Finite Group Databasebeta

← C67×#20127 #30699378

Subgroups — C67×#20127 #30699378

Order 17152.

560 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3372C211yesmaximal, characteristic, complemented
2C67×#3364C211yesmaximal, characteristic, complemented
3C67×#3357C211yesmaximal, characteristic
4C67×#3340C211yesmaximal, characteristic
5C67×#1718C211yesmaximal, characteristic, complemented
6C67×#1699C211yesmaximal, characteristic, complemented
7C67×#1692C211yesmaximal, characteristic, complemented
8C67×#1683C211yesmaximal, characteristic, complemented
9C67×#1676C211yesmaximal, characteristic, complemented
10C67×#1646C211yesmaximal, characteristic
11C67×#1644C211yesmaximal, characteristic, complemented
12C67×#1638C211yesmaximal, characteristic, complemented
13C67×#1636C211yesmaximal, characteristic, complemented
14C67×#1588C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16#557×C67C2211yescharacteristic
17#556×C67C2211yescharacteristic
18#555×C67C2211yescharacteristic
19#554×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#491×C67—21no—
23#488×C67—21no—
24#487×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#487×C67C2211yescharacteristic
28#485×C67—21no—
29#485×C67—21no—
30#482×C67—21no—
31#480×C67—21no—
32#479×C67—21no—
33#479×C67—21no—
34#477×C67C2211yescharacteristic
35#477×C67C2211yescharacteristic
36#477×C67C2211yescharacteristic
37#476×C67C2211yescharacteristic
38#474×C67C2211yescharacteristic
39#474×C67C2211yescharacteristic
40#468×C67—21no—
41#465×C67—21no—
42#464×C67—21no—
43#462×C67C2211yescharacteristic
44#461×C67—21no—
45#461×C67C2211yescharacteristic
46#446×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48#440×C67—21no—
49C4×Q16×C67—21no—
50C4×SD16×C67C2211yescharacteristic
51C4×SD16×C67C2211yescharacteristic
52#436×C67C2211yescharacteristic
53C8×D4×C67C2211yescharacteristic
54#428×C67—21no—
55#427×C67—21no—
56#335×C67C2211yescharacteristic
57#335×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×Q16×C67—21no—
68C2×SD16×C67—21no—
69C2×SD16×C67D411yescharacteristic
70C2×SD16×C67D411yescharacteristic
71C2×M16×C67—21no—
72C22×C8×C67—21no—
73C4⋊Q8×C67—21no—
74C4⋊Q8×C67C2311yescharacteristic
75C4⋊Q8×C67C2311yescharacteristic
76C4⋊Q8×C67C2311yescharacteristic
77C4⋊Q8×C67C2311yescharacteristic
78#126×C67—41no—
79#125×C67—21no—
80#125×C67—21no—
81#125×C67—41no—
82#124×C67—21no—
83#124×C67—41no—
84C22⋊Q8×C67—21no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—41no—
87#121×C67—21no—
88#121×C67—21no—
89C4×Q8×C67—21no—
90C4×Q8×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67C2311yescharacteristic
93C4×Q8×C67C2311yescharacteristic
94C4×D4×C67—21no—
95C4×D4×C67C2311yescharacteristic
96C4×D4×C67C2311yescharacteristic
97#117×C67—21no—
98#117×C67—21no—
99#117×C67—21no—
100C2×C4⋊C4×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#106×C67—21no—
109#106×C67—21no—
110#106×C67—21no—
111#106×C67D411yescharacteristic
112#106×C67D411yescharacteristic
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—41no—
129#103×C67—41no—
130#103×C67D411yescharacteristic
131#103×C67D411yescharacteristic
132#102×C67—21no—
133#102×C67—21no—
134#102×C67—21no—
135#102×C67D411yescharacteristic
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138C22⋊C8×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×C67C2×D411yescharacteristic
149C2×Q8×C67C2×D411yescharacteristic
150C2×D4×C67—41no—
151C2×D4×C67C2×D411yescharacteristic
152C2×D4×C67C2×D411yescharacteristic
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67—41no—
156Q16×C67—42no—
157SD16×C67—41no—
158SD16×C67—41no—
159SD16×C67—42no—
160M16×C67—42no—
161C2×C8×C67—21no—
162C2×C8×C67—21no—
163C2×C8×C67—21no—
164C2×C8×C67—21no—
165C2×C8×C67—21no—
166C2×C8×C67—42no—
167C2×C8×C67C2×D411yescharacteristic
168C2×C8×C67C2×D411yescharacteristic
169C2×C8×C67C2×D411yescharacteristic
170C2×C8×C67C2×D411yescharacteristic
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×C67C2×D411yescharacteristic
