Finite Group Databasebeta

← C67×#19939 #30699190

Subgroups — C67×#19939 #30699190

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3369C211yesmaximal, characteristic, complemented
2C67×#3368C211yesmaximal, characteristic
3C67×#3363C211yesmaximal, characteristic, complemented
4C67×#3304C211yesmaximal, characteristic
5C67×#1708C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1702C211yesmaximal, characteristic, complemented
8C67×#1683C211yesmaximal, characteristic, complemented
9C67×#1682C211yesmaximal, characteristic, complemented
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1652C211yesmaximal, characteristic, complemented
12C67×#1631C211yesmaximal, characteristic
13C67×#1623C211yesmaximal, characteristic, complemented
14C67×#1591C211yesmaximal, characteristic, complemented
15C8×SD16×C67C211yesmaximal, characteristic, complemented
16Q82×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19#550×C67C2211yescharacteristic
20#498×C67C2211yescharacteristic
21#492×C67C2211yescharacteristic
22#490×C67—21no—
23#489×C67—21no—
24#487×C67—21no—
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#482×C67—21no—
30#478×C67—21no—
31#477×C67—21no—
32#477×C67C2211yescharacteristic
33#476×C67C2211yescharacteristic
34#475×C67—21no—
35#475×C67C2211yescharacteristic
36#474×C67C2211yescharacteristic
37#465×C67—21no—
38#464×C67—21no—
39#462×C67—21no—
40#462×C67C2211yescharacteristic
41#462×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#440×C67—21no—
51C4×Q16×C67—21no—
52C4×Q16×C67C2211yescharacteristic
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
66C82×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×Q16×C67—21no—
69C2×Q16×C67—21no—
70C2×Q16×C67D411yescharacteristic
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—41no—
74C2×SD16×C67D411yescharacteristic
75C22×C8×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67—41no—
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#127×C67—21no—
84#127×C67C2311yescharacteristic
85#126×C67—41no—
86#125×C67—21no—
87#125×C67—41no—
88#124×C67—21no—
89C22⋊Q8×C67—21no—
90#121×C67—21no—
91#121×C67—41no—
92C4×Q8×C67—21no—
93C4×Q8×C67—21no—
94C4×Q8×C67—21no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67C2311yescharacteristic
98C4×Q8×C67C2311yescharacteristic
99C4×Q8×C67C2311yescharacteristic
100C4×D4×C67—21no—
101C4×D4×C67C2311yescharacteristic
102#117×C67—21no—
103#107×C67—21no—
104#107×C67—21no—
105#107×C67D411yescharacteristic
106#106×C67—21no—
107#106×C67—21no—
108#106×C67—41no—
109#106×C67D411yescharacteristic
110C4⋊C8×C67—21no—
111C4⋊C8×C67—21no—
112C4⋊C8×C67—21no—
113C4⋊C8×C67C2311yescharacteristic
114C4⋊C8×C67C2311yescharacteristic
115C4⋊C8×C67C2311yescharacteristic
116C4⋊C8×C67C2311yescharacteristic
117#103×C67—21no—
118#103×C67—21no—
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×C67D411yescharacteristic
126#103×C67D411yescharacteristic
127#103×C67D411yescharacteristic
128#102×C67—21no—
129#102×C67—21no—
130#102×C67—21no—
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—41no—
162Q16×C67—42no—
163Q16×C67—42no—
164SD16×C67—22no—
165SD16×C67—42no—
166SD16×C67—82no—
167SD16×C67—81no—
168C2×C8×C67—21no—
169C2×C8×C67—21no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—81no—
174C2×C8×C67C2×D411yescharacteristic
175C2×C8×C67C2×D411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
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—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—41no—
188C4⋊C4×C67—41no—
189C4⋊C4×C67—41no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67C2×D411yescharacteristic
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C22⋊C4×C67—21no—
198C22⋊C4×C67—41no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67—41no—
