Finite Group Databasebeta

← C67×#20213 #30699464

Subgroups — C67×#20213 #30699464

Order 17152.

558 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3387C211yesmaximal, characteristic, complemented
2C67×#3385C211yesmaximal, characteristic
3C67×#3372C211yesmaximal, characteristic
4C67×#3338C211yesmaximal, characteristic
5C67×#1723C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1696C211yesmaximal, characteristic, complemented
8C67×#1696C211yesmaximal, characteristic, complemented
9C67×#1671C211yesmaximal, characteristic
10C67×#1643C211yesmaximal, characteristic
11C67×#1636C211yesmaximal, characteristic, complemented
12C67×#1633C211yesmaximal, characteristic, complemented
13C67×#1632C211yesmaximal, characteristic
14C67×#1593C211yesmaximal, characteristic, complemented
15C67×#1582C211yesmaximal, characteristic
16Q82×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19#554×C67C2211yescharacteristic
20#501×C67—21no—
21#501×C67—21no—
22#500×C67C2211yescharacteristic
23#500×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#494×C67C2211yescharacteristic
26#491×C67—21no—
27#487×C67C2211yescharacteristic
28#487×C67C2211yescharacteristic
29#484×C67—21no—
30#484×C67—21no—
31#479×C67—21no—
32#479×C67—21no—
33#479×C67—21no—
34#477×C67C2211yescharacteristic
35#477×C67C2211yescharacteristic
36#477×C67C2211yescharacteristic
37#477×C67C2211yescharacteristic
38#477×C67C2211yescharacteristic
39#476×C67C2211yescharacteristic
40#475×C67—21no—
41#475×C67C2211yescharacteristic
42#474×C67—21no—
43#467×C67—21no—
44#461×C67—21no—
45#461×C67C2211yescharacteristic
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48#446×C67C2211yescharacteristic
49#441×C67—21no—
50#440×C67—21no—
51#440×C67—21no—
52C4×Q16×C67C2211yescharacteristic
53C4×SD16×C67C2211yescharacteristic
54C8×D4×C67C2211yescharacteristic
55C2×#107×C67—21no—
56#334×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#333×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×Q16×C67—41no—
68C2×Q16×C67D411yescharacteristic
69C2×SD16×C67—21no—
70C2×SD16×C67D411yescharacteristic
71C22×C8×C67—21no—
72C4⋊Q8×C67—21no—
73C4⋊Q8×C67—21no—
74C4⋊Q8×C67—21no—
75C4⋊Q8×C67—41no—
76C4⋊Q8×C67C2311yescharacteristic
77C4⋊Q8×C67C2311yescharacteristic
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80#125×C67—21no—
81#125×C67—21no—
82#125×C67—21no—
83#125×C67—41no—
84C22⋊Q8×C67—21no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—41no—
87C22⋊Q8×C67—41no—
88C4×Q8×C67—21no—
89C4×Q8×C67—21no—
90C4×Q8×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67C2311yescharacteristic
93C4×Q8×C67C2311yescharacteristic
94C4×Q8×C67C2311yescharacteristic
95C4×D4×C67—21no—
96C4×D4×C67—21no—
97C4×D4×C67C2311yescharacteristic
98C2×C4⋊C4×C67—21no—
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×C67—21no—
106#107×C67—21no—
107#107×C67—21no—
108#107×C67—21no—
109#107×C67D411yescharacteristic
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—21no—
113#106×C67—21no—
114#106×C67D411yescharacteristic
115C4⋊C8×C67—21no—
116C4⋊C8×C67—21no—
117C4⋊C8×C67—21no—
118C4⋊C8×C67C2311yescharacteristic
119C4⋊C8×C67C2311yescharacteristic
120C4⋊C8×C67C2311yescharacteristic
121C4⋊C8×C67C2311yescharacteristic
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—21no—
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—
140#97×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×C67C2×D411yescharacteristic
150C2×Q8×C67C2×D411yescharacteristic
151C2×Q8×C67C2×D411yescharacteristic
152C2×D4×C67C2×D411yescharacteristic
153C22×C4×C67—21no—
154C22×C4×C67—41no—
155C22×C4×C67—41no—
156Q16×C67—22no—
157Q16×C67—82no—
158SD16×C67—22no—
159SD16×C67—42no—
160C2×C8×C67—21no—
161C2×C8×C67—21no—
162C2×C8×C67—21no—
163C2×C8×C67—21no—
164C2×C8×C67—21no—
165C2×C8×C67—42no—
166C2×C8×C67C2×D411yescharacteristic
167C2×C8×C67C2×D411yescharacteristic
168C2×C8×C67C2×D411yescharacteristic
