Finite Group Databasebeta

← C67×#19837 #30699088

Subgroups — C67×#19837 #30699088

Order 17152.

628 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3373C211yesmaximal, characteristic
2C67×#3365C211yesmaximal, characteristic, complemented
3C67×#3359C211yesmaximal, characteristic, complemented
4D4×Q16×C67C211yesmaximal, characteristic, complemented
5C67×#1720C211yesmaximal, characteristic, complemented
6C67×#1700C211yesmaximal, characteristic, complemented
7C67×#1682C211yesmaximal, characteristic, complemented
8C67×#1681C211yesmaximal, characteristic
9C67×#1670C211yesmaximal, characteristic, complemented
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1651C211yesmaximal, characteristic, complemented
12C67×#1640C211yesmaximal, characteristic, complemented
13C67×#1627C211yesmaximal, characteristic
14C67×#1584C211yesmaximal, characteristic, complemented
15C67×SD16⋊C8C211yesmaximal, characteristic, complemented
16C22×Q16×C67—21no—
17#559×C67C2211yescharacteristic
18#557×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20D4×Q8×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#494×C67C2211yescharacteristic
23#489×C67—21no—
24#488×C67—21no—
25#487×C67—21no—
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#486×C67C2211yescharacteristic
30#478×C67—21no—
31#476×C67—21no—
32#475×C67—21no—
33#475×C67C2211yescharacteristic
34#475×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#467×C67—21no—
37#464×C67—21no—
38#463×C67—21no—
39#462×C67C2211yescharacteristic
40#462×C67C2211yescharacteristic
41#461×C67—21no—
42#461×C67C2211yescharacteristic
43#460×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#451×C67—21no—
46#451×C67—21no—
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49C8×Q8×C67C2211yescharacteristic
50#442×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67—21no—
53C4×SD16×C67—21no—
54C4×SD16×C67C2211yescharacteristic
55C8×D4×C67C2211yescharacteristic
56#335×C67C2211yescharacteristic
57#335×C67C2211yescharacteristic
58#332×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#331×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C22×Q8×C67—21no—
68C22×Q8×C67—21no—
69C2×Q16×C67—21no—
70C2×Q16×C67—21no—
71C2×Q16×C67—21no—
72C2×Q16×C67—21no—
73C2×Q16×C67—44no—
74C2×Q16×C67D411yescharacteristic
75C2×SD16×C67—21no—
76C2×SD16×C67—21no—
77C2×SD16×C67D411yescharacteristic
78C2×D8×C67—21no—
79C22×C8×C67—21no—
80C4⋊Q8×C67—21no—
81C4⋊Q8×C67—21no—
82C4⋊Q8×C67—21no—
83C4⋊Q8×C67C2311yescharacteristic
84C4⋊Q8×C67C2311yescharacteristic
85C4⋊Q8×C67C2311yescharacteristic
86#127×C67C2311yescharacteristic
87#125×C67—21no—
88#125×C67—41no—
89#124×C67—21no—
90#124×C67—21no—
91#123×C67—41no—
92C22⋊Q8×C67—21no—
93C22⋊Q8×C67—21no—
94C22⋊Q8×C67—41no—
95C22⋊Q8×C67—41no—
96C4×Q8×C67—21no—
97C4×Q8×C67—21no—
98C4×Q8×C67C2311yescharacteristic
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×D4×C67—21no—
102C4×D4×C67—21no—
103C4×D4×C67—21no—
104C4×D4×C67—21no—
105C4×D4×C67C2311yescharacteristic
106#107×C67D411yescharacteristic
107#106×C67—21no—
108#106×C67—21no—
109#106×C67—21no—
110#106×C67—41no—
111#106×C67D411yescharacteristic
112C4⋊C8×C67—21no—
113C4⋊C8×C67—21no—
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×C67D411yescharacteristic
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—21no—
135#102×C67—21no—
136#102×C67—41no—
137#102×C67D411yescharacteristic
138C22⋊C8×C67—21no—
139#97×C67—21no—
140C4×C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C4×C8×C67C2311yescharacteristic
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67—41no—
149C2×Q8×C67—41no—
150C2×Q8×C67—42no—
151C2×Q8×C67—42no—
152C2×Q8×C67—81no—
153C2×Q8×C67—81no—
154C2×Q8×C67C2×D411yescharacteristic
155C2×Q8×C67C2×D411yescharacteristic
156C2×Q8×C67C2×D411yescharacteristic
157C2×D4×C67—21no—
158C2×D4×C67—21no—
159C2×D4×C67—21no—
160C2×D4×C67C2×D411yescharacteristic
161C22×C4×C67—21no—
162C22×C4×C67—41no—
163C22×C4×C67—41no—
164C22×C4×C67—41no—
165Q16×C67—22no—
166Q16×C67—42no—
167Q16×C67—42no—
168Q16×C67—42no—
169Q16×C67—81no—
170SD16×C67—22no—
171SD16×C67—42no—
172SD16×C67—42no—
173D8×C67—42no—
174C2×C8×C67—21no—
175C2×C8×C67—21no—
