Finite Group Databasebeta

← C67×#19924 #30699175

Subgroups — C67×#19924 #30699175

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3392C211yesmaximal, characteristic
2C67×#3381C211yesmaximal, characteristic, complemented
3C67×#3374C211yesmaximal, characteristic, complemented
4C67×#3333C211yesmaximal, characteristic
5C67×#1708C211yesmaximal, characteristic, complemented
6C67×#1699C211yesmaximal, characteristic, complemented
7C67×#1698C211yesmaximal, characteristic, complemented
8C67×#1672C211yesmaximal, characteristic, complemented
9C67×#1666C211yesmaximal, characteristic, complemented
10C67×#1643C211yesmaximal, characteristic, complemented
11C67×#1641C211yesmaximal, characteristic
12C67×#1636C211yesmaximal, characteristic, complemented
13C67×#1626C211yesmaximal, characteristic, complemented
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×Q16×C67C211yesmaximal, characteristic, complemented
16#557×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19#550×C67C2211yescharacteristic
20#499×C67—21no—
21#498×C67C2211yescharacteristic
22#492×C67—21no—
23#492×C67C2211yescharacteristic
24#491×C67—21no—
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#480×C67—21no—
30#479×C67—21no—
31#479×C67—21no—
32#478×C67—21no—
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#474×C67C2211yescharacteristic
37#468×C67—21no—
38#462×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#446×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#441×C67—21no—
50C4×Q16×C67C2211yescharacteristic
51C4×SD16×C67—21no—
52C4×SD16×C67—21no—
53C4×SD16×C67C2211yescharacteristic
54#436×C67C2211yescharacteristic
55#417×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×C67D411yescharacteristic
70C2×SD16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—21no—
74C2×SD16×C67—21no—
75C2×SD16×C67D411yescharacteristic
76C2×M16×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82#127×C67—21no—
83#127×C67C2311yescharacteristic
84#125×C67—21no—
85#125×C67—21no—
86#125×C67—21no—
87#125×C67—41no—
88#125×C67—41no—
89C22⋊Q8×C67—21no—
90C22⋊Q8×C67—41no—
91#121×C67—21no—
92#121×C67—41no—
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×C67—21no—
102C4×D4×C67C2311yescharacteristic
103C2×C4⋊C4×C67—21no—
104#107×C67—21no—
105#107×C67—21no—
106#107×C67—21no—
107#107×C67D411yescharacteristic
108#106×C67—21no—
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×C67—21no—
126#103×C67—41no—
127#103×C67D411yescharacteristic
128#103×C67D411yescharacteristic
129#103×C67D411yescharacteristic
130#102×C67—21no—
131#102×C67—21no—
132#102×C67—21no—
133#102×C67—21no—
134#102×C67—21no—
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
144#41×C67—42no—
145#41×C67—81no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67—21no—
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—41no—
156C2×D4×C67—41no—
157C2×D4×C67C2×D411yescharacteristic
158C22×C4×C67—21no—
159C22×C4×C67—41no—
160C22×C4×C67—41no—
161Q16×C67—22no—
162Q16×C67—42no—
163SD16×C67—22no—
164SD16×C67—42no—
165SD16×C67—81no—
166SD16×C67—81no—
167SD16×C67—81no—
168SD16×C67—81no—
169M16×C67—81no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67C2×D411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
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—21no—
188C4⋊C4×C67—41no—
189C4⋊C4×C67—41no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67—41no—
194C4⋊C4×C67—41no—
195C4⋊C4×C67—41no—
196C4⋊C4×C67C2×D411yescharacteristic
197C4⋊C4×C67C2×D411yescharacteristic
198C4⋊C4×C67C2×D411yescharacteristic
199C4⋊C4×C67C2×D411yescharacteristic
200C22⋊C4×C67—21no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67—41no—
207C42×C67—41no—
