Finite Group Databasebeta

← C67×#20046 #30699297

Subgroups — C67×#20046 #30699297

Order 17152.

594 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×#3385C211yesmaximal, characteristic, complemented
2Q8×SD16×C67C211yesmaximal, characteristic, complemented
3C67×#3331C211yesmaximal, characteristic
4C67×#3298C211yesmaximal, characteristic
5C67×#1718C211yesmaximal, characteristic, complemented
6C67×#1700C211yesmaximal, characteristic, complemented
7C67×#1691C211yesmaximal, characteristic, complemented
8C67×#1684C211yesmaximal, characteristic, complemented
9C67×#1675C211yesmaximal, characteristic, complemented
10C67×#1645C211yesmaximal, characteristic, complemented
11C67×#1643C211yesmaximal, characteristic, complemented
12C67×#1637C211yesmaximal, characteristic, complemented
13C67×#1635C211yesmaximal, characteristic
14C67×#1588C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16Q82×C67C2211yescharacteristic
17#554×C67C2211yescharacteristic
18D4×Q8×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#501×C67—21no—
21#500×C67C2211yescharacteristic
22#498×C67—21no—
23#498×C67C2211yescharacteristic
24#487×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#487×C67C2211yescharacteristic
28#483×C67—21no—
29#483×C67—21no—
30#479×C67—21no—
31#477×C67C2211yescharacteristic
32#476×C67—21no—
33#476×C67C2211yescharacteristic
34#476×C67C2211yescharacteristic
35#476×C67C2211yescharacteristic
36#475×C67—21no—
37#475×C67C2211yescharacteristic
38#475×C67C2211yescharacteristic
39#474×C67—21no—
40#470×C67—21no—
41#465×C67—21no—
42#462×C67C2211yescharacteristic
43#461×C67C2211yescharacteristic
44#452×C67—21no—
45#448×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#417×C67—21no—
55C2×#103×C67—21no—
56#335×C67C2211yescharacteristic
57#335×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C22×Q8×C67—21no—
69C2×Q16×C67—21no—
70C2×SD16×C67—21no—
71C2×SD16×C67D411yescharacteristic
72C2×SD16×C67D411yescharacteristic
73C2×M16×C67—21no—
74C22×C8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—41no—
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#125×C67—21no—
84#123×C67—41no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—21no—
88C22⋊Q8×C67—41no—
89C22⋊Q8×C67—41no—
90#121×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67—21no—
93C4×Q8×C67C2311yescharacteristic
94C4×Q8×C67C2311yescharacteristic
95C4×D4×C67—21no—
96C4×D4×C67—21no—
97C4×D4×C67C2311yescharacteristic
98C4×D4×C67C2311yescharacteristic
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67—21no—
101#107×C67—21no—
102#107×C67—21no—
103#107×C67—21no—
104#106×C67—21no—
105#106×C67—21no—
106#106×C67—21no—
107#106×C67—21no—
108#106×C67—21no—
109#106×C67—21no—
110#106×C67—21no—
111#106×C67D411yescharacteristic
112#106×C67D411yescharacteristic
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—21no—
127#103×C67—21no—
128#103×C67—41no—
129#103×C67—41no—
130#103×C67D411yescharacteristic
131#103×C67D411yescharacteristic
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—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×D4×C67—41no—
157C2×D4×C67C2×D411yescharacteristic
158C2×D4×C67C2×D411yescharacteristic
159C22×C4×C67—21no—
160C22×C4×C67—21no—
161C22×C4×C67—41no—
162C22×C4×C67—41no—
163C22×C4×C67—41no—
164Q16×C67—81no—
165SD16×C67—22no—
166SD16×C67—22no—
167SD16×C67—81no—
168M16×C67—81no—
169C2×C8×C67—21no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—81no—
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—21no—
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×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C4⋊C4×C67C2×D411yescharacteristic
198C4⋊C4×C67C2×D411yescharacteristic
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C22⋊C4×C67—41no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67—41no—