205C42×C67—41no—
206C42×C67—41no—
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209C23×C67—41no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—41no—
216Q8×C67—41no—
217Q8×C67—41no—
218Q8×C67—41no—
219Q8×C67—81no—
220Q8×C67—161no—
221Q8×C67#13512yes—
222D4×C67—21no—
223D4×C67—21no—
224D4×C67—41no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—81no—
229D4×C67—82no—
230D4×C67—161no—
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×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67H(5,𝔽2)11yescharacteristic
247C8×C67—22no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C8×C67—41no—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—81no—
260C22×C67—81no—
261C22×C67—81no—
262C22×C67D4211yescharacteristic
263C4×C67—22no—
264C4×C67—21no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—81no—
272C4×C67—81no—
273C4×C67—81no—
274C4×C67C2×#13511yescharacteristic
275C4×C67#57511yescharacteristic
276C4×C67#57711yescharacteristic
277C4×C67#57811yescharacteristic
278#19939C6711yesmaximal, characteristic, direct product factor, Sylow
279C2×C67—81no—
280C2×C67—161no—
281C2×C67#327511yescharacteristic
282C2×C67#328511yescharacteristic
283C2×C67#329111yescharacteristic
284#3369C2×C6711yescharacteristic, complemented
285#3368C2×C6711yescharacteristic
286#3363C2×C6711yescharacteristic, complemented
287#3304C2×C6711yescharacteristic
288#1708C2×C6711yescharacteristic, complemented
289#1704C2×C6711yescharacteristic, complemented
290#1702C2×C6711yescharacteristic, complemented
291#1683C2×C6711yescharacteristic, complemented
292#1682C2×C6711yescharacteristic, complemented
293#1653C2×C6711yescharacteristic, complemented
294#1652C2×C6711yescharacteristic, complemented
295#1631C2×C6711yescharacteristic
296#1623C2×C6711yescharacteristic, complemented
297#1591C2×C6711yescharacteristic, complemented
298C8×SD16C2×C6711yescharacteristic, complemented
299C67#1993911yescharacteristic, direct product factor, Sylow
300Q82C22×C6711yescharacteristic
301#557C22×C6711yescharacteristic
302#556C22×C6711yescharacteristic
303#550C22×C6711yescharacteristic
304#498C22×C6711yescharacteristic
305#492C22×C6711yescharacteristic
306#490—21no—
307#489—21no—
308#487—21no—
309#487C22×C6711yescharacteristic
310#487C22×C6711yescharacteristic
311#486C22×C6711yescharacteristic
312#486C22×C6711yescharacteristic
313#482—21no—
314#478—21no—
315#477—21no—
316#477C22×C6711yescharacteristic
317#476C22×C6711yescharacteristic
318#475—21no—
319#475C22×C6711yescharacteristic
320#474C22×C6711yescharacteristic
321#465—21no—
322#464—21no—
323#462—21no—
324#462C22×C6711yescharacteristic
325#462C22×C6711yescharacteristic
326#460—21no—
327#460C22×C6711yescharacteristic
328#460C22×C6711yescharacteristic
329#452—21no—
330#446C22×C6711yescharacteristic
331#446C22×C6711yescharacteristic
332C8×Q8C22×C6711yescharacteristic
333#441—21no—
334#440—21no—
335C4×Q16—21no—
336C4×Q16C22×C6711yescharacteristic
337C4×SD16C22×C6711yescharacteristic
338C8×D4C22×C6711yescharacteristic
339#416—21no—
340#335C22×C6711yescharacteristic
341#334C22×C6711yescharacteristic
342#333C22×C6711yescharacteristic
343#332C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345#331C22×C6711yescharacteristic
346Q8⋊C8C22×C6711yescharacteristic
347Q8⋊C8C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349D4⋊C8C22×C6711yescharacteristic
350C82C22×C6711yescharacteristic
351C2×#41—21no—
352C2×Q16—21no—
353C2×Q16—21no—
354C2×Q16D4×C6711yescharacteristic
355C2×SD16—21no—
356C2×SD16—21no—
357C2×SD16—41no—
358C2×SD16D4×C6711yescharacteristic
359C22×C8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—41no—
364C4⋊Q8C23×C6711yescharacteristic
365C4⋊Q8C23×C6711yescharacteristic
366C4⋊Q8C23×C6711yescharacteristic
