Finite Group Databasebeta

← C67×#20148 #30699399

Subgroups — C67×#20148 #30699399

Order 17152.

604 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3396C211yesmaximal, characteristic, complemented
2C67×#3394C211yesmaximal, characteristic, complemented
3C67×#3373C211yesmaximal, characteristic, complemented
4C67×#3328C211yesmaximal, characteristic, complemented
5C67×#1717C211yesmaximal, characteristic, complemented
6C67×#1687C211yesmaximal, characteristic
7C67×#1686C211yesmaximal, characteristic, complemented
8C67×#1682C211yesmaximal, characteristic, complemented
9C67×#1680C211yesmaximal, characteristic, complemented
10C67×#1640C211yesmaximal, characteristic, complemented
11C67×#1640C211yesmaximal, characteristic, complemented
12C67×#1628C211yesmaximal, characteristic, complemented
13C67×#1627C211yesmaximal, characteristic, complemented
14C67×#1588C211yesmaximal, characteristic, complemented
15C67×#1587C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17#559×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#495×C67—21no—
21#495×C67—21no—
22#493×C67C2211yescharacteristic
23#492×C67C2211yescharacteristic
24#489×C67—21no—
25#487×C67C2211yescharacteristic
26#486×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#485×C67—21no—
30#476×C67—21no—
31#475×C67C2211yescharacteristic
32#475×C67C2211yescharacteristic
33#470×C67—21no—
34#464×C67—21no—
35#463×C67—21no—
36#463×C67—21no—
37#463×C67—21no—
38#463×C67—21no—
39#462×C67C2211yescharacteristic
40#462×C67C2211yescharacteristic
41#460×C67C2211yescharacteristic
42#460×C67C2211yescharacteristic
43#459×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#451×C67—21no—
46#446×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#442×C67—21no—
50C4×Q16×C67C2211yescharacteristic
51C4×SD16×C67—21no—
52C4×SD16×C67C2211yescharacteristic
53C4×D8×C67—21no—
54#436×C67C2211yescharacteristic
55#419×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#332×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#331×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—21no—
75C2×SD16×C67D411yescharacteristic
76C2×D8×C67—21no—
77C2×D8×C67—21no—
78C2×D8×C67—21no—
79C2×D8×C67—21no—
80C2×M16×C67—21no—
81C4⋊Q8×C67—21no—
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84#127×C67C2311yescharacteristic
85#127×C67C2311yescharacteristic
86#126×C67—41no—
87#124×C67—21no—
88#124×C67—21no—
89#124×C67—21no—
90#124×C67—21no—
91#124×C67—21no—
92#123×C67—41no—
93#123×C67—41no—
94C22⋊Q8×C67—21no—
95C22⋊Q8×C67—21no—
96C22⋊Q8×C67—41no—
97C4×Q8×C67—21no—
98C4×Q8×C67C2311yescharacteristic
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×D4×C67—21no—
102C4×D4×C67—21no—
103C4×D4×C67—21no—
104C4×D4×C67—21no—
105C4×D4×C67—21no—
106C4×D4×C67C2311yescharacteristic
107#117×C67—21no—
108#107×C67D411yescharacteristic
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×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—41no—
135#102×C67—41no—
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138#97×C67—21no—
139C4×C8×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—41no—
147C2×Q8×C67—42no—
148C2×Q8×C67—81no—
149C2×Q8×C67C2×D411yescharacteristic
150C2×Q8×C67C2×D411yescharacteristic
151C2×Q8×C67C2×D411yescharacteristic
152C2×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67—21no—
155C2×D4×C67—21no—
156C2×D4×C67—21no—
157C2×D4×C67—21no—
158C2×D4×C67C2×D411yescharacteristic
159C22×C4×C67—21no—
160C22×C4×C67—41no—
161C22×C4×C67—41no—
162C22×C4×C67—41no—
163Q16×C67—41no—
164Q16×C67—81no—
165Q16×C67—81no—
166SD16×C67—41no—
167SD16×C67—42no—
168SD16×C67—81no—
169SD16×C67—81no—
170SD16×C67—81no—
171D8×C67—42no—
172D8×C67—42no—
173D8×C67—81no—
174D8×C67—81no—
175M16×C67—81no—
176C2×C8×C67—21no—
177C2×C8×C67—21no—
178C2×C8×C67—21no—
179C2×C8×C67—21no—
180C2×C8×C67—21no—
181C2×C8×C67C2×D411yescharacteristic
182C2×C8×C67C2×D411yescharacteristic
183C2×C8×C67C2×D411yescharacteristic
184C2×C8×C67C2×D411yescharacteristic
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67—41no—
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C4⋊C4×C67C2×D411yescharacteristic
198C22⋊C4×C67—21no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C22⋊C4×C67—41no—
204C22⋊C4×C67—41no—
205C22⋊C4×C67—41no—
