Finite Group Databasebeta

← C67×#20047 #30699298

Subgroups — C67×#20047 #30699298

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3387C211yesmaximal, characteristic, complemented
2C67×#3385C211yesmaximal, characteristic
3C67×#3359C211yesmaximal, characteristic
4C67×#3299C211yesmaximal, characteristic
5C67×#1718C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1700C211yesmaximal, characteristic, complemented
8C67×#1676C211yesmaximal, characteristic, complemented
9C67×#1671C211yesmaximal, characteristic
10C67×#1645C211yesmaximal, characteristic, complemented
11C67×#1643C211yesmaximal, characteristic
12C67×#1637C211yesmaximal, characteristic
13C67×#1636C211yesmaximal, characteristic, complemented
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic
16Q82×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#501×C67—21no—
21#501×C67—21no—
22#500×C67C2211yescharacteristic
23#498×C67C2211yescharacteristic
24#487×C67—21no—
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#487×C67C2211yescharacteristic
28#487×C67C2211yescharacteristic
29#484×C67—21no—
30#479×C67—21no—
31#479×C67—21no—
32#477×C67C2211yescharacteristic
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#476×C67C2211yescharacteristic
36#475×C67—21no—
37#475×C67—21no—
38#475×C67C2211yescharacteristic
39#475×C67C2211yescharacteristic
40#474×C67—21no—
41#467×C67—21no—
42#462×C67C2211yescharacteristic
43#461×C67—21no—
44#461×C67C2211yescharacteristic
45#451×C67—21no—
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#441×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#333×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×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×C67D411yescharacteristic
70C2×SD16×C67—21no—
71C2×SD16×C67D411yescharacteristic
72C22×C8×C67—21no—
73C4⋊Q8×C67—21no—
74C4⋊Q8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—41no—
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82#125×C67—21no—
83#125×C67—21no—
84#125×C67—41no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—41no—
88C22⋊Q8×C67—41no—
89C4×Q8×C67—21no—
90C4×Q8×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67—21no—
93C4×Q8×C67C2311yescharacteristic
94C4×Q8×C67C2311yescharacteristic
95C4×Q8×C67C2311yescharacteristic
96C4×D4×C67—21no—
97C4×D4×C67—21no—
98C4×D4×C67C2311yescharacteristic
99C2×C4⋊C4×C67—21no—
100#107×C67—21no—
101#107×C67—21no—
102#107×C67—21no—
103#107×C67—21no—
104#107×C67—21no—
105#107×C67D411yescharacteristic
106#106×C67—21no—
107#106×C67—21no—
108#106×C67—21no—
109#106×C67—21no—
110#106×C67—21no—
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×C67—21no—
128#103×C67—21no—
129#103×C67—21no—
130#103×C67—41no—
131#103×C67—41no—
132#103×C67D411yescharacteristic
133#103×C67D411yescharacteristic
134#103×C67D411yescharacteristic
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—21no—
148C2×Q8×C67—41no—
149C2×Q8×C67—41no—
150C2×Q8×C67—42no—
151C2×Q8×C67—81no—
152C2×Q8×C67C2×D411yescharacteristic
153C2×Q8×C67C2×D411yescharacteristic
154C2×Q8×C67C2×D411yescharacteristic
155C2×D4×C67C2×D411yescharacteristic
156C22×C4×C67—21no—
157C22×C4×C67—41no—
158C22×C4×C67—41no—
159Q16×C67—22no—
160Q16×C67—81no—
161SD16×C67—22no—
162SD16×C67—42no—
163C2×C8×C67—21no—
164C2×C8×C67—21no—
165C2×C8×C67—21no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—81no—
169C2×C8×C67C2×D411yescharacteristic
170C2×C8×C67C2×D411yescharacteristic
171C2×C8×C67C2×D411yescharacteristic
172C2×C8×C67C2×D411yescharacteristic
173C4⋊C4×C67—21no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—21no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—21no—
179C4⋊C4×C67—21no—
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
184C4⋊C4×C67—41no—
185C4⋊C4×C67—41no—
186C4⋊C4×C67—41no—
187C4⋊C4×C67—41no—
188C4⋊C4×C67—41no—