188C4⋊C4×C67C2×D411yescharacteristic
189C4⋊C4×C67C2×D411yescharacteristic
190C4⋊C4×C67C2×D411yescharacteristic
191C22⋊C4×C67—21no—
192C22⋊C4×C67—21no—
193C22⋊C4×C67—41no—
194C22⋊C4×C67—41no—
195C22⋊C4×C67—41no—
196C22⋊C4×C67—41no—
197C22⋊C4×C67—41no—
198C42×C67—21no—
199C42×C67—21no—
200C42×C67—41no—
201C42×C67—41no—
202C42×C67—41no—
203C42×C67C2411yescharacteristic
204C23×C67—21no—
205C23×C67—21no—
206Q8×C67—21no—
207Q8×C67—21no—
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—41no—
211Q8×C67—41no—
212Q8×C67—41no—
213Q8×C67—41no—
214Q8×C67—81no—
215Q8×C67—81no—
216D4×C67—21no—
217D4×C67—21no—
218D4×C67—21no—
219D4×C67—21no—
220D4×C67—161no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
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—161no—
236C2×C4×C67C22×D411yescharacteristic
237C2×C4×C67C22×D411yescharacteristic
238C2×C4×C67H(5,𝔽2)11yescharacteristic
239C8×C67—21no—
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—41no—
244C8×C67—41no—
245C8×C67—41no—
246C8×C67—41no—
247C8×C67—41no—
248C22×C67—41no—
249C22×C67—41no—
250C22×C67—41no—
251C22×C67—41no—
252C22×C67—41no—
253C22×C67—41no—
254C22×C67D4211yescharacteristic
255C4×C67—21no—
256C4×C67—41no—
257C4×C67—41no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67—41no—
261C4×C67—41no—
262C4×C67—81no—
263C4×C67—81no—
264C4×C67—81no—
265C4×C67—81no—
266C4×C67#57511yescharacteristic
267C4×C67#57511yescharacteristic
268C4×C67#57811yescharacteristic
269C4×C67#57811yescharacteristic
270#20127C6711yesmaximal, characteristic, direct product factor, Sylow
271C2×C67—81no—
272C2×C67—81no—
273C2×C67#326911yescharacteristic
274C2×C67#326911yescharacteristic
275C2×C67#329011yescharacteristic
276#3372C2×C6711yescharacteristic, complemented
277#3364C2×C6711yescharacteristic, complemented
278#3357C2×C6711yescharacteristic
279#3340C2×C6711yescharacteristic
280#1718C2×C6711yescharacteristic, complemented
281#1699C2×C6711yescharacteristic, complemented
282#1692C2×C6711yescharacteristic, complemented
283#1683C2×C6711yescharacteristic, complemented
284#1676C2×C6711yescharacteristic, complemented
285#1646C2×C6711yescharacteristic
286#1644C2×C6711yescharacteristic, complemented
287#1638C2×C6711yescharacteristic, complemented
288#1636C2×C6711yescharacteristic, complemented
289#1588C2×C6711yescharacteristic, complemented
290#1585C2×C6711yescharacteristic, complemented
291C67#2012711yescharacteristic, direct product factor, Sylow
292#557C22×C6711yescharacteristic
293#556C22×C6711yescharacteristic
294#555C22×C6711yescharacteristic
295#554C22×C6711yescharacteristic
296#500C22×C6711yescharacteristic
297#498C22×C6711yescharacteristic
298#491—21no—
299#488—21no—
300#487C22×C6711yescharacteristic
301#487C22×C6711yescharacteristic
302#487C22×C6711yescharacteristic
303#487C22×C6711yescharacteristic
304#485—21no—
305#485—21no—
306#482—21no—
307#480—21no—
308#479—21no—
309#479—21no—
310#477C22×C6711yescharacteristic
311#477C22×C6711yescharacteristic
312#477C22×C6711yescharacteristic
313#476C22×C6711yescharacteristic
314#474C22×C6711yescharacteristic
315#474C22×C6711yescharacteristic
316#468—21no—
317#465—21no—
318#464—21no—
319#462C22×C6711yescharacteristic
320#461—21no—
321#461C22×C6711yescharacteristic
322#446C22×C6711yescharacteristic
323C8×Q8C22×C6711yescharacteristic
324#440—21no—
325C4×Q16—21no—
326C4×SD16C22×C6711yescharacteristic
327C4×SD16C22×C6711yescharacteristic
328#436C22×C6711yescharacteristic
329C8×D4C22×C6711yescharacteristic
330#428—21no—
331#427—21no—
332#335C22×C6711yescharacteristic
333#335C22×C6711yescharacteristic
334#333C22×C6711yescharacteristic
335#333C22×C6711yescharacteristic
336#332C22×C6711yescharacteristic
337#332C22×C6711yescharacteristic
338Q8⋊C8C22×C6711yescharacteristic
339Q8⋊C8C22×C6711yescharacteristic
340D4⋊C8C22×C6711yescharacteristic
341D4⋊C8C22×C6711yescharacteristic
342#322C22×C6711yescharacteristic
343C2×Q16—21no—
344C2×SD16—21no—
345C2×SD16D4×C6711yescharacteristic
