Finite Group Databasebeta

← C67×#20051 #30699302

Subgroups — C67×#20051 #30699302

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3392C211yesmaximal, characteristic
2C67×#3387C211yesmaximal, characteristic, complemented
3C67×#3369C211yesmaximal, characteristic, complemented
4C67×#3304C211yesmaximal, characteristic
5C67×#1714C211yesmaximal, characteristic, complemented
6C67×#1702C211yesmaximal, characteristic, complemented
7C67×#1686C211yesmaximal, characteristic, complemented
8C67×#1684C211yesmaximal, characteristic, complemented
9C67×#1671C211yesmaximal, characteristic, complemented
10C67×#1650C211yesmaximal, characteristic, complemented
11C67×#1648C211yesmaximal, characteristic, complemented
12C67×#1631C211yesmaximal, characteristic
13C67×#1623C211yesmaximal, characteristic, complemented
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16Q82×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19#550×C67C2211yescharacteristic
20#501×C67—21no—
21#500×C67C2211yescharacteristic
22#498×C67C2211yescharacteristic
23#492×C67—21no—
24#492×C67C2211yescharacteristic
25#492×C67C2211yescharacteristic
26#489×C67—21no—
27#487×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#482×C67—21no—
30#479×C67—21no—
31#477×C67C2211yescharacteristic
32#476×C67C2211yescharacteristic
33#475×C67—21no—
34#475×C67—21no—
35#475×C67C2211yescharacteristic
36#475×C67C2211yescharacteristic
37#465×C67—21no—
38#464×C67—21no—
39#462×C67C2211yescharacteristic
40#461×C67—21no—
41#461×C67C2211yescharacteristic
42#460×C67—21no—
43#460×C67C2211yescharacteristic
44#460×C67C2211yescharacteristic
45#452×C67—21no—
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#441×C67—21no—
50#441×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67—21no—
53C4×SD16×C67C2211yescharacteristic
54C8×D4×C67C2211yescharacteristic
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
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×Q16×C67—21no—
69C2×Q16×C67D411yescharacteristic
70C2×SD16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—21no—
75C2×SD16×C67—41no—
76C2×SD16×C67D411yescharacteristic
77C22×C8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67—21no—
81C4⋊Q8×C67—41no—
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84C4⋊Q8×C67C2311yescharacteristic
85#127×C67—21no—
86#127×C67C2311yescharacteristic
87#126×C67—41no—
88#125×C67—21no—
89#125×C67—41no—
90#124×C67—21no—
91C22⋊Q8×C67—21no—
92#121×C67—21no—
93#121×C67—41no—
94C4×Q8×C67—21no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67—21no—
98C4×Q8×C67—21no—
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×Q8×C67C2311yescharacteristic
102C4×D4×C67—21no—
103C4×D4×C67C2311yescharacteristic
104#117×C67—21no—
105#107×C67—21no—
106#107×C67—41no—
107#107×C67D411yescharacteristic
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×C67—21no—
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×C67—21no—
128#103×C67D411yescharacteristic
129#103×C67D411yescharacteristic
130#103×C67D411yescharacteristic
131#102×C67—21no—
132#102×C67—21no—
133#102×C67—21no—
134#102×C67—21no—
135#102×C67—21no—
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138#97×C67—21no—
139#97×C67—21no—
140C4×C8×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—41no—
150C2×Q8×C67C2×D411yescharacteristic
151C2×Q8×C67C2×D411yescharacteristic
152C2×Q8×C67C2×D411yescharacteristic
153C2×D4×C67—21no—
154C2×D4×C67—21no—
155C2×D4×C67—21no—
156C2×D4×C67—41no—
157C2×D4×C67—41no—
158C2×D4×C67C2×D411yescharacteristic
159C22×C4×C67—21no—
160C22×C4×C67—41no—
161Q16×C67—22no—
162Q16×C67—42no—
163SD16×C67—41no—
164SD16×C67—42no—
165SD16×C67—81no—
166SD16×C67—82no—
167SD16×C67—81no—
168SD16×C67—81no—
169SD16×C67—81no—
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×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
179C2×C8×C67C2×D411yescharacteristic
180C4⋊C4×C67—21no—
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—41no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67—41no—
