Finite Group Databasebeta

← C67×#20050 #30699301

Subgroups — C67×#20050 #30699301

Order 17152.

600 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3385C211yesmaximal, characteristic, complemented
2C67×#3362C211yesmaximal, characteristic, complemented
3C67×#3357C211yesmaximal, characteristic, complemented
4C67×#3305C211yesmaximal, characteristic
5C67×#1722C211yesmaximal, characteristic, complemented
6C67×#1698C211yesmaximal, characteristic, complemented
7C67×#1697C211yesmaximal, characteristic, complemented
8C67×#1676C211yesmaximal, characteristic, complemented
9C67×#1675C211yesmaximal, characteristic, complemented
10C67×#1634C211yesmaximal, characteristic, complemented
11C67×#1633C211yesmaximal, characteristic, complemented
12C67×#1631C211yesmaximal, characteristic, complemented
13C67×#1622C211yesmaximal, characteristic, complemented
14C67×#1588C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16Q82×C67C2211yescharacteristic
17#555×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19#550×C67C2211yescharacteristic
20#501×C67—21no—
21#500×C67C2211yescharacteristic
22#500×C67C2211yescharacteristic
23#494×C67C2211yescharacteristic
24#492×C67C2211yescharacteristic
25#486×C67—21no—
26#486×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#485×C67—21no—
29#482×C67—21no—
30#480×C67—21no—
31#479×C67—21no—
32#477×C67C2211yescharacteristic
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#476×C67—21no—
36#476×C67C2211yescharacteristic
37#476×C67C2211yescharacteristic
38#476×C67C2211yescharacteristic
39#474×C67—21no—
40#468×C67—21no—
41#466×C67—21no—
42#460×C67—21no—
43#460×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#449×C67—21no—
46#446×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48#440×C67—21no—
49C4×SD16×C67—21no—
50C4×SD16×C67C2211yescharacteristic
51C4×SD16×C67C2211yescharacteristic
52#436×C67C2211yescharacteristic
53C8×D4×C67C2211yescharacteristic
54#428×C67—21no—
55#416×C67—21no—
56#335×C67C2211yescharacteristic
57#335×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#333×C67C2211yescharacteristic
61#331×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×Q16×C67—41no—
69C2×SD16×C67—21no—
70C2×SD16×C67—41no—
71C2×SD16×C67D411yescharacteristic
72C2×SD16×C67D411yescharacteristic
73C2×D8×C67—21no—
74C2×M16×C67—21no—
75C22×C8×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—41no—
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82#127×C67—21no—
83#127×C67C2311yescharacteristic
84#125×C67—21no—
85#124×C67—41no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—41no—
88#121×C67—21no—
89#121×C67—21no—
90#121×C67—21no—
91#121×C67—41no—
92C4×Q8×C67—21no—
93C4×Q8×C67—21no—
94C4×Q8×C67C2311yescharacteristic
95C4×Q8×C67C2311yescharacteristic
96C4×D4×C67—21no—
97C4×D4×C67—21no—
98C4×D4×C67C2311yescharacteristic
99C4×D4×C67C2311yescharacteristic
100#117×C67—21no—
101#117×C67—21no—
102C2×C4⋊C4×C67—21no—
103#107×C67—21no—
104#107×C67—21no—
105#107×C67—21no—
106#107×C67—21no—
107#107×C67—21no—
108#107×C67—21no—
109#107×C67—21no—
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—21no—
113#106×C67D411yescharacteristic
114#106×C67D411yescharacteristic
115C4⋊C8×C67—21no—
116C4⋊C8×C67—21no—
117C4⋊C8×C67C2311yescharacteristic
118C4⋊C8×C67C2311yescharacteristic
119C4⋊C8×C67C2311yescharacteristic
120C4⋊C8×C67C2311yescharacteristic
121#103×C67—21no—
122#103×C67—21no—
123#103×C67—21no—
124#103×C67D411yescharacteristic
125#103×C67D411yescharacteristic
126#102×C67—21no—
127#102×C67—21no—
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×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
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×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67—21no—
155C2×D4×C67—41no—
156C2×D4×C67—41no—
157C2×D4×C67—41no—
158C2×D4×C67C2×D411yescharacteristic
159C2×D4×C67C2×D411yescharacteristic
160C22×C4×C67—21no—
161C22×C4×C67—21no—
162C22×C4×C67—41no—
163C22×C4×C67—41no—
164Q16×C67—82no—
165SD16×C67—22no—
166SD16×C67—41no—
167SD16×C67—42no—
168SD16×C67—82no—
169D8×C67—81no—
170M16×C67—42no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67—21no—
176C2×C8×C67—81no—
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
179C2×C8×C67C2×D411yescharacteristic
