Finite Group Databasebeta

← C67×#20115 #30699366

Subgroups — C67×#20115 #30699366

Order 17152.

594 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×#3400C211yesmaximal, characteristic, complemented
2C67×#3381C211yesmaximal, characteristic, complemented
3C67×#3373C211yesmaximal, characteristic
4C67×#3299C211yesmaximal, characteristic, complemented
5C67×#1714C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1692C211yesmaximal, characteristic, complemented
8C67×#1682C211yesmaximal, characteristic, complemented
9C67×#1665C211yesmaximal, characteristic
10C67×#1649C211yesmaximal, characteristic, complemented
11C67×#1648C211yesmaximal, characteristic, complemented
12C67×#1630C211yesmaximal, characteristic, complemented
13C67×#1622C211yesmaximal, characteristic
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#499×C67—21no—
22#498×C67C2211yescharacteristic
23#497×C67—21no—
24#492×C67C2211yescharacteristic
25#492×C67C2211yescharacteristic
26#489×C67—21no—
27#487×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#484×C67—21no—
30#479×C67—21no—
31#478×C67—21no—
32#477×C67C2211yescharacteristic
33#477×C67C2211yescharacteristic
34#476×C67—21no—
35#475×C67C2211yescharacteristic
36#474×C67C2211yescharacteristic
37#467×C67—21no—
38#464×C67—21no—
39#463×C67—21no—
40#462×C67C2211yescharacteristic
41#462×C67C2211yescharacteristic
42#461×C67—21no—
43#460×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#451×C67—21no—
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#442×C67—21no—
50#440×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67—21no—
53C4×SD16×C67C2211yescharacteristic
54C8×D4×C67C2211yescharacteristic
55C2×#103×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
67C22×Q8×C67—21no—
68C2×Q16×C67—21no—
69C2×Q16×C67—21no—
70C2×Q16×C67D411yescharacteristic
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—41no—
75C2×SD16×C67D411yescharacteristic
76C2×D8×C67—21no—
77C22×C8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#127×C67C2311yescharacteristic
84#125×C67—21no—
85#125×C67—21no—
86#125×C67—41no—
87#124×C67—21no—
88#124×C67—21no—
89#123×C67—41no—
90C22⋊Q8×C67—21no—
91C22⋊Q8×C67—21no—
92C22⋊Q8×C67—41no—
93C22⋊Q8×C67—41no—
94C4×Q8×C67—21no—
95C4×Q8×C67—21no—
96C4×Q8×C67C2311yescharacteristic
97C4×Q8×C67C2311yescharacteristic
98C4×Q8×C67C2311yescharacteristic
99C4×D4×C67—21no—
100C4×D4×C67—21no—
101C4×D4×C67—21no—
102C4×D4×C67—21no—
103C4×D4×C67C2311yescharacteristic
104C2×C4⋊C4×C67—21no—
105#107×C67—21no—
106#107×C67—21no—
107#107×C67—21no—
108#107×C67—41no—
109#107×C67D411yescharacteristic
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
144C2×Q8×C67—21no—
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—41no—
148C2×Q8×C67—42no—
149C2×Q8×C67—81no—
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×C67C2×D411yescharacteristic
157C22×C4×C67—21no—
158C22×C4×C67—41no—
159C22×C4×C67—41no—
160C22×C4×C67—41no—
161Q16×C67—41no—
162Q16×C67—81no—
163Q16×C67—81no—
164SD16×C67—22no—
165SD16×C67—42no—
166SD16×C67—81no—
167SD16×C67—82no—
168SD16×C67—81no—
169D8×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×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—
203C22⋊C4×C67—41no—
204C22⋊C4×C67—41no—
205C42×C67—21no—
206C42×C67—21no—
207C42×C67—21no—
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—21no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—81no—
223Q8×C67—82no—
224D4×C67—21no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—41no—
229D4×C67—81no—
230D4×C67#13712yes—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—21no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—82no—
244C2×C4×C67—161no—
245C2×C4×C67—161no—
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—22no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—81no—
262C22×C67—81no—
263C22×C67—81no—
264C22×C67D4211yescharacteristic
265C4×C67—21no—
266C4×C67—41no—
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#57611yescharacteristic
279C4×C67#57811yescharacteristic
280#20115C6711yesmaximal, characteristic, direct product factor, Sylow
281C2×C67—42no—
282C2×C67—161no—
283C2×C67#326811yescharacteristic
284C2×C67#327011yescharacteristic
285C2×C67#329111yescharacteristic
286#3400C2×C6711yescharacteristic, complemented
287#3381C2×C6711yescharacteristic, complemented
288#3373C2×C6711yescharacteristic
289#3299C2×C6711yescharacteristic, complemented
290#1714C2×C6711yescharacteristic, complemented
291#1704C2×C6711yescharacteristic, complemented
292#1692C2×C6711yescharacteristic, complemented
293#1682C2×C6711yescharacteristic, complemented
294#1665C2×C6711yescharacteristic