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67—41no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—41no—
217Q8×C67—41no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—81no—
221Q8×C67#13512yes—
222D4×C67—21no—
223D4×C67—21no—
224D4×C67—41no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—82no—
229D4×C67—81no—
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×C67—161no—
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67H(5,𝔽2)11yescharacteristic
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67#13712yes—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—81no—
261C22×C67—81no—
262C22×C67—81no—
263C22×C67D4211yescharacteristic
264C4×C67—22no—
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—81no—
273C4×C67—81no—
274C4×C67—81no—
275C4×C67C2×#13511yescharacteristic
276C4×C67C2×#13711yescharacteristic
277C4×C67#57711yescharacteristic
278C4×C67#57711yescharacteristic
279#19924C6711yesmaximal, characteristic, direct product factor, Sylow
280C2×C67—81no—
281C2×C67—161no—
282C2×C67#327811yescharacteristic
283C2×C67#328211yescharacteristic
284C2×C67#329011yescharacteristic
285#3392C2×C6711yescharacteristic
286#3381C2×C6711yescharacteristic, complemented
287#3374C2×C6711yescharacteristic, complemented
288#3333C2×C6711yescharacteristic
289#1708C2×C6711yescharacteristic, complemented
290#1699C2×C6711yescharacteristic, complemented
291#1698C2×C6711yescharacteristic, complemented
292#1672C2×C6711yescharacteristic, complemented
293#1666C2×C6711yescharacteristic, complemented
294#1643C2×C6711yescharacteristic, complemented
295#1641C2×C6711yescharacteristic
296#1636C2×C6711yescharacteristic, complemented
297#1626C2×C6711yescharacteristic, complemented
298#1590C2×C6711yescharacteristic, complemented
299C8×Q16C2×C6711yescharacteristic, complemented
300C67#1992411yescharacteristic, direct product factor, Sylow
301#557C22×C6711yescharacteristic
302#557C22×C6711yescharacteristic
303#554C22×C6711yescharacteristic
304#550C22×C6711yescharacteristic
305#499—21no—
306#498C22×C6711yescharacteristic
307#492—21no—
308#492C22×C6711yescharacteristic
309#491—21no—
310#487C22×C6711yescharacteristic
311#487C22×C6711yescharacteristic
312#486C22×C6711yescharacteristic
313#486C22×C6711yescharacteristic
314#480—21no—
315#479—21no—
316#479—21no—
317#478—21no—
318#477C22×C6711yescharacteristic
319#477C22×C6711yescharacteristic
320#475C22×C6711yescharacteristic
321#474C22×C6711yescharacteristic
322#468—21no—
323#462—21no—
324#462C22×C6711yescharacteristic
325#461—21no—
326#461C22×C6711yescharacteristic
327#460—21no—
328#460C22×C6711yescharacteristic
329#460C22×C6711yescharacteristic
330#452—21no—
331#446C22×C6711yescharacteristic
332C8×Q8C22×C6711yescharacteristic
333C8×Q8C22×C6711yescharacteristic
334#441—21no—
335C4×Q16C22×C6711yescharacteristic
336C4×SD16—21no—
337C4×SD16—21no—
338C4×SD16C22×C6711yescharacteristic
339#436C22×C6711yescharacteristic
340#417—21no—
341#335C22×C6711yescharacteristic
342#334C22×C6711yescharacteristic
343#333C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345#332C22×C6711yescharacteristic
346#331C22×C6711yescharacteristic
347Q8⋊C8C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349Q8⋊C8C22×C6711yescharacteristic
350D4⋊C8C22×C6711yescharacteristic
351C82C22×C6711yescharacteristic
352C2×#41—21no—
353C2×Q16—21no—
354C2×Q16D4×C6711yescharacteristic
355C2×SD16—21no—
356C2×SD16—21no—
357C2×SD16—21no—
358C2×SD16—21no—
359C2×SD16—21no—
360C2×SD16D4×C6711yescharacteristic
361C2×M16—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8C23×C6711yescharacteristic
365C4⋊Q8C23×C6711yescharacteristic
366C4⋊Q8C23×C6711yescharacteristic