176C2×C8×C67—21no—
177C2×C8×C67—21no—
178C2×C8×C67—21no—
179C2×C8×C67—42no—
180C2×C8×C67C2×D411yescharacteristic
181C2×C8×C67C2×D411yescharacteristic
182C2×C8×C67C2×D411yescharacteristic
183C2×C8×C67C2×D411yescharacteristic
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—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—41no—
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—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—21no—
211C42×C67—41no—
212C42×C67C2411yescharacteristic
213C23×C67—21no—
214C23×C67—41no—
215Q8×C67—21no—
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—
226Q8×C67—82no—
227Q8×C67—82no—
228Q8×C67AGL(1,ℤ/8)12yes—
229D4×C67—21no—
230D4×C67—41no—
231D4×C67—41no—
232D4×C67—41no—
233D4×C67—41no—
234D4×C67—81no—
235D4×C67#13512yes—
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—21no—
243C2×C4×C67—41no—
244C2×C4×C67—41no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—82no—
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—21no—
258C8×C67—41no—
259C8×C67—41no—
260C8×C67—41no—
261C8×C67—41no—
262C8×C67—41no—
263C22×C67—22no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—81no—
267C22×C67—81no—
268C22×C67—81no—
269C22×C67D4211yescharacteristic
270C4×C67—22no—
271C4×C67—21no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—81no—
279C4×C67—81no—
280C4×C67—81no—
281C4×C67C2×#13511yescharacteristic
282C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
283C4×C67#57711yescharacteristic
284C4×C67#57811yescharacteristic
285#19837C6711yesmaximal, characteristic, direct product factor, Sylow
286C2×C67—42no—
287C2×C67—161no—
288C2×C67#326511yescharacteristic
289C2×C67#328511yescharacteristic
290C2×C67#328911yescharacteristic
291#3373C2×C6711yescharacteristic
292#3365C2×C6711yescharacteristic, complemented
293#3359C2×C6711yescharacteristic, complemented
294D4×Q16C2×C6711yescharacteristic, complemented
295#1720C2×C6711yescharacteristic, complemented
296#1700C2×C6711yescharacteristic, complemented
297#1682C2×C6711yescharacteristic, complemented
298#1681C2×C6711yescharacteristic
299#1670C2×C6711yescharacteristic, complemented
300#1653C2×C6711yescharacteristic, complemented
301#1651C2×C6711yescharacteristic, complemented
302#1640C2×C6711yescharacteristic, complemented
303#1627C2×C6711yescharacteristic
304#1584C2×C6711yescharacteristic, complemented
305SD16⋊C8C2×C6711yescharacteristic, complemented
306C67#1983711yescharacteristic, direct product factor, Sylow
307C22×Q16—21no—
308#559C22×C6711yescharacteristic
309#557C22×C6711yescharacteristic
310D4×Q8C22×C6711yescharacteristic
311D4×Q8C22×C6711yescharacteristic
312#498C22×C6711yescharacteristic
313#494C22×C6711yescharacteristic
314#489—21no—
315#488—21no—
316#487—21no—
317#487C22×C6711yescharacteristic
318#486C22×C6711yescharacteristic
319#486C22×C6711yescharacteristic
320#486C22×C6711yescharacteristic
321#478—21no—
322#476—21no—
323#475—21no—
324#475C22×C6711yescharacteristic
325#475C22×C6711yescharacteristic
326#475C22×C6711yescharacteristic
327#467—21no—
328#464—21no—
329#463—21no—
330#462C22×C6711yescharacteristic
331#462C22×C6711yescharacteristic
332#461—21no—
333#461C22×C6711yescharacteristic
334#460C22×C6711yescharacteristic
335#459C22×C6711yescharacteristic
336#451—21no—
337#451—21no—
338#446C22×C6711yescharacteristic
339C8×Q8C22×C6711yescharacteristic
340C8×Q8C22×C6711yescharacteristic
341#442—21no—
342C4×Q16C22×C6711yescharacteristic
343C4×SD16—21no—
344C4×SD16—21no—
345C4×SD16C22×C6711yescharacteristic
346C8×D4C22×C6711yescharacteristic
347#335C22×C6711yescharacteristic
348#335C22×C6711yescharacteristic
349#332C22×C6711yescharacteristic
350#332C22×C6711yescharacteristic
351#332C22×C6711yescharacteristic
352#331C22×C6711yescharacteristic
353Q8⋊C8C22×C6711yescharacteristic
354Q8⋊C8C22×C6711yescharacteristic
355Q8⋊C8C22×C6711yescharacteristic
356D4⋊C8C22×C6711yescharacteristic
357#322C22×C6711yescharacteristic
358C22×Q8—21no—
359C22×Q8—21no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—21no—
364C2×Q16—44no—
365C2×Q16D4×C6711yescharacteristic
366C2×SD16—21no—
367C2×SD16—21no—
368C2×SD16D4×C6711yescharacteristic
369C2×D8—21no—