189C4⋊C4×C67—41no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67C2×D411yescharacteristic
193C4⋊C4×C67C2×D411yescharacteristic
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C22⋊C4×C67—21no—
197C22⋊C4×C67—41no—
198C22⋊C4×C67—41no—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—41no—
203C42×C67—41no—
204C42×C67C2411yescharacteristic
205C23×C67—21no—
206Q8×C67—21no—
207Q8×C67—21no—
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—41no—
212Q8×C67—41no—
213Q8×C67—41no—
214Q8×C67—41no—
215Q8×C67—82no—
216Q8×C67—81no—
217Q8×C67—81no—
218Q8×C67—161no—
219Q8×C67#13712yes—
220D4×C67—21no—
221D4×C67#13712yes—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—82no—
236C2×C4×C67—161no—
237C2×C4×C67C22×D411yescharacteristic
238C2×C4×C67C22×D411yescharacteristic
239C2×C4×C67H(5,𝔽2)11yescharacteristic
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—41no—
245C8×C67—41no—
246C8×C67—41no—
247C8×C67—41no—
248C8×C67—41no—
249C22×C67—22no—
250C22×C67—41no—
251C22×C67—41no—
252C22×C67D4211yescharacteristic
253C4×C67—22no—
254C4×C67—21no—
255C4×C67—41no—
256C4×C67—41no—
257C4×C67—41no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67—41no—
261C4×C67—81no—
262C4×C67—81no—
263C4×C67—81no—
264C4×C67—81no—
265C4×C67C2×#13711yescharacteristic
266C4×C67C2×#13711yescharacteristic
267C4×C67#57711yescharacteristic
268C4×C67#57811yescharacteristic
269#20047C6711yesmaximal, characteristic, direct product factor, Sylow
270C2×C67—42no—
271C2×C67#326611yescharacteristic
272C2×C67#327011yescharacteristic
273C2×C67#329011yescharacteristic
274#3387C2×C6711yescharacteristic, complemented
275#3385C2×C6711yescharacteristic
276#3359C2×C6711yescharacteristic
277#3299C2×C6711yescharacteristic
278#1718C2×C6711yescharacteristic, complemented
279#1704C2×C6711yescharacteristic, complemented
280#1700C2×C6711yescharacteristic, complemented
281#1676C2×C6711yescharacteristic, complemented
282#1671C2×C6711yescharacteristic
283#1645C2×C6711yescharacteristic, complemented
284#1643C2×C6711yescharacteristic
285#1637C2×C6711yescharacteristic
286#1636C2×C6711yescharacteristic, complemented
287#1587C2×C6711yescharacteristic, complemented
288#1585C2×C6711yescharacteristic
289C67#2004711yescharacteristic, direct product factor, Sylow
290Q82C22×C6711yescharacteristic
291#557C22×C6711yescharacteristic
292#554C22×C6711yescharacteristic
293D4×Q8C22×C6711yescharacteristic
294#501—21no—
295#501—21no—
296#500C22×C6711yescharacteristic
297#498C22×C6711yescharacteristic
298#487—21no—
299#487C22×C6711yescharacteristic
300#487C22×C6711yescharacteristic
301#487C22×C6711yescharacteristic
302#487C22×C6711yescharacteristic
303#484—21no—
304#479—21no—
305#479—21no—
306#477C22×C6711yescharacteristic
307#477C22×C6711yescharacteristic
308#477C22×C6711yescharacteristic
309#476C22×C6711yescharacteristic
310#475—21no—
311#475—21no—
312#475C22×C6711yescharacteristic
313#475C22×C6711yescharacteristic
314#474—21no—
315#467—21no—
316#462C22×C6711yescharacteristic
317#461—21no—
318#461C22×C6711yescharacteristic
319#451—21no—
320#446C22×C6711yescharacteristic
321#446C22×C6711yescharacteristic
322C8×Q8C22×C6711yescharacteristic
323#441—21no—
324#440—21no—
325C4×Q16C22×C6711yescharacteristic
326C4×SD16—21no—
327C4×SD16C22×C6711yescharacteristic
328C8×D4C22×C6711yescharacteristic
329C2×#103—21no—
330#335C22×C6711yescharacteristic
331#334C22×C6711yescharacteristic
332#333C22×C6711yescharacteristic
333#333C22×C6711yescharacteristic
334#332C22×C6711yescharacteristic
335#332C22×C6711yescharacteristic
336Q8⋊C8C22×C6711yescharacteristic
337Q8⋊C8C22×C6711yescharacteristic
338Q8⋊C8C22×C6711yescharacteristic
339D4⋊C8C22×C6711yescharacteristic
340#322C22×C6711yescharacteristic
341C22×Q8—21no—
342C2×Q16—21no—
343C2×Q16D4×C6711yescharacteristic