206C22⋊C4×C67—41no—
207C22⋊C4×C67—41no—
208C42×C67—21no—
209C42×C67—21no—
210C42×C67—21no—
211C42×C67—41no—
212C42×C67C2411yescharacteristic
213C23×C67—21no—
214C23×C67—41no—
215C23×C67—41no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—21no—
219Q8×C67—21no—
220Q8×C67—21no—
221Q8×C67—21no—
222Q8×C67—41no—
223Q8×C67—41no—
224Q8×C67—41no—
225Q8×C67—41no—
226Q8×C67—82no—
227D4×C67—21no—
228D4×C67—21no—
229D4×C67—41no—
230D4×C67—41no—
231D4×C67—41no—
232D4×C67—41no—
233D4×C67—41no—
234D4×C67—41no—
235D4×C67—41no—
236D4×C67—41no—
237D4×C67—81no—
238D4×C67—81no—
239C2×C4×C67—21no—
240C2×C4×C67—21no—
241C2×C4×C67—21no—
242C2×C4×C67—21no—
243C2×C4×C67—21no—
244C2×C4×C67—21no—
245C2×C4×C67—21no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—41no—
250C2×C4×C67—82no—
251C2×C4×C67—161no—
252C2×C4×C67—161no—
253C2×C4×C67C22×D411yescharacteristic
254C2×C4×C67C22×D411yescharacteristic
255C2×C4×C67H(5,𝔽2)11yescharacteristic
256C8×C67—21no—
257C8×C67—21no—
258C8×C67—21no—
259C8×C67—21no—
260C8×C67—41no—
261C8×C67—41no—
262C8×C67—41no—
263C8×C67—41no—
264C8×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—81no—
269C22×C67—81no—
270C22×C67—81no—
271C22×C67—81no—
272C22×C67—81no—
273C22×C67—81no—
274C22×C67D4211yescharacteristic
275C4×C67—21no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67—41no—
281C4×C67—41no—
282C4×C67—41no—
283C4×C67—81no—
284C4×C67—81no—
285C4×C67#57511yescharacteristic
286C4×C67#57511yescharacteristic
287C4×C67#57611yescharacteristic
288C4×C67#57711yescharacteristic
289#20148C6711yesmaximal, characteristic, direct product factor, Sylow
290C2×C67—81no—
291C2×C67—161no—
292C2×C67—161no—
293C2×C67#326811yescharacteristic
294C2×C67#327111yescharacteristic
295C2×C67#329011yescharacteristic
296#3396C2×C6711yescharacteristic, complemented
297#3394C2×C6711yescharacteristic, complemented
298#3373C2×C6711yescharacteristic, complemented
299#3328C2×C6711yescharacteristic, complemented
300#1717C2×C6711yescharacteristic, complemented
301#1687C2×C6711yescharacteristic
302#1686C2×C6711yescharacteristic, complemented
303#1682C2×C6711yescharacteristic, complemented
304#1680C2×C6711yescharacteristic, complemented
305#1640C2×C6711yescharacteristic, complemented
306#1640C2×C6711yescharacteristic, complemented
307#1628C2×C6711yescharacteristic, complemented
308#1627C2×C6711yescharacteristic, complemented
309#1588C2×C6711yescharacteristic, complemented
310#1587C2×C6711yescharacteristic, complemented
311C67#2014811yescharacteristic, direct product factor, Sylow
312#559C22×C6711yescharacteristic
313#559C22×C6711yescharacteristic
314#556C22×C6711yescharacteristic
315D4×Q8C22×C6711yescharacteristic
316#495—21no—
317#495—21no—
318#493C22×C6711yescharacteristic
319#492C22×C6711yescharacteristic
320#489—21no—
321#487C22×C6711yescharacteristic
322#486C22×C6711yescharacteristic
323#486C22×C6711yescharacteristic
324#486C22×C6711yescharacteristic
325#485—21no—
326#476—21no—
327#475C22×C6711yescharacteristic
328#475C22×C6711yescharacteristic
329#470—21no—
330#464—21no—
331#463—21no—
332#463—21no—
333#463—21no—
334#463—21no—
335#462C22×C6711yescharacteristic
336#462C22×C6711yescharacteristic
337#460C22×C6711yescharacteristic
338#460C22×C6711yescharacteristic
339#459C22×C6711yescharacteristic
340#459C22×C6711yescharacteristic
341#451—21no—
342#446C22×C6711yescharacteristic
343C8×Q8C22×C6711yescharacteristic
344C8×Q8C22×C6711yescharacteristic
345#442—21no—
346C4×Q16C22×C6711yescharacteristic
347C4×SD16—21no—
348C4×SD16C22×C6711yescharacteristic
349C4×D8—21no—
350#436C22×C6711yescharacteristic
351#419—21no—
352#335C22×C6711yescharacteristic
353#334C22×C6711yescharacteristic
354#332C22×C6711yescharacteristic
355#332C22×C6711yescharacteristic
356#331C22×C6711yescharacteristic
357#331C22×C6711yescharacteristic
358Q8⋊C8C22×C6711yescharacteristic
359Q8⋊C8C22×C6711yescharacteristic
360Q8⋊C8C22×C6711yescharacteristic
361D4⋊C8C22×C6711yescharacteristic
362#322C22×C6711yescharacteristic
363C22×Q8—21no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×Q16D4×C6711yescharacteristic
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16—21no—
371C2×SD16D4×C6711yescharacteristic
372C2×D8—21no—
373C2×D8—21no—
374C2×D8—21no—
375C2×D8—21no—