367#127—21no—
368#127C23×C6711yescharacteristic
369#125—21no—
370#125—21no—
371#125—21no—
372#125—41no—
373#125—41no—
374C22⋊Q8—21no—
375C22⋊Q8—41no—
376#121—21no—
377#121—41no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C23×C6711yescharacteristic
383C4×Q8C23×C6711yescharacteristic
384C4×Q8C23×C6711yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4C23×C6711yescharacteristic
388C2×C4⋊C4—21no—
389#107—21no—
390#107—21no—
391#107—21no—
392#107D4×C6711yescharacteristic
393#106—21no—
394#106—21no—
395#106—21no—
396#106D4×C6711yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8C23×C6711yescharacteristic
401C4⋊C8C23×C6711yescharacteristic
402C4⋊C8C23×C6711yescharacteristic
403C4⋊C8C23×C6711yescharacteristic
404#103—21no—
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—21no—
410#103—21no—
411#103—41no—
412#103D4×C6711yescharacteristic
413#103D4×C6711yescharacteristic
414#103D4×C6711yescharacteristic
415#102—21no—
416#102—21no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102—41no—
421#102D4×C6711yescharacteristic
422C22⋊C8—21no—
423#97—21no—
424C4×C8—21no—
425C4×C8—21no—
426C4×C8C23×C6711yescharacteristic
427C4×C8C23×C6711yescharacteristic
428C4×C8C23×C6711yescharacteristic
429#41—42no—
430#41—81no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
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—
445C22×C4—41no—
446Q16—22no—
447Q16—42no—
448SD16—22no—
449SD16—42no—
450SD16—81no—
451SD16—81no—
452SD16—81no—
453SD16—81no—
454M16—81no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C2×C8C2×D4×C6711yescharacteristic
463C2×C8C2×D4×C6711yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4C2×D4×C6711yescharacteristic
482C4⋊C4C2×D4×C6711yescharacteristic
483C4⋊C4C2×D4×C6711yescharacteristic
484C4⋊C4C2×D4×C6711yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—41no—
492C42—41no—
493C42C24×C6711yescharacteristic
494C23—21no—
495C23—41no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—81no—
506Q8#135×C6712yes—
507D4—21no—
508D4—21no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—82no—
514D4—81no—
515D4—161no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—161no—
530C2×C4C22×D4×C6711yescharacteristic
531C2×C4C22×D4×C6711yescharacteristic
532C2×C4H(5,𝔽2)×C6711yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8#137×C6712yes—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22D42×C6711yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4C2×#135×C6711yescharacteristic
561C4C2×#137×C6711yescharacteristic
562C4#577×C6711yescharacteristic
563C4#577×C6711yescharacteristic
564C2—81no—
565C2—161no—
566C2C67×#327811yescharacteristic
567C2C67×#328211yescharacteristic
568C2C67×#329011yescharacteristic
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×#3381C2semidirect
3. C67×#3374C2semidirect
5. C67×#1708C2semidirect
6. C67×#1699C2semidirect
7. C67×#1698C2semidirect
8. C67×#1672C2semidirect
9. C67×#1666C2semidirect
10. C67×#1643C2semidirect
12. C67×#1636C2semidirect
13. C67×#1626C2semidirect
14. C67×#1590C2semidirect
15. C8×Q16×C67C2semidirect
279. #19924C67direct (N × H)
286. #3381C2×C67semidirect
287. #3374C2×C67semidirect
289. #1708C2×C67semidirect
290. #1699C2×C67semidirect
291. #1698C2×C67semidirect
292. #1672C2×C67semidirect
293. #1666C2×C67semidirect
294. #1643C2×C67semidirect
296. #1636C2×C67semidirect
297. #1626C2×C67semidirect
298. #1590C2×C67semidirect
299. C8×Q16C2×C67semidirect
300. C67#19924direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33922
C67×#33812
C67×#33742
C67×#33332
C67×#17082
C67×#16992
C67×#16982
C67×#16722
C67×#16662
C67×#16432
C67×#16412
C67×#16362
C67×#16262
C67×#15902
C8×Q16×C672
#1992467