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209C23×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—41no—
214Q8×C67—41no—
215Q8×C67—41no—
216Q8×C67—41no—
217Q8×C67—82no—
218Q8×C67—81no—
219Q8×C67—81no—
220Q8×C67—161no—
221Q8×C67#13712yes—
222D4×C67—21no—
223D4×C67—21no—
224D4×C67—21no—
225D4×C67—82no—
226D4×C67#13712yes—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—82no—
242C2×C4×C67—82no—
243C2×C4×C67—161no—
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67H(5,𝔽2)11yescharacteristic
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C8×C67—41no—
256C22×C67—22no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
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—81no—
271C4×C67—81no—
272C4×C67—81no—
273C4×C67—81no—
274C4×C67C2×#13711yescharacteristic
275C4×C67C2×#13711yescharacteristic
276C4×C67#57711yescharacteristic
277C4×C67#57711yescharacteristic
278#20046C6711yesmaximal, characteristic, direct product factor, Sylow
279C2×C67—42no—
280C2×C67—81no—
281C2×C67#326611yescharacteristic
282C2×C67#326611yescharacteristic
283C2×C67#329011yescharacteristic
284#3385C2×C6711yescharacteristic, complemented
285Q8×SD16C2×C6711yescharacteristic, complemented
286#3331C2×C6711yescharacteristic
287#3298C2×C6711yescharacteristic
288#1718C2×C6711yescharacteristic, complemented
289#1700C2×C6711yescharacteristic, complemented
290#1691C2×C6711yescharacteristic, complemented
291#1684C2×C6711yescharacteristic, complemented
292#1675C2×C6711yescharacteristic, complemented
293#1645C2×C6711yescharacteristic, complemented
294#1643C2×C6711yescharacteristic, complemented
295#1637C2×C6711yescharacteristic, complemented
296#1635C2×C6711yescharacteristic
297#1588C2×C6711yescharacteristic, complemented
298#1585C2×C6711yescharacteristic, complemented
299C67#2004611yescharacteristic, direct product factor, Sylow
300Q82C22×C6711yescharacteristic
301#554C22×C6711yescharacteristic
302D4×Q8C22×C6711yescharacteristic
303#547C22×C6711yescharacteristic
304#501—21no—
305#500C22×C6711yescharacteristic
306#498—21no—
307#498C22×C6711yescharacteristic
308#487C22×C6711yescharacteristic
309#487C22×C6711yescharacteristic
310#487C22×C6711yescharacteristic
311#487C22×C6711yescharacteristic
312#483—21no—
313#483—21no—
314#479—21no—
315#477C22×C6711yescharacteristic
316#476—21no—
317#476C22×C6711yescharacteristic
318#476C22×C6711yescharacteristic
319#476C22×C6711yescharacteristic
320#475—21no—
321#475C22×C6711yescharacteristic
322#475C22×C6711yescharacteristic
323#474—21no—
324#470—21no—
325#465—21no—
326#462C22×C6711yescharacteristic
327#461C22×C6711yescharacteristic
328#452—21no—
329#448—21no—
330#446C22×C6711yescharacteristic
331C8×Q8C22×C6711yescharacteristic
332#440—21no—
333C4×SD16—21no—
334C4×SD16C22×C6711yescharacteristic
335C4×SD16C22×C6711yescharacteristic
336#436C22×C6711yescharacteristic
337C8×D4C22×C6711yescharacteristic
338#417—21no—
339C2×#103—21no—
340#335C22×C6711yescharacteristic
341#335C22×C6711yescharacteristic
342#333C22×C6711yescharacteristic
343#333C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345#332C22×C6711yescharacteristic
346Q8⋊C8C22×C6711yescharacteristic
347Q8⋊C8C22×C6711yescharacteristic
348D4⋊C8C22×C6711yescharacteristic
349D4⋊C8C22×C6711yescharacteristic
350#322C22×C6711yescharacteristic
351C2×#41—21no—
352C22×Q8—21no—
353C2×Q16—21no—
354C2×SD16—21no—
355C2×SD16D4×C6711yescharacteristic
356C2×SD16D4×C6711yescharacteristic
357C2×M16—21no—
358C22×C8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—41no—
363C4⋊Q8C23×C6711yescharacteristic
364C4⋊Q8C23×C6711yescharacteristic
365C4⋊Q8C23×C6711yescharacteristic
366C4⋊Q8C23×C6711yescharacteristic
367#125—21no—
