Finite Group Databasebeta

← C67×#19921 #30699172

Subgroups — C67×#19921 #30699172

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×#3394C211yesmaximal, characteristic
2Q8×SD16×C67C211yesmaximal, characteristic, complemented
3C67×#3375C211yesmaximal, characteristic, complemented
4C67×#3328C211yesmaximal, characteristic, complemented
5C67×#1708C211yesmaximal, characteristic, complemented
6C67×#1700C211yesmaximal, characteristic, complemented
7C67×#1697C211yesmaximal, characteristic
8C67×#1672C211yesmaximal, characteristic, complemented
9C67×#1666C211yesmaximal, characteristic, complemented
10C67×#1645C211yesmaximal, characteristic, complemented
11C67×#1640C211yesmaximal, characteristic, complemented
12C67×#1638C211yesmaximal, characteristic, complemented
13C67×#1627C211yesmaximal, characteristic
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×Q16×C67C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17Q82×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#498×C67—21no—
21#498×C67C2211yescharacteristic
22#495×C67—21no—
23#492×C67C2211yescharacteristic
24#489×C67—21no—
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#485×C67—21no—
30#477×C67C2211yescharacteristic
31#476×C67—21no—
32#476×C67C2211yescharacteristic
33#475×C67—21no—
34#475×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#474×C67—21no—
37#470×C67—21no—
38#464×C67—21no—
39#464×C67—21no—
40#463×C67—21no—
41#462×C67C2211yescharacteristic
42#462×C67C2211yescharacteristic
43#460×C67C2211yescharacteristic
44#459×C67C2211yescharacteristic
45#451×C67—21no—
46#446×C67C2211yescharacteristic
47C8×Q8×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#440×C67—21no—
50C4×Q16×C67C2211yescharacteristic
51C4×SD16×C67—21no—
52C4×SD16×C67—21no—
53C4×SD16×C67C2211yescharacteristic
54#436×C67C2211yescharacteristic
55#419×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
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×C67D411yescharacteristic
75C2×D8×C67—21no—
76C2×M16×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67—41no—
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84#127×C67C2311yescharacteristic
85#126×C67—41no—
86#124×C67—21no—
87#124×C67—21no—
88#124×C67—21no—
89#123×C67—41no—
90C22⋊Q8×C67—21no—
91C22⋊Q8×C67—21no—
92C22⋊Q8×C67—41no—
93C4×Q8×C67—21no—
94C4×Q8×C67—21no—
95C4×Q8×C67—21no—
96C4×Q8×C67C2311yescharacteristic
97C4×Q8×C67C2311yescharacteristic
98C4×Q8×C67C2311yescharacteristic
99C4×D4×C67—21no—
100C4×D4×C67—21no—
101C4×D4×C67—21no—
102C4×D4×C67C2311yescharacteristic
103#117×C67—21no—
104#107×C67—21no—
105#107×C67D411yescharacteristic
106#106×C67—21no—
107#106×C67—21no—
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
144C2×Q8×C67—21no—
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—41no—
148C2×Q8×C67—41no—
149C2×Q8×C67—42no—
150C2×Q8×C67—81no—
151C2×Q8×C67C2×D411yescharacteristic
152C2×Q8×C67C2×D411yescharacteristic
153C2×Q8×C67C2×D411yescharacteristic
154C2×D4×C67—21no—
155C2×D4×C67—21no—
156C2×D4×C67—21no—
157C2×D4×C67C2×D411yescharacteristic
158C22×C4×C67—21no—
159C22×C4×C67—41no—
160C22×C4×C67—41no—
161Q16×C67—22no—
162Q16×C67—81no—
163Q16×C67—81no—
164SD16×C67—22no—
165SD16×C67—42no—
166SD16×C67—42no—
167SD16×C67—81no—
168D8×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×C67C2×D411yescharacteristic
193C4⋊C4×C67C2×D411yescharacteristic
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C22⋊C4×C67—21no—
197C22⋊C4×C67—41no—
198C22⋊C4×C67—41no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
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—82no—
221Q8×C67—81no—
222Q8×C67—161no—
223Q8×C67#13512yes—
224D4×C67—21no—
225D4×C67—21no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—41no—
229D4×C67—41no—
230D4×C67—81no—
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#19921C6711yesmaximal, characteristic, direct product factor, Sylow
280C2×C67—81no—
281C2×C67—161no—
282C2×C67#327811yescharacteristic
283C2×C67#328211yescharacteristic
284C2×C67#329011yescharacteristic
285#3394C2×C6711yescharacteristic
286Q8×SD16C2×C6711yescharacteristic, complemented
287#3375C2×C6711yescharacteristic, complemented
288#3328C2×C6711yescharacteristic, complemented
289#1708C2×C6711yescharacteristic, complemented
290#1700C2×C6711yescharacteristic, complemented
291#1697C2×C6711yescharacteristic
