Finite Group Databasebeta

← C67×#19897 #30699148

Subgroups — C67×#19897 #30699148

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3392C211yesmaximal, characteristic
2C67×#3387C211yesmaximal, characteristic, complemented
3Q8×SD16×C67C211yesmaximal, characteristic, complemented
4C67×#3278C211yesmaximal, characteristic
5C67×#1705C211yesmaximal, characteristic, complemented
6C67×#1691C211yesmaximal, characteristic, complemented
7C67×#1686C211yesmaximal, characteristic, complemented
8C67×#1672C211yesmaximal, characteristic, complemented
9C67×#1666C211yesmaximal, characteristic, complemented
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1651C211yesmaximal, characteristic, complemented
12C67×#1631C211yesmaximal, characteristic
13C67×#1623C211yesmaximal, characteristic, complemented
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×Q16×C67C211yesmaximal, characteristic, complemented
16C2×#137×C67—21no—
17Q82×C67C2211yescharacteristic
18#557×C67C2211yescharacteristic
19#550×C67C2211yescharacteristic
20D4×Q8×C67C2211yescharacteristic
21#501×C67—21no—
22#498×C67—21no—
23#498×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#498×C67C2211yescharacteristic
26#492×C67—21no—
27#492×C67C2211yescharacteristic
28#492×C67C2211yescharacteristic
29#492×C67C2211yescharacteristic
30#479×C67—21no—
31#477×C67C2211yescharacteristic
32#476×C67—21no—
33#476×C67C2211yescharacteristic
34#475×C67—21no—
35#475×C67—21no—
36#475×C67C2211yescharacteristic
37#475×C67C2211yescharacteristic
38#468×C67—21no—
39#462×C67C2211yescharacteristic
40#461×C67—21no—
41#461×C67C2211yescharacteristic
42#460×C67—21no—
43#460×C67C2211yescharacteristic
44#460×C67C2211yescharacteristic
45#452×C67—21no—
46#448×C67—21no—
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49C8×Q8×C67C2211yescharacteristic
50#441×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67—21no—
53C4×SD16×C67—21no—
54C4×SD16×C67C2211yescharacteristic
55#436×C67C2211yescharacteristic
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—
68C22×Q8×C67—21no—
69#137×C67—44no—
70C2×Q16×C67—21no—
71C2×Q16×C67D411yescharacteristic
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—21no—
75C2×SD16×C67—21no—
76C2×SD16×C67—21no—
77C2×SD16×C67—21no—
78C2×SD16×C67—21no—
79C2×SD16×C67—41no—
80C2×SD16×C67D411yescharacteristic
81C2×M16×C67—21no—
82C4⋊Q8×C67—21no—
83C4⋊Q8×C67—21no—
84C4⋊Q8×C67—21no—
85C4⋊Q8×C67—21no—
86C4⋊Q8×C67—41no—
87C4⋊Q8×C67C2311yescharacteristic
88C4⋊Q8×C67C2311yescharacteristic
89C4⋊Q8×C67C2311yescharacteristic
90#127×C67—21no—
91#127×C67C2311yescharacteristic
92#125×C67—21no—
93#125×C67—41no—
94C22⋊Q8×C67—21no—
95C22⋊Q8×C67—41no—
96#121×C67—21no—
97#121×C67—41no—
98C4×Q8×C67—21no—
99C4×Q8×C67—21no—
100C4×Q8×C67—21no—
101C4×Q8×C67—21no—
102C4×Q8×C67C2311yescharacteristic
103C4×Q8×C67C2311yescharacteristic
104C4×Q8×C67C2311yescharacteristic
105C4×D4×C67—21no—
106C4×D4×C67—21no—
107C4×D4×C67C2311yescharacteristic
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×C67—21no—
115#106×C67—21no—
116#106×C67—21no—
117#106×C67—41no—
118#106×C67D411yescharacteristic
119C4⋊C8×C67—21no—
120C4⋊C8×C67—21no—
121C4⋊C8×C67—21no—
122C4⋊C8×C67C2311yescharacteristic
123C4⋊C8×C67C2311yescharacteristic
124C4⋊C8×C67C2311yescharacteristic
125C4⋊C8×C67C2311yescharacteristic
126#103×C67—21no—
127#103×C67—21no—
128#103×C67—21no—