367#127—21no—
368#127C23×C6711yescharacteristic
369#126—41no—
370#125—21no—
371#125—41no—
372#124—21no—
373C22⋊Q8—21no—
374#121—21no—
375#121—41no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8C23×C6711yescharacteristic
382C4×Q8C23×C6711yescharacteristic
383C4×Q8C23×C6711yescharacteristic
384C4×D4—21no—
385C4×D4C23×C6711yescharacteristic
386#117—21no—
387#107—21no—
388#107—21no—
389#107D4×C6711yescharacteristic
390#106—21no—
391#106—21no—
392#106—41no—
393#106D4×C6711yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8C23×C6711yescharacteristic
398C4⋊C8C23×C6711yescharacteristic
399C4⋊C8C23×C6711yescharacteristic
400C4⋊C8C23×C6711yescharacteristic
401#103—21no—
402#103—21no—
403#103—21no—
404#103—21no—
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103D4×C6711yescharacteristic
410#103D4×C6711yescharacteristic
411#103D4×C6711yescharacteristic
412#102—21no—
413#102—21no—
414#102—21no—
415#102—21no—
416#102—21no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102D4×C6711yescharacteristic
421C22⋊C8—21no—
422#97—21no—
423#97—21no—
424C4×C8—21no—
425C4×C8C23×C6711yescharacteristic
426C4×C8C23×C6711yescharacteristic
427C4×C8C23×C6711yescharacteristic
428#41—42no—
429#41—81no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8C2×D4×C6711yescharacteristic
435C2×Q8C2×D4×C6711yescharacteristic
436C2×Q8C2×D4×C6711yescharacteristic
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4C2×D4×C6711yescharacteristic
443C22×C4—21no—
444C22×C4—41no—
445Q16—41no—
446Q16—42no—
447Q16—42no—
448SD16—22no—
449SD16—42no—
450SD16—82no—
451SD16—81no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—81no—
458C2×C8C2×D4×C6711yescharacteristic
459C2×C8C2×D4×C6711yescharacteristic
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4C2×D4×C6711yescharacteristic
478C4⋊C4C2×D4×C6711yescharacteristic
479C4⋊C4C2×D4×C6711yescharacteristic
480C4⋊C4C2×D4×C6711yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42C24×C6711yescharacteristic
492C23—21no—
493C23—41no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—81no—
504Q8—161no—
505Q8#135×C6712yes—
506D4—21no—
507D4—21no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—81no—
513D4—82no—
514D4—161no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4C22×D4×C6711yescharacteristic
529C2×C4C22×D4×C6711yescharacteristic
530C2×C4H(5,𝔽2)×C6711yescharacteristic
531C8—22no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22D42×C6711yescharacteristic
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4C2×#135×C6711yescharacteristic
559C4#575×C6711yescharacteristic
560C4#577×C6711yescharacteristic
561C4#578×C6711yescharacteristic
562C2—81no—
563C2—161no—
564C2C67×#327511yescharacteristic
565C2C67×#328511yescharacteristic
566C2C67×#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
1. C67×#3369C2semidirect
3. C67×#3363C2semidirect
5. C67×#1708C2semidirect
6. C67×#1704C2semidirect
7. C67×#1702C2semidirect
8. C67×#1683C2semidirect
9. C67×#1682C2semidirect
10. C67×#1653C2semidirect
11. C67×#1652C2semidirect
13. C67×#1623C2semidirect
14. C67×#1591C2semidirect
15. C8×SD16×C67C2semidirect
278. #19939C67direct (N × H)
284. #3369C2×C67semidirect
286. #3363C2×C67semidirect
288. #1708C2×C67semidirect
289. #1704C2×C67semidirect
290. #1702C2×C67semidirect
291. #1683C2×C67semidirect
292. #1682C2×C67semidirect
293. #1653C2×C67semidirect
294. #1652C2×C67semidirect
296. #1623C2×C67semidirect
297. #1591C2×C67semidirect
298. C8×SD16C2×C67semidirect
299. C67#19939direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33692
C67×#33682
C67×#33632
C67×#33042
C67×#17082
C67×#17042
C67×#17022
C67×#16832
C67×#16822
C67×#16532
C67×#16522
C67×#16312
C67×#16232
C67×#15912
C8×SD16×C672
#1993967