C2×D4×C6711yescharacteristic
487C4⋊C4C2×D4×C6711yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C42—21no—
497C42—41no—
498C42C24×C6711yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502Q8—21no—
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—81no—
509Q8—81no—
510Q8—82no—
511D4—21no—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—82no—
517D4—161no—
518D4#137×C6712yes—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—161no—
537C2×C4C22×D4×C6711yescharacteristic
538C2×C4C22×D4×C6711yescharacteristic
539C2×C4H(5,𝔽2)×C6711yescharacteristic
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—41no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22D42×C6711yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4C2×#137×C6711yescharacteristic
570C4#575×C6711yescharacteristic
571C4#577×C6711yescharacteristic
572C4#577×C6711yescharacteristic
573C2—42no—
574C2—81no—
575C2—81no—
576C2C67×#326611yescharacteristic
577C2C67×#326711yescharacteristic
578C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3303C2
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×#3384C2semidirect
2. C67×#3357C2semidirect
4. C67×#3298C2semidirect
5. C67×#1718C2semidirect
6. C67×#1704C2semidirect
7. C67×#1699C2semidirect
8. C67×#1675C2semidirect
9. C67×#1671C2semidirect
10. C67×#1646C2semidirect
11. C67×#1643C2semidirect
12. C67×#1637C2semidirect
13. C67×#1635C2semidirect
14. C67×#1586C2semidirect
15. C67×#1585C2semidirect
283. #20041C67direct (N × H)
290. #3384C2×C67semidirect
291. #3357C2×C67semidirect
293. #3298C2×C67semidirect
294. #1718C2×C67semidirect
295. #1704C2×C67semidirect
296. #1699C2×C67semidirect
297. #1675C2×C67semidirect
298. #1671C2×C67semidirect
299. #1646C2×C67semidirect
300. #1643C2×C67semidirect
301. #1637C2×C67semidirect
302. #1635C2×C67semidirect
303. #1586C2×C67semidirect
304. #1585C2×C67semidirect
305. C67#20041direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33842
C67×#33572
C67×#33032
C67×#32982
C67×#17182
C67×#17042
C67×#16992
C67×#16752
C67×#16712
C67×#16462
C67×#16432
C67×#16372
C67×#16352
C67×#15862
C67×#15852
#2004167