129#103×C67—21no—
130#103×C67—21no—
131#103×C67D411yescharacteristic
132#103×C67D411yescharacteristic
133#103×C67D411yescharacteristic
134#102×C67—21no—
135#102×C67—21no—
136#102×C67—21no—
137#102×C67D411yescharacteristic
138C22⋊C8×C67—21no—
139#97×C67—21no—
140C4×C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C4×C8×C67C2311yescharacteristic
145#41×C67—42no—
146#41×C67—81no—
147C2×Q8×C67—21no—
148C2×Q8×C67—21no—
149C2×Q8×C67—21no—
150C2×Q8×C67—41no—
151C2×Q8×C67—41no—
152C2×Q8×C67—42no—
153C2×Q8×C67—81no—
154C2×Q8×C67C2×D411yescharacteristic
155C2×Q8×C67C2×D411yescharacteristic
156C2×Q8×C67C2×D411yescharacteristic
157C2×D4×C67—21no—
158C2×D4×C67—21no—
159C2×D4×C67—21no—
160C2×D4×C67—41no—
161C2×D4×C67—41no—
162C2×D4×C67C2×D411yescharacteristic
163C22×C4×C67—21no—
164C22×C4×C67—41no—
165C22×C4×C67—41no—
166Q16×C67—42no—
167Q16×C67SD1614yescomplemented
168SD16×C67—22no—
169SD16×C67—42no—
170SD16×C67—42no—
171SD16×C67—42no—
172SD16×C67—81no—
173SD16×C67—81no—
174SD16×C67—82no—
175SD16×C67—81no—
176SD16×C67—81no—
177M16×C67—42no—
178C2×C8×C67—21no—
179C2×C8×C67—21no—
180C2×C8×C67—21no—
181C2×C8×C67—21no—
182C2×C8×C67—21no—
183C2×C8×C67C2×D411yescharacteristic
184C2×C8×C67C2×D411yescharacteristic
185C2×C8×C67C2×D411yescharacteristic
186C2×C8×C67C2×D411yescharacteristic
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—41no—
197C4⋊C4×C67—41no—
198C4⋊C4×C67—41no—
199C4⋊C4×C67—41no—
200C4⋊C4×C67—41no—
201C4⋊C4×C67—41no—
202C4⋊C4×C67C2×D411yescharacteristic
203C4⋊C4×C67C2×D411yescharacteristic
204C4⋊C4×C67C2×D411yescharacteristic
205C4⋊C4×C67C2×D411yescharacteristic
206C22⋊C4×C67—21no—
207C22⋊C4×C67—41no—
208C22⋊C4×C67—41no—
209C42×C67—21no—
210C42×C67—21no—
211C42×C67—21no—
212C42×C67—41no—
213C42×C67—41no—
214C42×C67C2411yescharacteristic
215C23×C67—21no—
216C23×C67—41no—
217Q8×C67—21no—
218Q8×C67—21no—
219Q8×C67—21no—
220Q8×C67—21no—
221Q8×C67—41no—
222Q8×C67—41no—
223Q8×C67—41no—
224Q8×C67—41no—
225Q8×C67—82no—
226Q8×C67—81no—
227Q8×C67—161no—
228Q8×C67C2×SD1612yes—
229Q8×C67C2×SD1612yes—
230D4×C67—21no—
231D4×C67—21no—
232D4×C67—41no—
233D4×C67—41no—
234D4×C67—41no—
235D4×C67—41no—
236D4×C67—82no—
237D4×C67—81no—
238D4×C67—161no—
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—41no—
251C2×C4×C67—82no—
252C2×C4×C67—82no—
253C2×C4×C67C22×D411yescharacteristic
254C2×C4×C67C22×D411yescharacteristic
255C2×C4×C67H(5,𝔽2)11yescharacteristic
256C8×C67—22no—
257C8×C67—21no—
258C8×C67—21no—
259C8×C67—41no—
260C8×C67—41no—
261C8×C67—41no—
262C8×C67—41no—
263C8×C67C2×SD1612yes—
264C8×C67#13712yescomplemented
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—81no—
269C22×C67—81no—
270C22×C67—81no—
271C22×C67D4211yescharacteristic
272C4×C67—22no—
273C4×C67—22no—
274C4×C67—21no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67—81no—
281C4×C67—81no—
282C4×C67—81no—
283C4×C67C22×SD1611yescharacteristic
284C4×C67C2×#13711yescharacteristic
285C4×C67#57711yescharacteristic
286C4×C67#57711yescharacteristic
287#19897C6711yesmaximal, characteristic, direct product factor, Sylow
288C2×C67—81no—
289C2×C67—161no—
290C2×C67#327811yescharacteristic
291C2×C67D4×SD1611yescharacteristic
292C2×C67#329111yescharacteristic
293#3392C2×C6711yescharacteristic
294#3387C2×C6711yescharacteristic, complemented
295Q8×SD16C2×C6711yescharacteristic, complemented
296#3278C2×C6711yescharacteristic