376C2×M16—21no—
377C4⋊Q8—21no—
378C4⋊Q8C23×C6711yescharacteristic
379C4⋊Q8C23×C6711yescharacteristic
380#127C23×C6711yescharacteristic
381#127C23×C6711yescharacteristic
382#126—41no—
383#124—21no—
384#124—21no—
385#124—21no—
386#124—21no—
387#124—21no—
388#123—41no—
389#123—41no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—41no—
393C4×Q8—21no—
394C4×Q8C23×C6711yescharacteristic
395C4×Q8C23×C6711yescharacteristic
396C4×Q8C23×C6711yescharacteristic
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4C23×C6711yescharacteristic
403#117—21no—
404#107D4×C6711yescharacteristic
405#106—21no—
406#106—21no—
407#106D4×C6711yescharacteristic
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8C23×C6711yescharacteristic
412C4⋊C8C23×C6711yescharacteristic
413C4⋊C8C23×C6711yescharacteristic
414C4⋊C8C23×C6711yescharacteristic
415#103—21no—
416#103—21no—
417#103—21no—
418#103—21no—
419#103—21no—
420#103—21no—
421#103D4×C6711yescharacteristic
422#103D4×C6711yescharacteristic
423#103D4×C6711yescharacteristic
424#102—21no—
425#102—21no—
426#102—21no—
427#102—21no—
428#102—21no—
429#102—21no—
430#102—41no—
431#102—41no—
432#102D4×C6711yescharacteristic
433C22⋊C8—21no—
434#97—21no—
435C4×C8—21no—
436C4×C8—21no—
437C4×C8C23×C6711yescharacteristic
438C4×C8C23×C6711yescharacteristic
439C4×C8C23×C6711yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—41no—
443C2×Q8—42no—
444C2×Q8—81no—
445C2×Q8C2×D4×C6711yescharacteristic
446C2×Q8C2×D4×C6711yescharacteristic
447C2×Q8C2×D4×C6711yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4C2×D4×C6711yescharacteristic
455C22×C4—21no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459Q16—41no—
460Q16—81no—
461Q16—81no—
462SD16—41no—
463SD16—42no—
464SD16—81no—
465SD16—81no—
466SD16—81no—
467D8—42no—
468D8—42no—
469D8—81no—
470D8—81no—
471M16—81no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8C2×D4×C6711yescharacteristic
478C2×C8C2×D4×C6711yescharacteristic
479C2×C8C2×D4×C6711yescharacteristic
480C2×C8C2×D4×C6711yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4C2×D4×C6711yescharacteristic
491C4⋊C4C2×D4×C6711yescharacteristic
492C4⋊C4C2×D4×C6711yescharacteristic
493C4⋊C4C2×D4×C6711yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42C24×C6711yescharacteristic
509C23—21no—
510C23—41no—
511C23—41no—
512Q8—21no—
513Q8—21no—
514Q8—21no—
515Q8—21no—
516Q8—21no—
517Q8—21no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—82no—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—81no—
534D4—81no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—82no—
547C2×C4—161no—
548C2×C4—161no—
549C2×C4C22×D4×C6711yescharacteristic
550C2×C4C22×D4×C6711yescharacteristic
551C2×C4H(5,𝔽2)×C6711yescharacteristic
552C8—21no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22D42×C6711yescharacteristic
571C4—21no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4#575×C6711yescharacteristic
582C4#575×C6711yescharacteristic
583C4#576×C6711yescharacteristic
584C4#577×C6711yescharacteristic
585C2—81no—
586C2—161no—
587C2—161no—
588C2C67×#326811yescharacteristic
589C2C67×#327111yescharacteristic
590C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#1687C2
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×#3396C2semidirect
2. C67×#3394C2semidirect
3. C67×#3373C2semidirect
4. C67×#3328C2semidirect
5. C67×#1717C2semidirect
7. C67×#1686C2semidirect
8. C67×#1682C2semidirect
9. C67×#1680C2semidirect
10. C67×#1640C2semidirect
11. C67×#1640C2semidirect
12. C67×#1628C2semidirect
13. C67×#1627C2semidirect
14. C67×#1588C2semidirect
15. C67×#1587C2semidirect
289. #20148C67direct (N × H)
296. #3396C2×C67semidirect
297. #3394C2×C67semidirect
298. #3373C2×C67semidirect
299. #3328C2×C67semidirect
300. #1717C2×C67semidirect
302. #1686C2×C67semidirect
303. #1682C2×C67semidirect
304. #1680C2×C67semidirect
305. #1640C2×C67semidirect
306. #1640C2×C67semidirect
307. #1628C2×C67semidirect
308. #1627C2×C67semidirect
309. #1588C2×C67semidirect
310. #1587C2×C67semidirect
311. C67#20148direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33962
C67×#33942
C67×#33732
C67×#33282
C67×#17172
C67×#16872
C67×#16862
C67×#16822
C67×#16802
C67×#16402
C67×#16282
C67×#16272
C67×#15882
C67×#15872
#2014867