295#1649C2×C6711yescharacteristic, complemented
296#1648C2×C6711yescharacteristic, complemented
297#1630C2×C6711yescharacteristic, complemented
298#1622C2×C6711yescharacteristic
299#1587C2×C6711yescharacteristic, complemented
300#1585C2×C6711yescharacteristic, complemented
301C67#2011511yescharacteristic, direct product factor, Sylow
302#559C22×C6711yescharacteristic
303#557C22×C6711yescharacteristic
304#554C22×C6711yescharacteristic
305D4×Q8C22×C6711yescharacteristic
306#500C22×C6711yescharacteristic
307#499—21no—
308#498C22×C6711yescharacteristic
309#497—21no—
310#492C22×C6711yescharacteristic
311#492C22×C6711yescharacteristic
312#489—21no—
313#487C22×C6711yescharacteristic
314#486C22×C6711yescharacteristic
315#484—21no—
316#479—21no—
317#478—21no—
318#477C22×C6711yescharacteristic
319#477C22×C6711yescharacteristic
320#476—21no—
321#475C22×C6711yescharacteristic
322#474C22×C6711yescharacteristic
323#467—21no—
324#464—21no—
325#463—21no—
326#462C22×C6711yescharacteristic
327#462C22×C6711yescharacteristic
328#461—21no—
329#460C22×C6711yescharacteristic
330#459C22×C6711yescharacteristic
331#451—21no—
332#446C22×C6711yescharacteristic
333#446C22×C6711yescharacteristic
334C8×Q8C22×C6711yescharacteristic
335#442—21no—
336#440—21no—
337C4×Q16C22×C6711yescharacteristic
338C4×SD16—21no—
339C4×SD16C22×C6711yescharacteristic
340C8×D4C22×C6711yescharacteristic
341C2×#103—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
353C22×Q8—21no—
354C2×Q16—21no—
355C2×Q16—21no—
356C2×Q16D4×C6711yescharacteristic
357C2×SD16—21no—
358C2×SD16—21no—
359C2×SD16—21no—
360C2×SD16—41no—
361C2×SD16D4×C6711yescharacteristic
362C2×D8—21no—
363C22×C8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8C23×C6711yescharacteristic
367C4⋊Q8C23×C6711yescharacteristic
368C4⋊Q8C23×C6711yescharacteristic
369#127C23×C6711yescharacteristic
370#125—21no—
371#125—21no—
372#125—41no—
373#124—21no—
374#124—21no—
375#123—41no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—41no—
379C22⋊Q8—41no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C23×C6711yescharacteristic
383C4×Q8C23×C6711yescharacteristic
384C4×Q8C23×C6711yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4C23×C6711yescharacteristic
390C2×C4⋊C4—21no—
391#107—21no—
392#107—21no—
393#107—21no—
394#107—41no—
395#107D4×C6711yescharacteristic
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
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—81no—
436C2×Q8C2×D4×C6711yescharacteristic
437C2×Q8C2×D4×C6711yescharacteristic
438C2×Q8C2×D4×C6711yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×D4×C6711yescharacteristic
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447Q16—41no—
448Q16—81no—
449Q16—81no—
450SD16—22no—
451SD16—42no—
452SD16—81no—
453SD16—82no—
454SD16—81no—
455D8—42no—
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—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42C24×C6711yescharacteristic
496C23—21no—
497C23—41no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—81no—
509Q8—82no—
510D4—21no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—81no—
516D4#137×C6712yes—
517C2×C4—21no—
518C2×C4—21no—
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—82no—
530C2×C4—161no—
531C2×C4—161no—
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—22no—
545C22—41no—
546C22—41no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22D42×C6711yescharacteristic
551C4—21no—
552C4—41no—
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#576×C6711yescharacteristic
565C4#578×C6711yescharacteristic
566C2—42no—
567C2—161no—
568C2C67×#326811yescharacteristic
569C2C67×#327011yescharacteristic
570C2C67×#329111yescharacteristic
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
1. C67×#3400C2semidirect
2. C67×#3381C2semidirect
4. C67×#3299C2semidirect
5. C67×#1714C2semidirect
6. C67×#1704C2semidirect
7. C67×#1692C2semidirect
8. C67×#1682C2semidirect
10. C67×#1649C2semidirect
11. C67×#1648C2semidirect
12. C67×#1630C2semidirect
14. C67×#1587C2semidirect
15. C67×#1585C2semidirect
280. #20115C67direct (N × H)
286. #3400C2×C67semidirect
287. #3381C2×C67semidirect
289. #3299C2×C67semidirect
290. #1714C2×C67semidirect
291. #1704C2×C67semidirect
292. #1692C2×C67semidirect
293. #1682C2×C67semidirect
295. #1649C2×C67semidirect
296. #1648C2×C67semidirect
297. #1630C2×C67semidirect
299. #1587C2×C67semidirect
300. #1585C2×C67semidirect
301. C67#20115direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#34002
C67×#33812
C67×#33732
C67×#32992
C67×#17142
C67×#17042
C67×#16922
C67×#16822
C67×#16652
C67×#16492
C67×#16482
C67×#16302
C67×#16222
C67×#15872
C67×#15852
#2011567