194C4⋊C4×C67—41no—
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C4⋊C4×C67C2×D411yescharacteristic
198C4⋊C4×C67C2×D411yescharacteristic
199C22⋊C4×C67—21no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67—41no—
207C42×C67—41no—
208C42×C67—41no—
209C42×C67C2411yescharacteristic
210C23×C67—21no—
211C23×C67—41no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—41no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—41no—
221Q8×C67—81no—
222Q8×C67—161no—
223Q8×C67#13712yes—
224D4×C67—21no—
225D4×C67—21no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—41no—
229D4×C67—41no—
230D4×C67—82no—
231D4×C67—81no—
232D4×C67—161no—
233C2×C4×C67—21no—
234C2×C4×C67—21no—
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—82no—
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67H(5,𝔽2)11yescharacteristic
249C8×C67—22no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—21no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C8×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—81no—
262C22×C67—81no—
263C22×C67—81no—
264C22×C67D4211yescharacteristic
265C4×C67—22no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—81no—
274C4×C67—81no—
275C4×C67—81no—
276C4×C67C2×#13711yescharacteristic
277C4×C67#57511yescharacteristic
278C4×C67#57711yescharacteristic
279C4×C67#57711yescharacteristic
280#20051C6711yesmaximal, characteristic, direct product factor, Sylow
281C2×C67—81no—
282C2×C67—161no—
283C2×C67#326611yescharacteristic
284C2×C67#327111yescharacteristic
285C2×C67#329111yescharacteristic
286#3392C2×C6711yescharacteristic
287#3387C2×C6711yescharacteristic, complemented
288#3369C2×C6711yescharacteristic, complemented
289#3304C2×C6711yescharacteristic
290#1714C2×C6711yescharacteristic, complemented
291#1702C2×C6711yescharacteristic, complemented
292#1686C2×C6711yescharacteristic, complemented
293#1684C2×C6711yescharacteristic, complemented
294#1671C2×C6711yescharacteristic, complemented
295#1650C2×C6711yescharacteristic, complemented
296#1648C2×C6711yescharacteristic, complemented
297#1631C2×C6711yescharacteristic
298#1623C2×C6711yescharacteristic, complemented
299#1587C2×C6711yescharacteristic, complemented
300#1585C2×C6711yescharacteristic, complemented
301C67#2005111yescharacteristic, direct product factor, Sylow
302Q82C22×C6711yescharacteristic
303#557C22×C6711yescharacteristic
304#556C22×C6711yescharacteristic
305#550C22×C6711yescharacteristic
306#501—21no—
307#500C22×C6711yescharacteristic
308#498C22×C6711yescharacteristic
309#492—21no—
310#492C22×C6711yescharacteristic
311#492C22×C6711yescharacteristic
312#489—21no—
313#487C22×C6711yescharacteristic
314#486C22×C6711yescharacteristic
315#482—21no—
316#479—21no—
317#477C22×C6711yescharacteristic
318#476C22×C6711yescharacteristic
319#475—21no—
320#475—21no—
321#475C22×C6711yescharacteristic
322#475C22×C6711yescharacteristic
323#465—21no—
324#464—21no—
325#462C22×C6711yescharacteristic
326#461—21no—
327#461C22×C6711yescharacteristic
328#460—21no—
329#460C22×C6711yescharacteristic
330#460C22×C6711yescharacteristic
331#452—21no—
332#446C22×C6711yescharacteristic
333#446C22×C6711yescharacteristic
334C8×Q8C22×C6711yescharacteristic
335#441—21no—
336#441—21no—
337C4×Q16C22×C6711yescharacteristic
338C4×SD16—21no—
339C4×SD16C22×C6711yescharacteristic
340C8×D4C22×C6711yescharacteristic
341#416—21no—
342#335C22×C6711yescharacteristic
343#334C22×C6711yescharacteristic
344#333C22×C6711yescharacteristic
345#332C22×C6711yescharacteristic
346#332C22×C6711yescharacteristic
347#331C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349Q8⋊C8C22×C6711yescharacteristic
350Q8⋊C8C22×C6711yescharacteristic
351D4⋊C8C22×C6711yescharacteristic
352#322C22×C6711yescharacteristic
353C2×#41—21no—
354C2×Q16—21no—
355C2×Q16D4×C6711yescharacteristic
356C2×SD16—21no—
357C2×SD16—21no—
358C2×SD16—21no—
359C2×SD16—21no—
360C2×SD16—21no—
361C2×SD16—41no—
362C2×SD16D4×C6711yescharacteristic
363C22×C8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—41no—