344C2×SD16—21no—
345C2×SD16D4×C6711yescharacteristic
346C22×C8—21no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—41no—
352C4⋊Q8C23×C6711yescharacteristic
353C4⋊Q8C23×C6711yescharacteristic
354C4⋊Q8C23×C6711yescharacteristic
355C4⋊Q8C23×C6711yescharacteristic
356#125—21no—
357#125—21no—
358#125—41no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—41no—
362C22⋊Q8—41no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C23×C6711yescharacteristic
368C4×Q8C23×C6711yescharacteristic
369C4×Q8C23×C6711yescharacteristic
370C4×D4—21no—
371C4×D4—21no—
372C4×D4C23×C6711yescharacteristic
373C2×C4⋊C4—21no—
374#107—21no—
375#107—21no—
376#107—21no—
377#107—21no—
378#107—21no—
379#107D4×C6711yescharacteristic
380#106—21no—
381#106—21no—
382#106—21no—
383#106—21no—
384#106—21no—
385#106D4×C6711yescharacteristic
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8C23×C6711yescharacteristic
390C4⋊C8C23×C6711yescharacteristic
391C4⋊C8C23×C6711yescharacteristic
392C4⋊C8C23×C6711yescharacteristic
393#103—21no—
394#103—21no—
395#103—21no—
396#103—21no—
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103—21no—
404#103—41no—
405#103—41no—
406#103D4×C6711yescharacteristic
407#103D4×C6711yescharacteristic
408#103D4×C6711yescharacteristic
409#102—21no—
410#102D4×C6711yescharacteristic
411C22⋊C8—21no—
412#97—21no—
413#97—21no—
414C4×C8—21no—
415C4×C8C23×C6711yescharacteristic
416C4×C8C23×C6711yescharacteristic
417C4×C8C23×C6711yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—41no—
423C2×Q8—41no—
424C2×Q8—42no—
425C2×Q8—81no—
426C2×Q8C2×D4×C6711yescharacteristic
427C2×Q8C2×D4×C6711yescharacteristic
428C2×Q8C2×D4×C6711yescharacteristic
429C2×D4C2×D4×C6711yescharacteristic
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433Q16—22no—
434Q16—81no—
435SD16—22no—
436SD16—42no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—81no—
443C2×C8C2×D4×C6711yescharacteristic
444C2×C8C2×D4×C6711yescharacteristic
445C2×C8C2×D4×C6711yescharacteristic
446C2×C8C2×D4×C6711yescharacteristic
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4C2×D4×C6711yescharacteristic
467C4⋊C4C2×D4×C6711yescharacteristic
468C4⋊C4C2×D4×C6711yescharacteristic
469C4⋊C4C2×D4×C6711yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42C24×C6711yescharacteristic
479C23—21no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—82no—
490Q8—81no—
491Q8—81no—
492Q8—161no—
493Q8#137×C6712yes—
494D4—21no—
495D4#137×C6712yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—82no—
510C2×C4—161no—
511C2×C4C22×D4×C6711yescharacteristic
512C2×C4C22×D4×C6711yescharacteristic
513C2×C4H(5,𝔽2)×C6711yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22D42×C6711yescharacteristic
527C4—22no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4C2×#137×C6711yescharacteristic
540C4C2×#137×C6711yescharacteristic
541C4#577×C6711yescharacteristic
542C4#578×C6711yescharacteristic
543C2—42no—
544C2C67×#326611yescharacteristic
545C2C67×#327011yescharacteristic
546C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC2×D4×C67C2×D4
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×#3387C2semidirect
5. C67×#1718C2semidirect
6. C67×#1704C2semidirect
7. C67×#1700C2semidirect
8. C67×#1676C2semidirect
10. C67×#1645C2semidirect
13. C67×#1636C2semidirect
14. C67×#1587C2semidirect
269. #20047C67direct (N × H)
274. #3387C2×C67semidirect
278. #1718C2×C67semidirect
279. #1704C2×C67semidirect
280. #1700C2×C67semidirect
281. #1676C2×C67semidirect
283. #1645C2×C67semidirect
286. #1636C2×C67semidirect
287. #1587C2×C67semidirect
289. C67#20047direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33872
C67×#33852
C67×#33592
C67×#32992
C67×#17182
C67×#17042
C67×#17002
C67×#16762
C67×#16712
C67×#16452
C67×#16432
C67×#16372
C67×#16362
C67×#15872
C67×#15852
#2004767