368#123—41no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373C22⋊Q8—41no—
374#121—21no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8C23×C6711yescharacteristic
378C4×Q8C23×C6711yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4C23×C6711yescharacteristic
382C4×D4C23×C6711yescharacteristic
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385#107—21no—
386#107—21no—
387#107—21no—
388#106—21no—
389#106—21no—
390#106—21no—
391#106—21no—
392#106—21no—
393#106—21no—
394#106—21no—
395#106D4×C6711yescharacteristic
396#106D4×C6711yescharacteristic
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8C23×C6711yescharacteristic
400C4⋊C8C23×C6711yescharacteristic
401C4⋊C8C23×C6711yescharacteristic
402C4⋊C8C23×C6711yescharacteristic
403#103—21no—
404#103—21no—
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—21no—
410#103—21no—
411#103—21no—
412#103—41no—
413#103—41no—
414#103D4×C6711yescharacteristic
415#103D4×C6711yescharacteristic
416#102—21no—
417#102—21no—
418#102—21no—
419#102D4×C6711yescharacteristic
420#102D4×C6711yescharacteristic
421C22⋊C8—21no—
422C22⋊C8—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—21no—
434C2×Q8—41no—
435C2×Q8—41no—
436C2×Q8—42no—
437C2×Q8—81no—
438C2×Q8C2×D4×C6711yescharacteristic
439C2×Q8C2×D4×C6711yescharacteristic
440C2×D4—41no—
441C2×D4C2×D4×C6711yescharacteristic
442C2×D4C2×D4×C6711yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448Q16—81no—
449SD16—22no—
450SD16—22no—
451SD16—81no—
452M16—81no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—81no—
459C2×C8C2×D4×C6711yescharacteristic
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C2×C8C2×D4×C6711yescharacteristic
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—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C2×D4×C6711yescharacteristic
480C4⋊C4C2×D4×C6711yescharacteristic
481C4⋊C4C2×D4×C6711yescharacteristic
482C4⋊C4C2×D4×C6711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42C24×C6711yescharacteristic
492C23—21no—
493C23—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—82no—
502Q8—81no—
503Q8—81no—
504Q8—161no—
505Q8#137×C6712yes—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—82no—
510D4#137×C6712yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—161no—
528C2×C4C22×D4×C6711yescharacteristic
529C2×C4C22×D4×C6711yescharacteristic
530C2×C4H(5,𝔽2)×C6711yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22D42×C6711yescharacteristic
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4C2×#137×C6711yescharacteristic
559C4C2×#137×C6711yescharacteristic
560C4#577×C6711yescharacteristic
561C4#577×C6711yescharacteristic
562C2—42no—
563C2—81no—
564C2C67×#326611yescharacteristic
565C2C67×#326611yescharacteristic
566C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#547×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×#3385C2semidirect
2. Q8×SD16×C67C2semidirect
5. C67×#1718C2semidirect
6. C67×#1700C2semidirect
7. C67×#1691C2semidirect
8. C67×#1684C2semidirect
9. C67×#1675C2semidirect
10. C67×#1645C2semidirect
11. C67×#1643C2semidirect
12. C67×#1637C2semidirect
14. C67×#1588C2semidirect
15. C67×#1585C2semidirect
278. #20046C67direct (N × H)
284. #3385C2×C67semidirect
285. Q8×SD16C2×C67semidirect
288. #1718C2×C67semidirect
289. #1700C2×C67semidirect
290. #1691C2×C67semidirect
291. #1684C2×C67semidirect
292. #1675C2×C67semidirect
293. #1645C2×C67semidirect
294. #1643C2×C67semidirect
295. #1637C2×C67semidirect
297. #1588C2×C67semidirect
298. #1585C2×C67semidirect
299. C67#20046direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33852
Q8×SD16×C672
C67×#33312
C67×#32982
C67×#17182
C67×#17002
C67×#16912
C67×#16842
C67×#16752
C67×#16452
C67×#16432
C67×#16372
C67×#16352
C67×#15882
C67×#15852
#2004667