368C4⋊Q8C23×C6711yescharacteristic
369C4⋊Q8C23×C6711yescharacteristic
370C4⋊Q8C23×C6711yescharacteristic
371#127—21no—
372#127C23×C6711yescharacteristic
373#126—41no—
374#125—21no—
375#125—41no—
376#124—21no—
377C22⋊Q8—21no—
378#121—21no—
379#121—41no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8C23×C6711yescharacteristic
386C4×Q8C23×C6711yescharacteristic
387C4×Q8C23×C6711yescharacteristic
388C4×D4—21no—
389C4×D4C23×C6711yescharacteristic
390#117—21no—
391#107—21no—
392#107—41no—
393#107D4×C6711yescharacteristic
394#106—21no—
395#106—21no—
396#106—21no—
397#106—21no—
398#106—21no—
399#106D4×C6711yescharacteristic
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8C23×C6711yescharacteristic
404C4⋊C8C23×C6711yescharacteristic
405C4⋊C8C23×C6711yescharacteristic
406C4⋊C8C23×C6711yescharacteristic
407#103—21no—
408#103—21no—
409#103—21no—
410#103—21no—
411#103—21no—
412#103—21no—
413#103—21no—
414#103D4×C6711yescharacteristic
415#103D4×C6711yescharacteristic
416#103D4×C6711yescharacteristic
417#102—21no—
418#102—21no—
419#102—21no—
420#102—21no—
421#102—21no—
422#102D4×C6711yescharacteristic
423C22⋊C8—21no—
424#97—21no—
425#97—21no—
426C4×C8—21no—
427C4×C8C23×C6711yescharacteristic
428C4×C8C23×C6711yescharacteristic
429C4×C8C23×C6711yescharacteristic
430#41—42no—
431#41—81no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—41no—
436C2×Q8C2×D4×C6711yescharacteristic
437C2×Q8C2×D4×C6711yescharacteristic
438C2×Q8C2×D4×C6711yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4C2×D4×C6711yescharacteristic
445C22×C4—21no—
446C22×C4—41no—
447Q16—22no—
448Q16—42no—
449SD16—41no—
450SD16—42no—
451SD16—81no—
452SD16—82no—
453SD16—81no—
454SD16—81no—
455SD16—81no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—81no—
462C2×C8C2×D4×C6711yescharacteristic
463C2×C8C2×D4×C6711yescharacteristic
464C2×C8C2×D4×C6711yescharacteristic
465C2×C8C2×D4×C6711yescharacteristic
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4C2×D4×C6711yescharacteristic
482C4⋊C4C2×D4×C6711yescharacteristic
483C4⋊C4C2×D4×C6711yescharacteristic
484C4⋊C4C2×D4×C6711yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42C24×C6711yescharacteristic
496C23—21no—
497C23—41no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—81no—
508Q8—161no—
509Q8#137×C6712yes—
510D4—21no—
511D4—21no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—82no—
517D4—81no—
518D4—161no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4C22×D4×C6711yescharacteristic
533C2×C4C22×D4×C6711yescharacteristic
534C2×C4H(5,𝔽2)×C6711yescharacteristic
535C8—22no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22D42×C6711yescharacteristic
551C4—22no—
552C4—21no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4C2×#137×C6711yescharacteristic
563C4#575×C6711yescharacteristic
564C4#577×C6711yescharacteristic
565C4#577×C6711yescharacteristic
566C2—81no—
567C2—161no—
568C2C67×#326611yescharacteristic
569C2C67×#327111yescharacteristic
570C2C67×#329111yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#550×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×#3387C2semidirect
3. C67×#3369C2semidirect
5. C67×#1714C2semidirect
6. C67×#1702C2semidirect
7. C67×#1686C2semidirect
8. C67×#1684C2semidirect
9. C67×#1671C2semidirect
10. C67×#1650C2semidirect
11. C67×#1648C2semidirect
13. C67×#1623C2semidirect
14. C67×#1587C2semidirect
15. C67×#1585C2semidirect
280. #20051C67direct (N × H)
287. #3387C2×C67semidirect
288. #3369C2×C67semidirect
290. #1714C2×C67semidirect
291. #1702C2×C67semidirect
292. #1686C2×C67semidirect
293. #1684C2×C67semidirect
294. #1671C2×C67semidirect
295. #1650C2×C67semidirect
296. #1648C2×C67semidirect
298. #1623C2×C67semidirect
299. #1587C2×C67semidirect
300. #1585C2×C67semidirect
301. C67#20051direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33922
C67×#33872
C67×#33692
C67×#33042
C67×#17142
C67×#17022
C67×#16862
C67×#16842
C67×#16712
C67×#16502
C67×#16482
C67×#16312
C67×#16232
C67×#15872
C67×#15852
#2005167