180C2×C8×C67C2×D411yescharacteristic
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—21no—
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—21no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—41no—
206C42×C67—41no—
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209C23×C67—21no—
210C23×C67—41no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—81no—
215Q8×C67—81no—
216Q8×C67—81no—
217Q8×C67—161no—
218Q8×C67#13712yes—
219D4×C67—21no—
220D4×C67—21no—
221D4×C67—21no—
222D4×C67—21no—
223D4×C67—41no—
224D4×C67—41no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—82no—
228D4×C67—81no—
229D4×C67—161no—
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—41no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—82no—
245C2×C4×C67—161no—
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67H(5,𝔽2)11yescharacteristic
249C8×C67—22no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—21no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C8×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67—41no—
264C22×C67—81no—
265C22×C67—81no—
266C22×C67—81no—
267C22×C67D4211yescharacteristic
268C4×C67—22no—
269C4×C67—21no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67—81no—
276C4×C67—81no—
277C4×C67—81no—
278C4×C67C2×#13711yescharacteristic
279C4×C67#57511yescharacteristic
280C4×C67#57711yescharacteristic
281C4×C67#57811yescharacteristic
282#20050C6711yesmaximal, characteristic, direct product factor, Sylow
283C2×C67—81no—
284C2×C67—81no—
285C2×C67—161no—
286C2×C67#326611yescharacteristic
287C2×C67#326911yescharacteristic
288C2×C67#329211yescharacteristic
289#3385C2×C6711yescharacteristic, complemented
290#3362C2×C6711yescharacteristic, complemented
291#3357C2×C6711yescharacteristic, complemented
292#3305C2×C6711yescharacteristic
293#1722C2×C6711yescharacteristic, complemented
294#1698C2×C6711yescharacteristic, complemented
295#1697C2×C6711yescharacteristic, complemented
296#1676C2×C6711yescharacteristic, complemented
297#1675C2×C6711yescharacteristic, complemented
298#1634C2×C6711yescharacteristic, complemented
299#1633C2×C6711yescharacteristic, complemented
300#1631C2×C6711yescharacteristic, complemented
301#1622C2×C6711yescharacteristic, complemented
302#1588C2×C6711yescharacteristic, complemented
303#1585C2×C6711yescharacteristic, complemented
304C67#2005011yescharacteristic, direct product factor, Sylow
305Q82C22×C6711yescharacteristic
306#555C22×C6711yescharacteristic
307#554C22×C6711yescharacteristic
308#550C22×C6711yescharacteristic
309#501—21no—
310#500C22×C6711yescharacteristic
311#500C22×C6711yescharacteristic
312#494C22×C6711yescharacteristic
313#492C22×C6711yescharacteristic
314#486—21no—
315#486C22×C6711yescharacteristic
316#486C22×C6711yescharacteristic
317#485—21no—
318#482—21no—
319#480—21no—
320#479—21no—
321#477C22×C6711yescharacteristic
322#477C22×C6711yescharacteristic
323#477C22×C6711yescharacteristic
324#476—21no—
325#476C22×C6711yescharacteristic
326#476C22×C6711yescharacteristic
327#476C22×C6711yescharacteristic
328#474—21no—
329#468—21no—
330#466—21no—
331#460—21no—
332#460C22×C6711yescharacteristic
333#459C22×C6711yescharacteristic
334#449—21no—
335#446C22×C6711yescharacteristic
336C8×Q8C22×C6711yescharacteristic
337#440—21no—
338C4×SD16—21no—
339C4×SD16C22×C6711yescharacteristic
340C4×SD16C22×C6711yescharacteristic
341#436C22×C6711yescharacteristic
342C8×D4C22×C6711yescharacteristic
343#428—21no—
344#416—21no—
345#335C22×C6711yescharacteristic
346#335C22×C6711yescharacteristic
347#333C22×C6711yescharacteristic
348#333C22×C6711yescharacteristic
349#333C22×C6711yescharacteristic
350#331C22×C6711yescharacteristic
351Q8⋊C8C22×C6711yescharacteristic
352Q8⋊C8C22×C6711yescharacteristic
353D4⋊C8C22×C6711yescharacteristic
354D4⋊C8C22×C6711yescharacteristic
355#322C22×C6711yescharacteristic
356C2×#41—21no—
357C2×Q16—41no—
358C2×SD16—21no—
359C2×SD16—41no—
360C2×SD16D4×C6711yescharacteristic
361C2×SD16D4×C6711yescharacteristic
362C2×D8—21no—
363C2×M16—21no—
364C22×C8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—41no—
368C4⋊Q8C23×C6711yescharacteristic
369C4⋊Q8C23×C6711yescharacteristic
370C4⋊Q8C23×C6711yescharacteristic
371#127—21no—
372#127C23×C6711yescharacteristic
373#125—21no—