370C22×C8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8C23×C6711yescharacteristic
375C4⋊Q8C23×C6711yescharacteristic
376C4⋊Q8C23×C6711yescharacteristic
377#127C23×C6711yescharacteristic
378#125—21no—
379#125—41no—
380#124—21no—
381#124—21no—
382#123—41no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—41no—
386C22⋊Q8—41no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8C23×C6711yescharacteristic
390C4×Q8C23×C6711yescharacteristic
391C4×Q8C23×C6711yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4C23×C6711yescharacteristic
397#107D4×C6711yescharacteristic
398#106—21no—
399#106—21no—
400#106—21no—
401#106—41no—
402#106D4×C6711yescharacteristic
403C4⋊C8—21no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8C23×C6711yescharacteristic
407C4⋊C8C23×C6711yescharacteristic
408C4⋊C8C23×C6711yescharacteristic
409C4⋊C8C23×C6711yescharacteristic
410#103—21no—
411#103—21no—
412#103—21no—
413#103—21no—
414#103—21no—
415#103—21no—
416#103—21no—
417#103—21no—
418#103D4×C6711yescharacteristic
419#103D4×C6711yescharacteristic
420#103D4×C6711yescharacteristic
421#102—21no—
422#102—21no—
423#102—21no—
424#102—21no—
425#102—21no—
426#102—21no—
427#102—41no—
428#102D4×C6711yescharacteristic
429C22⋊C8—21no—
430#97—21no—
431C4×C8—21no—
432C4×C8—21no—
433C4×C8C23×C6711yescharacteristic
434C4×C8C23×C6711yescharacteristic
435C4×C8C23×C6711yescharacteristic
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—81no—
445C2×Q8C2×D4×C6711yescharacteristic
446C2×Q8C2×D4×C6711yescharacteristic
447C2×Q8C2×D4×C6711yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4C2×D4×C6711yescharacteristic
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456Q16—22no—
457Q16—42no—
458Q16—42no—
459Q16—42no—
460Q16—81no—
461SD16—22no—
462SD16—42no—
463SD16—42no—
464D8—42no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—42no—
471C2×C8C2×D4×C6711yescharacteristic
472C2×C8C2×D4×C6711yescharacteristic
473C2×C8C2×D4×C6711yescharacteristic
474C2×C8C2×D4×C6711yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C6711yescharacteristic
490C4⋊C4C2×D4×C6711yescharacteristic
491C4⋊C4C2×D4×C6711yescharacteristic
492C4⋊C4C2×D4×C6711yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—41no—
503C42C24×C6711yescharacteristic
504C23—21no—
505C23—41no—
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—82no—
518Q8—82no—
519Q8AGL(1,ℤ/8)×C6712yes—
520D4—21no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—81no—
526D4#135×C6712yes—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—161no—
542C2×C4C22×D4×C6711yescharacteristic
543C2×C4C22×D4×C6711yescharacteristic
544C2×C4H(5,𝔽2)×C6711yescharacteristic
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—41no—
553C8—41no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22D42×C6711yescharacteristic
561C4—22no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4C2×#135×C6711yescharacteristic
573C4C2×AGL(1,ℤ/8)×C6711yescharacteristic
574C4#577×C6711yescharacteristic
575C4#578×C6711yescharacteristic
576C2—42no—
577C2—161no—
578C2C67×#326511yescharacteristic
579C2C67×#328511yescharacteristic
580C2C67×#328911yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#459×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
2. C67×#3365C2semidirect
3. C67×#3359C2semidirect
4. D4×Q16×C67C2semidirect
5. C67×#1720C2semidirect
6. C67×#1700C2semidirect
7. C67×#1682C2semidirect
9. C67×#1670C2semidirect
10. C67×#1653C2semidirect
11. C67×#1651C2semidirect
12. C67×#1640C2semidirect
14. C67×#1584C2semidirect
15. C67×SD16⋊C8C2semidirect
285. #19837C67direct (N × H)
292. #3365C2×C67semidirect
293. #3359C2×C67semidirect
294. D4×Q16C2×C67semidirect
295. #1720C2×C67semidirect
296. #1700C2×C67semidirect
297. #1682C2×C67semidirect
299. #1670C2×C67semidirect
300. #1653C2×C67semidirect
301. #1651C2×C67semidirect
302. #1640C2×C67semidirect
304. #1584C2×C67semidirect
305. SD16⋊C8C2×C67semidirect
306. C67#19837direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33732
C67×#33652
C67×#33592
D4×Q16×C672
C67×#17202
C67×#17002
C67×#16822
C67×#16812
C67×#16702
C67×#16532
C67×#16512
C67×#16402
C67×#16272
C67×#15842
C67×SD16⋊C82
#1983767