169C2×C8×C67C2×D411yescharacteristic
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—21no—
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—41no—
182C4⋊C4×C67—41no—
183C4⋊C4×C67—41no—
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×C67C2×D411yescharacteristic
191C4⋊C4×C67C2×D411yescharacteristic
192C4⋊C4×C67C2×D411yescharacteristic
193C4⋊C4×C67C2×D411yescharacteristic
194C22⋊C4×C67—21no—
195C22⋊C4×C67—41no—
196C22⋊C4×C67—41no—
197C42×C67—21no—
198C42×C67—21no—
199C42×C67—21no—
200C42×C67—41no—
201C42×C67—41no—
202C42×C67C2411yescharacteristic
203C23×C67—21no—
204Q8×C67—21no—
205Q8×C67—21no—
206Q8×C67—21no—
207Q8×C67—21no—
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—41no—
211Q8×C67—41no—
212Q8×C67—81no—
213Q8×C67—81no—
214Q8×C67—81no—
215Q8×C67—161no—
216D4×C67—21no—
217D4×C67#13712yes—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—161no—
232C2×C4×C67—161no—
233C2×C4×C67C22×D411yescharacteristic
234C2×C4×C67C22×D411yescharacteristic
235C2×C4×C67H(5,𝔽2)11yescharacteristic
236C8×C67—21no—
237C8×C67—21no—
238C8×C67—21no—
239C8×C67—41no—
240C8×C67—41no—
241C8×C67—41no—
242C8×C67—41no—
243C8×C67—41no—
244C8×C67#13712yes—
245C22×C67—22no—
246C22×C67—41no—
247C22×C67—41no—
248C22×C67D4211yescharacteristic
249C4×C67—21no—
250C4×C67—41no—
251C4×C67—41no—
252C4×C67—41no—
253C4×C67—41no—
254C4×C67—41no—
255C4×C67—41no—
256C4×C67—41no—
257C4×C67—81no—
258C4×C67—81no—
259C4×C67—81no—
260C4×C67—81no—
261C4×C67C2×#13711yescharacteristic
262C4×C67C2×#13711yescharacteristic
263C4×C67#57811yescharacteristic
264C4×C67#57811yescharacteristic
265#20213C6711yesmaximal, characteristic, direct product factor, Sylow
266C2×C67—42no—
267C2×C67#327011yescharacteristic
268C2×C67#327611yescharacteristic
269C2×C67#328811yescharacteristic
270#3387C2×C6711yescharacteristic, complemented
271#3385C2×C6711yescharacteristic
272#3372C2×C6711yescharacteristic
273#3338C2×C6711yescharacteristic
274#1723C2×C6711yescharacteristic, complemented
275#1704C2×C6711yescharacteristic, complemented
276#1696C2×C6711yescharacteristic, complemented
277#1696C2×C6711yescharacteristic, complemented
278#1671C2×C6711yescharacteristic
279#1643C2×C6711yescharacteristic
280#1636C2×C6711yescharacteristic, complemented
281#1633C2×C6711yescharacteristic, complemented
282#1632C2×C6711yescharacteristic
283#1593C2×C6711yescharacteristic, complemented
284#1582C2×C6711yescharacteristic
285C67#2021311yescharacteristic, direct product factor, Sylow
286Q82C22×C6711yescharacteristic
287#557C22×C6711yescharacteristic
288#554C22×C6711yescharacteristic
289#554C22×C6711yescharacteristic
290#501—21no—
291#501—21no—
292#500C22×C6711yescharacteristic
293#500C22×C6711yescharacteristic
294#498C22×C6711yescharacteristic
295#494C22×C6711yescharacteristic
296#491—21no—
297#487C22×C6711yescharacteristic
298#487C22×C6711yescharacteristic
299#484—21no—
300#484—21no—
301#479—21no—
302#479—21no—
303#479—21no—
304#477C22×C6711yescharacteristic
305#477C22×C6711yescharacteristic
306#477C22×C6711yescharacteristic
307#477C22×C6711yescharacteristic
308#477C22×C6711yescharacteristic
309#476C22×C6711yescharacteristic
310#475—21no—
311#475C22×C6711yescharacteristic
312#474—21no—
313#467—21no—
314#461—21no—
315#461C22×C6711yescharacteristic
316#446C22×C6711yescharacteristic
317#446C22×C6711yescharacteristic
318#446C22×C6711yescharacteristic
319#441—21no—
320#440—21no—
321#440—21no—
322C4×Q16C22×C6711yescharacteristic
323C4×SD16C22×C6711yescharacteristic
324C8×D4C22×C6711yescharacteristic
325C2×#107—21no—
326#334C22×C6711yescharacteristic
327#334C22×C6711yescharacteristic
328#333C22×C6711yescharacteristic
329#333C22×C6711yescharacteristic
330#333C22×C6711yescharacteristic
331#332C22×C6711yescharacteristic
332Q8⋊C8C22×C6711yescharacteristic
333Q8⋊C8C22×C6711yescharacteristic
334Q8⋊C8C22×C6711yescharacteristic