292#1672C2×C6711yescharacteristic, complemented
293#1666C2×C6711yescharacteristic, complemented
294#1645C2×C6711yescharacteristic, complemented
295#1640C2×C6711yescharacteristic, complemented
296#1638C2×C6711yescharacteristic, complemented
297#1627C2×C6711yescharacteristic
298#1590C2×C6711yescharacteristic, complemented
299C8×Q16C2×C6711yescharacteristic, complemented
300C67#1992111yescharacteristic, direct product factor, Sylow
301#559C22×C6711yescharacteristic
302Q82C22×C6711yescharacteristic
303#556C22×C6711yescharacteristic
304D4×Q8C22×C6711yescharacteristic
305#498—21no—
306#498C22×C6711yescharacteristic
307#495—21no—
308#492C22×C6711yescharacteristic
309#489—21no—
310#487C22×C6711yescharacteristic
311#487C22×C6711yescharacteristic
312#486C22×C6711yescharacteristic
313#486C22×C6711yescharacteristic
314#485—21no—
315#477C22×C6711yescharacteristic
316#476—21no—
317#476C22×C6711yescharacteristic
318#475—21no—
319#475C22×C6711yescharacteristic
320#475C22×C6711yescharacteristic
321#474—21no—
322#470—21no—
323#464—21no—
324#464—21no—
325#463—21no—
326#462C22×C6711yescharacteristic
327#462C22×C6711yescharacteristic
328#460C22×C6711yescharacteristic
329#459C22×C6711yescharacteristic
330#451—21no—
331#446C22×C6711yescharacteristic
332C8×Q8C22×C6711yescharacteristic
333C8×Q8C22×C6711yescharacteristic
334#440—21no—
335C4×Q16C22×C6711yescharacteristic
336C4×SD16—21no—
337C4×SD16—21no—
338C4×SD16C22×C6711yescharacteristic
339#436C22×C6711yescharacteristic
340#419—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
352C22×Q8—21no—
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16D4×C6711yescharacteristic
356C2×SD16—21no—
357C2×SD16—21no—
358C2×SD16—21no—
359C2×SD16D4×C6711yescharacteristic
360C2×D8—21no—
361C2×M16—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—41no—
366C4⋊Q8C23×C6711yescharacteristic
367C4⋊Q8C23×C6711yescharacteristic
368C4⋊Q8C23×C6711yescharacteristic
369#127C23×C6711yescharacteristic
370#126—41no—
371#124—21no—
372#124—21no—
373#124—21no—
374#123—41no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—41no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8C23×C6711yescharacteristic
382C4×Q8C23×C6711yescharacteristic
383C4×Q8C23×C6711yescharacteristic
384C4×D4—21no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4C23×C6711yescharacteristic
388#117—21no—
389#107—21no—
390#107D4×C6711yescharacteristic
391#106—21no—
392#106—21no—
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
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—81no—
436C2×Q8C2×D4×C6711yescharacteristic
437C2×Q8C2×D4×C6711yescharacteristic
438C2×Q8C2×D4×C6711yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×D4×C6711yescharacteristic
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446Q16—22no—
447Q16—81no—
448Q16—81no—
449SD16—22no—
450SD16—42no—
451SD16—42no—
452SD16—81no—
453D8—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⋊C4C2×D4×C6711yescharacteristic
478C4⋊C4C2×D4×C6711yescharacteristic
479C4⋊C4C2×D4×C6711yescharacteristic
480C4⋊C4C2×D4×C6711yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
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—82no—
506Q8—81no—
507Q8—161no—
508Q8#135×C6712yes—
509D4—21no—
510D4—21no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—81no—
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#459×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. Q8×SD16×C67C2semidirect
3. C67×#3375C2semidirect
4. C67×#3328C2semidirect
5. C67×#1708C2semidirect
6. C67×#1700C2semidirect
8. C67×#1672C2semidirect
9. C67×#1666C2semidirect
10. C67×#1645C2semidirect
11. C67×#1640C2semidirect
12. C67×#1638C2semidirect
14. C67×#1590C2semidirect
15. C8×Q16×C67C2semidirect
279. #19921C67direct (N × H)
286. Q8×SD16C2×C67semidirect
287. #3375C2×C67semidirect
288. #3328C2×C67semidirect
289. #1708C2×C67semidirect
290. #1700C2×C67semidirect
292. #1672C2×C67semidirect
293. #1666C2×C67semidirect
294. #1645C2×C67semidirect
295. #1640C2×C67semidirect
296. #1638C2×C67semidirect
298. #1590C2×C67semidirect
299. C8×Q16C2×C67semidirect
300. C67#19921direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33942
Q8×SD16×C672
C67×#33752
C67×#33282
C67×#17082
C67×#17002
C67×#16972
C67×#16722
C67×#16662
C67×#16452
C67×#16402
C67×#16382
C67×#16272
C67×#15902
C8×Q16×C672
#1992167