346C2×SD16D4×C6711yescharacteristic
347C2×M16—21no—
348C22×C8—21no—
349C4⋊Q8—21no—
350C4⋊Q8C23×C6711yescharacteristic
351C4⋊Q8C23×C6711yescharacteristic
352C4⋊Q8C23×C6711yescharacteristic
353C4⋊Q8C23×C6711yescharacteristic
354#126—41no—
355#125—21no—
356#125—21no—
357#125—41no—
358#124—21no—
359#124—41no—
360C22⋊Q8—21no—
361C22⋊Q8—21no—
362C22⋊Q8—41no—
363#121—21no—
364#121—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8C23×C6711yescharacteristic
369C4×Q8C23×C6711yescharacteristic
370C4×D4—21no—
371C4×D4C23×C6711yescharacteristic
372C4×D4C23×C6711yescharacteristic
373#117—21no—
374#117—21no—
375#117—21no—
376C2×C4⋊C4—21no—
377#107—21no—
378#107—21no—
379#107—21no—
380#107—21no—
381#107—21no—
382#107—21no—
383#107—21no—
384#106—21no—
385#106—21no—
386#106—21no—
387#106D4×C6711yescharacteristic
388#106D4×C6711yescharacteristic
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8C23×C6711yescharacteristic
392C4⋊C8C23×C6711yescharacteristic
393C4⋊C8C23×C6711yescharacteristic
394C4⋊C8C23×C6711yescharacteristic
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#102—21no—
409#102—21no—
410#102—21no—
411#102D4×C6711yescharacteristic
412#102D4×C6711yescharacteristic
413C22⋊C8—21no—
414C22⋊C8—21no—
415#97—21no—
416C4×C8—21no—
417C4×C8C23×C6711yescharacteristic
418C4×C8C23×C6711yescharacteristic
419C4×C8C23×C6711yescharacteristic
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8C2×D4×C6711yescharacteristic
425C2×Q8C2×D4×C6711yescharacteristic
426C2×D4—41no—
427C2×D4C2×D4×C6711yescharacteristic
428C2×D4C2×D4×C6711yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432Q16—42no—
433SD16—41no—
434SD16—41no—
435SD16—42no—
436M16—42no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—42no—
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—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4C2×D4×C6711yescharacteristic
464C4⋊C4C2×D4×C6711yescharacteristic
465C4⋊C4C2×D4×C6711yescharacteristic
466C4⋊C4C2×D4×C6711yescharacteristic
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42C24×C6711yescharacteristic
480C23—21no—
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—21no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—81no—
491Q8—81no—
492D4—21no—
493D4—21no—
494D4—21no—
495D4—21no—
496D4—161no—
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—41no—
510C2×C4—41no—
511C2×C4—161no—
512C2×C4C22×D4×C6711yescharacteristic
513C2×C4C22×D4×C6711yescharacteristic
514C2×C4H(5,𝔽2)×C6711yescharacteristic
515C8—21no—
516C8—21no—
517C8—21no—
518C8—21no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—41no—
523C8—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22D42×C6711yescharacteristic
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4#575×C6711yescharacteristic
543C4#575×C6711yescharacteristic
544C4#578×C6711yescharacteristic
545C4#578×C6711yescharacteristic
546C2—81no—
547C2—81no—
548C2C67×#326911yescharacteristic
549C2C67×#326911yescharacteristic
550C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#555×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×#3372C2semidirect
2. C67×#3364C2semidirect
5. C67×#1718C2semidirect
6. C67×#1699C2semidirect
7. C67×#1692C2semidirect
8. C67×#1683C2semidirect
9. C67×#1676C2semidirect
11. C67×#1644C2semidirect
12. C67×#1638C2semidirect
13. C67×#1636C2semidirect
14. C67×#1588C2semidirect
15. C67×#1585C2semidirect
270. #20127C67direct (N × H)
276. #3372C2×C67semidirect
277. #3364C2×C67semidirect
280. #1718C2×C67semidirect
281. #1699C2×C67semidirect
282. #1692C2×C67semidirect
283. #1683C2×C67semidirect
284. #1676C2×C67semidirect
286. #1644C2×C67semidirect
287. #1638C2×C67semidirect
288. #1636C2×C67semidirect
289. #1588C2×C67semidirect
290. #1585C2×C67semidirect
291. C67#20127direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33722
C67×#33642
C67×#33572
C67×#33402
C67×#17182
C67×#16992
C67×#16922
C67×#16832
C67×#16762
C67×#16462
C67×#16442
C67×#16382
C67×#16362
C67×#15882
C67×#15852
#2012767