297#1705C2×C6711yescharacteristic, complemented
298#1691C2×C6711yescharacteristic, complemented
299#1686C2×C6711yescharacteristic, complemented
300#1672C2×C6711yescharacteristic, complemented
301#1666C2×C6711yescharacteristic, complemented
302#1653C2×C6711yescharacteristic, complemented
303#1651C2×C6711yescharacteristic, complemented
304#1631C2×C6711yescharacteristic
305#1623C2×C6711yescharacteristic, complemented
306#1590C2×C6711yescharacteristic, complemented
307C8×Q16C2×C6711yescharacteristic, complemented
308C67#1989711yescharacteristic, direct product factor, Sylow
309C2×#137—21no—
310Q82C22×C6711yescharacteristic
311#557C22×C6711yescharacteristic
312#550C22×C6711yescharacteristic
313D4×Q8C22×C6711yescharacteristic
314#501—21no—
315#498—21no—
316#498C22×C6711yescharacteristic
317#498C22×C6711yescharacteristic
318#498C22×C6711yescharacteristic
319#492—21no—
320#492C22×C6711yescharacteristic
321#492C22×C6711yescharacteristic
322#492C22×C6711yescharacteristic
323#479—21no—
324#477C22×C6711yescharacteristic
325#476—21no—
326#476C22×C6711yescharacteristic
327#475—21no—
328#475—21no—
329#475C22×C6711yescharacteristic
330#475C22×C6711yescharacteristic
331#468—21no—
332#462C22×C6711yescharacteristic
333#461—21no—
334#461C22×C6711yescharacteristic
335#460—21no—
336#460C22×C6711yescharacteristic
337#460C22×C6711yescharacteristic
338#452—21no—
339#448—21no—
340#446C22×C6711yescharacteristic
341C8×Q8C22×C6711yescharacteristic
342C8×Q8C22×C6711yescharacteristic
343#441—21no—
344C4×Q16C22×C6711yescharacteristic
345C4×SD16—21no—
346C4×SD16—21no—
347C4×SD16C22×C6711yescharacteristic
348#436C22×C6711yescharacteristic
349#335C22×C6711yescharacteristic
350#334C22×C6711yescharacteristic
351#333C22×C6711yescharacteristic
352#332C22×C6711yescharacteristic
353#332C22×C6711yescharacteristic
354#331C22×C6711yescharacteristic
355Q8⋊C8C22×C6711yescharacteristic
356Q8⋊C8C22×C6711yescharacteristic
357Q8⋊C8C22×C6711yescharacteristic
358D4⋊C8C22×C6711yescharacteristic
359C82C22×C6711yescharacteristic
360C2×#41—21no—
361C22×Q8—21no—
362#137—44no—
363C2×Q16—21no—
364C2×Q16D4×C6711yescharacteristic
365C2×SD16—21no—
366C2×SD16—21no—
367C2×SD16—21no—
368C2×SD16—21no—
369C2×SD16—21no—
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—41no—
373C2×SD16D4×C6711yescharacteristic
374C2×M16—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—41no—
380C4⋊Q8C23×C6711yescharacteristic
381C4⋊Q8C23×C6711yescharacteristic
382C4⋊Q8C23×C6711yescharacteristic
383#127—21no—
384#127C23×C6711yescharacteristic
385#125—21no—
386#125—41no—
387C22⋊Q8—21no—
388C22⋊Q8—41no—
389#121—21no—
390#121—41no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—21no—
395C4×Q8C23×C6711yescharacteristic
396C4×Q8C23×C6711yescharacteristic
397C4×Q8C23×C6711yescharacteristic
398C4×D4—21no—
399C4×D4—21no—
400C4×D4C23×C6711yescharacteristic
401#107—21no—
402#107D4×C6711yescharacteristic
403#106—21no—
404#106—21no—
405#106—21no—
406#106—21no—
407#106—21no—
408#106—21no—
409#106—21no—
410#106—41no—
411#106D4×C6711yescharacteristic
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—21no—
415C4⋊C8C23×C6711yescharacteristic
416C4⋊C8C23×C6711yescharacteristic
417C4⋊C8C23×C6711yescharacteristic
418C4⋊C8C23×C6711yescharacteristic
419#103—21no—
420#103—21no—
421#103—21no—
422#103—21no—
423#103—21no—
424#103D4×C6711yescharacteristic