374#124—41no—
375C22⋊Q8—21no—
376C22⋊Q8—41no—
377#121—21no—
378#121—21no—
379#121—21no—
380#121—41no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×Q8C23×C6711yescharacteristic
384C4×Q8C23×C6711yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4C23×C6711yescharacteristic
388C4×D4C23×C6711yescharacteristic
389#117—21no—
390#117—21no—
391C2×C4⋊C4—21no—
392#107—21no—
393#107—21no—
394#107—21no—
395#107—21no—
396#107—21no—
397#107—21no—
398#107—21no—
399#106—21no—
400#106—21no—
401#106—21no—
402#106D4×C6711yescharacteristic
403#106D4×C6711yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8C23×C6711yescharacteristic
407C4⋊C8C23×C6711yescharacteristic
408C4⋊C8C23×C6711yescharacteristic
409C4⋊C8C23×C6711yescharacteristic
410#103—21no—
411#103—21no—
412#103—21no—
413#103D4×C6711yescharacteristic
414#103D4×C6711yescharacteristic
415#102—21no—
416#102—21no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102—21no—
421#102—21no—
422#102—21no—
423#102—21no—
424#102D4×C6711yescharacteristic
425#102D4×C6711yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—21no—
428#97—21no—
429C4×C8—21no—
430C4×C8C23×C6711yescharacteristic
431C4×C8C23×C6711yescharacteristic
432C4×C8C23×C6711yescharacteristic
433#41—42no—
434#41—81no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—41no—
439C2×Q8C2×D4×C6711yescharacteristic
440C2×Q8C2×D4×C6711yescharacteristic
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4C2×D4×C6711yescharacteristic
448C2×D4C2×D4×C6711yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453Q16—82no—
454SD16—22no—
455SD16—41no—
456SD16—42no—
457SD16—82no—
458D8—81no—
459M16—42no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—81no—
466C2×C8C2×D4×C6711yescharacteristic
467C2×C8C2×D4×C6711yescharacteristic
468C2×C8C2×D4×C6711yescharacteristic
469C2×C8C2×D4×C6711yescharacteristic
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4C2×D4×C6711yescharacteristic
483C4⋊C4C2×D4×C6711yescharacteristic
484C4⋊C4C2×D4×C6711yescharacteristic
485C4⋊C4C2×D4×C6711yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42C24×C6711yescharacteristic
497C23—21no—
498C23—21no—
499C23—41no—
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—81no—
504Q8—81no—
505Q8—81no—
506Q8—161no—
507Q8#137×C6712yes—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—21no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—82no—
517D4—81no—
518D4—161no—
519D4—161no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—82no—
534C2×C4—161no—
535C2×C4C22×D4×C6711yescharacteristic
536C2×C4C22×D4×C6711yescharacteristic
537C2×C4H(5,𝔽2)×C6711yescharacteristic
538C8—22no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—41no—
544C8—41no—
545C8—41no—
546C8—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22D42×C6711yescharacteristic
557C4—22no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4C2×#137×C6711yescharacteristic
568C4#575×C6711yescharacteristic
569C4#577×C6711yescharacteristic
570C4#578×C6711yescharacteristic
571C2—81no—
572C2—81no—
573C2—161no—
574C2C67×#326611yescharacteristic
575C2C67×#326911yescharacteristic
576C2C67×#329211yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3305C2
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×#3385C2semidirect
2. C67×#3362C2semidirect
3. C67×#3357C2semidirect
5. C67×#1722C2semidirect
6. C67×#1698C2semidirect
7. C67×#1697C2semidirect
8. C67×#1676C2semidirect
9. C67×#1675C2semidirect
10. C67×#1634C2semidirect
11. C67×#1633C2semidirect
12. C67×#1631C2semidirect
13. C67×#1622C2semidirect
14. C67×#1588C2semidirect
15. C67×#1585C2semidirect
282. #20050C67direct (N × H)
289. #3385C2×C67semidirect
290. #3362C2×C67semidirect
291. #3357C2×C67semidirect
293. #1722C2×C67semidirect
294. #1698C2×C67semidirect
295. #1697C2×C67semidirect
296. #1676C2×C67semidirect
297. #1675C2×C67semidirect
298. #1634C2×C67semidirect
299. #1633C2×C67semidirect
300. #1631C2×C67semidirect
301. #1622C2×C67semidirect
302. #1588C2×C67semidirect
303. #1585C2×C67semidirect
304. C67#20050direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33852
C67×#33622
C67×#33572
C67×#33052
C67×#17222
C67×#16982
C67×#16972
C67×#16762
C67×#16752
C67×#16342
C67×#16332
C67×#16312
C67×#16222
C67×#15882
C67×#15852
#2005067