335D4⋊C8C22×C6711yescharacteristic
336#322C22×C6711yescharacteristic
337C2×Q16—41no—
338C2×Q16D4×C6711yescharacteristic
339C2×SD16—21no—
340C2×SD16D4×C6711yescharacteristic
341C22×C8—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—21no—
345C4⋊Q8—41no—
346C4⋊Q8C23×C6711yescharacteristic
347C4⋊Q8C23×C6711yescharacteristic
348C4⋊Q8C23×C6711yescharacteristic
349C4⋊Q8C23×C6711yescharacteristic
350#125—21no—
351#125—21no—
352#125—21no—
353#125—41no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—41no—
357C22⋊Q8—41no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8C23×C6711yescharacteristic
363C4×Q8C23×C6711yescharacteristic
364C4×Q8C23×C6711yescharacteristic
365C4×D4—21no—
366C4×D4—21no—
367C4×D4C23×C6711yescharacteristic
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370#107—21no—
371#107—21no—
372#107—21no—
373#107—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#106D4×C6711yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8C23×C6711yescharacteristic
389C4⋊C8C23×C6711yescharacteristic
390C4⋊C8C23×C6711yescharacteristic
391C4⋊C8C23×C6711yescharacteristic
392#103—21no—
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—41no—
402#103D4×C6711yescharacteristic
403#103D4×C6711yescharacteristic
404#103D4×C6711yescharacteristic
405#102—21no—
406#102D4×C6711yescharacteristic
407C22⋊C8—21no—
408#97—21no—
409#97—21no—
410#97—21no—
411C4×C8C23×C6711yescharacteristic
412C4×C8C23×C6711yescharacteristic
413C4×C8C23×C6711yescharacteristic
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—41no—
419C2×Q8C2×D4×C6711yescharacteristic
420C2×Q8C2×D4×C6711yescharacteristic
421C2×Q8C2×D4×C6711yescharacteristic
422C2×D4C2×D4×C6711yescharacteristic
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—41no—
426Q16—22no—
427Q16—82no—
428SD16—22no—
429SD16—42no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—42no—
436C2×C8C2×D4×C6711yescharacteristic
437C2×C8C2×D4×C6711yescharacteristic
438C2×C8C2×D4×C6711yescharacteristic
439C2×C8C2×D4×C6711yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×D4×C6711yescharacteristic
461C4⋊C4C2×D4×C6711yescharacteristic
462C4⋊C4C2×D4×C6711yescharacteristic
463C4⋊C4C2×D4×C6711yescharacteristic
464C22⋊C4—21no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—41no—
471C42—41no—
472C42C24×C6711yescharacteristic
473C23—21no—
474Q8—21no—
475Q8—21no—
476Q8—21no—
477Q8—21no—
478Q8—21no—
479Q8—21no—
480Q8—41no—
481Q8—41no—
482Q8—81no—
483Q8—81no—
484Q8—81no—
485Q8—161no—
486D4—21no—
487D4#137×C6712yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—161no—
502C2×C4—161no—
503C2×C4C22×D4×C6711yescharacteristic
504C2×C4C22×D4×C6711yescharacteristic
505C2×C4H(5,𝔽2)×C6711yescharacteristic
506C8—21no—
507C8—21no—
508C8—21no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8#137×C6712yes—
515C22—22no—
516C22—41no—
517C22—41no—
518C22D42×C6711yescharacteristic
519C4—21no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4C2×#137×C6711yescharacteristic
532C4C2×#137×C6711yescharacteristic
533C4#578×C6711yescharacteristic
534C4#578×C6711yescharacteristic
535C2—42no—
536C2C67×#327011yescharacteristic
537C2C67×#327611yescharacteristic
538C2C67×#328811yescharacteristic
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×#1723C2semidirect
6. C67×#1704C2semidirect
7. C67×#1696C2semidirect
8. C67×#1696C2semidirect
11. C67×#1636C2semidirect
12. C67×#1633C2semidirect
14. C67×#1593C2semidirect
265. #20213C67direct (N × H)
270. #3387C2×C67semidirect
274. #1723C2×C67semidirect
275. #1704C2×C67semidirect
276. #1696C2×C67semidirect
277. #1696C2×C67semidirect
280. #1636C2×C67semidirect
281. #1633C2×C67semidirect
283. #1593C2×C67semidirect
285. C67#20213direct (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×#33722
C67×#33382
C67×#17232
C67×#17042
C67×#16962
C67×#16712
C67×#16432
C67×#16362
C67×#16332
C67×#16322
C67×#15932
C67×#15822
#2021367