425#103D4×C6711yescharacteristic
426#103D4×C6711yescharacteristic
427#102—21no—
428#102—21no—
429#102—21no—
430#102D4×C6711yescharacteristic
431C22⋊C8—21no—
432#97—21no—
433C4×C8—21no—
434C4×C8—21no—
435C4×C8C23×C6711yescharacteristic
436C4×C8C23×C6711yescharacteristic
437C4×C8C23×C6711yescharacteristic
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—42no—
446C2×Q8—81no—
447C2×Q8C2×D4×C6711yescharacteristic
448C2×Q8C2×D4×C6711yescharacteristic
449C2×Q8C2×D4×C6711yescharacteristic
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4C2×D4×C6711yescharacteristic
456C22×C4—21no—
457C22×C4—41no—
458C22×C4—41no—
459Q16—42no—
460Q16SD16×C6714yescomplemented
461SD16—22no—
462SD16—42no—
463SD16—42no—
464SD16—42no—
465SD16—81no—
466SD16—81no—
467SD16—82no—
468SD16—81no—
469SD16—81no—
470M16—42no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8C2×D4×C6711yescharacteristic
477C2×C8C2×D4×C6711yescharacteristic
478C2×C8C2×D4×C6711yescharacteristic
479C2×C8C2×D4×C6711yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4C2×D4×C6711yescharacteristic
496C4⋊C4C2×D4×C6711yescharacteristic
497C4⋊C4C2×D4×C6711yescharacteristic
498C4⋊C4C2×D4×C6711yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42—41no—
507C42C24×C6711yescharacteristic
508C23—21no—
509C23—41no—
510Q8—21no—
511Q8—21no—
512Q8—21no—
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—82no—
519Q8—81no—
520Q8—161no—
521Q8C2×SD16×C6712yes—
522Q8C2×SD16×C6712yes—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—82no—
530D4—81no—
531D4—161no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C22×D4×C6711yescharacteristic
547C2×C4C22×D4×C6711yescharacteristic
548C2×C4H(5,𝔽2)×C6711yescharacteristic
549C8—22no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8C2×SD16×C6712yes—
557C8#137×C6712yescomplemented
558C22—41no—
559C22—41no—
560C22—41no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22D42×C6711yescharacteristic
565C4—22no—
566C4—22no—
567C4—21no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C22×SD16×C6711yescharacteristic
577C4C2×#137×C6711yescharacteristic
578C4#577×C6711yescharacteristic
579C4#577×C6711yescharacteristic
580C2—81no—
581C2—161no—
582C2C67×#327811yescharacteristic
583C2D4×SD16×C6711yescharacteristic
584C2C67×#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
2. C67×#3387C2semidirect
3. Q8×SD16×C67C2semidirect
5. C67×#1705C2semidirect
6. C67×#1691C2semidirect
7. C67×#1686C2semidirect
8. C67×#1672C2semidirect
9. C67×#1666C2semidirect
10. C67×#1653C2semidirect
11. C67×#1651C2semidirect
13. C67×#1623C2semidirect
14. C67×#1590C2semidirect
15. C8×Q16×C67C2semidirect
167. Q16×C67SD16semidirect
264. C8×C67#137semidirect
287. #19897C67direct (N × H)
294. #3387C2×C67semidirect
295. Q8×SD16C2×C67semidirect
297. #1705C2×C67semidirect
298. #1691C2×C67semidirect
299. #1686C2×C67semidirect
300. #1672C2×C67semidirect
301. #1666C2×C67semidirect
302. #1653C2×C67semidirect
303. #1651C2×C67semidirect
305. #1623C2×C67semidirect
306. #1590C2×C67semidirect
307. C8×Q16C2×C67semidirect
308. C67#19897direct (N × H)
460. Q16SD16×C67semidirect
557. C8#137×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#33922
C67×#33872
Q8×SD16×C672
C67×#32782
C67×#17052
C67×#16912
C67×#16862
C67×#16722
C67×#16662
C67×#16532
C67×#16512
C67×#16312
C67×#16232
C67×#15902
C8×Q16×C672
#1989767