Finite Group Databasebeta

← C67×#19952 #30699203

Subgroups — C67×#19952 #30699203

Order 17152.

586 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×#3400C211yesmaximal, characteristic, complemented
2C67×#3394C211yesmaximal, characteristic
3C67×#3381C211yesmaximal, characteristic, complemented
4C67×#3356C211yesmaximal, characteristic, complemented
5C67×#1705C211yesmaximal, characteristic, complemented
6C67×#1692C211yesmaximal, characteristic, complemented
7C67×#1685C211yesmaximal, characteristic
8C67×#1672C211yesmaximal, characteristic, complemented
9C67×#1666C211yesmaximal, characteristic, complemented
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1652C211yesmaximal, characteristic, complemented
12C67×#1630C211yesmaximal, characteristic, complemented
13C67×#1622C211yesmaximal, characteristic
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×Q16×C67C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19#554×C67C2211yescharacteristic
20#499×C67—21no—
21#498×C67C2211yescharacteristic
22#498×C67C2211yescharacteristic
23#498×C67C2211yescharacteristic
24#497×C67—21no—
25#495×C67—21no—
26#492×C67C2211yescharacteristic
27#492×C67C2211yescharacteristic
28#492×C67C2211yescharacteristic
29#485×C67—21no—
30#483×C67—21no—
31#479×C67—21no—
32#478×C67—21no—
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#474×C67C2211yescharacteristic
37#470×C67—21no—
38#464×C67—21no—
39#464×C67—21no—
40#463×C67—21no—
41#462×C67C2211yescharacteristic
42#462×C67C2211yescharacteristic
43#461×C67—21no—
44#460×C67C2211yescharacteristic
45#459×C67C2211yescharacteristic
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#428×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×Q16×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×C67—41no—
76C2×SD16×C67D411yescharacteristic
77C2×D8×C67—21no—
78C2×M16×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#127×C67C2311yescharacteristic
84#126×C67—41no—
85#125×C67—21no—
86#125×C67—21no—
87#125×C67—41no—
88#124×C67—21no—
89#124×C67—21no—
90#124×C67—21no—
91#123×C67—41no—
92C22⋊Q8×C67—21no—
93C22⋊Q8×C67—21no—
94C22⋊Q8×C67—41no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67—21no—
98C4×Q8×C67C2311yescharacteristic
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×D4×C67—21no—
102C4×D4×C67—21no—
103C4×D4×C67—21no—
104C4×D4×C67C2311yescharacteristic
105#117×C67—21no—
106C2×C4⋊C4×C67—21no—
107#107×C67—21no—
108#107×C67—21no—
109#107×C67—21no—
110#107×C67D411yescharacteristic
111#106×C67—21no—
112#106×C67—21no—
113#106×C67—21no—
114#106×C67—21no—
115#106×C67—21no—
116#106×C67—41no—
117#106×C67D411yescharacteristic
118C4⋊C8×C67—21no—
119C4⋊C8×C67—21no—
120C4⋊C8×C67—21no—
121C4⋊C8×C67C2311yescharacteristic
122C4⋊C8×C67C2311yescharacteristic
123C4⋊C8×C67C2311yescharacteristic
124C4⋊C8×C67C2311yescharacteristic
125#103×C67—21no—
126#103×C67—21no—
127#103×C67—21no—
128#103×C67—21no—
129#103×C67—21no—
130#103×C67D411yescharacteristic
131#103×C67D411yescharacteristic
132#103×C67D411yescharacteristic
133#102×C67—21no—
134#102×C67—21no—
135#102×C67—21no—
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×C67C2×D411yescharacteristic
148C2×Q8×C67C2×D411yescharacteristic
149C2×Q8×C67C2×D411yescharacteristic
150C2×D4×C67—21no—
151C2×D4×C67—21no—
152C2×D4×C67—21no—
153C2×D4×C67C2×D411yescharacteristic
154C22×C4×C67—21no—
155C22×C4×C67—41no—
156C22×C4×C67—41no—
157Q16×C67—22no—
158Q16×C67—81no—
159Q16×C67—81no—
160SD16×C67—22no—
161SD16×C67—42no—
162SD16×C67—42no—
163SD16×C67—81no—
164SD16×C67—81no—
165SD16×C67—82no—
166SD16×C67—81no—
167D8×C67—81no—
168M16×C67—42no—
169C2×C8×C67—21no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67C2×D411yescharacteristic
175C2×C8×C67C2×D411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C4⋊C4×C67—21no—
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—41no—
188C4⋊C4×C67—41no—
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—41no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C22⋊C4×C67—41no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67—21no—
207C42×C67—41no—
208C42×C67—41no—
209C42×C67C2411yescharacteristic
210C23×C67—21no—
211C23×C67—41no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—81no—
223D4×C67—21no—
224D4×C67—21no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—41no—
228D4×C67—41no—
229D4×C67—81no—
230C2×C4×C67—21no—
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—161no—
243C2×C4×C67—161no—
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67H(5,𝔽2)11yescharacteristic
247C8×C67—22no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—41no—
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67#13512yes—
255C8×C67#13712yes—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—81no—
260C22×C67—81no—
261C22×C67—81no—
262C22×C67D4211yescharacteristic
263C4×C67—21no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—81no—
272C4×C67—81no—
273C4×C67—81no—
274C4×C67C2×#13511yescharacteristic
275C4×C67C2×#13711yescharacteristic
276C4×C67#57711yescharacteristic
277C4×C67#57711yescharacteristic
278#19952C6711yesmaximal, characteristic, direct product factor, Sylow
279C2×C67—81no—
280C2×C67—161no—
281C2×C67#327811yescharacteristic
282C2×C67#328411yescharacteristic
283C2×C67#329111yescharacteristic
284#3400C2×C6711yescharacteristic, complemented
285#3394C2×C6711yescharacteristic
286#3381C2×C6711yescharacteristic, complemented
287#3356C2×C6711yescharacteristic, complemented
288#1705C2×C6711yescharacteristic, complemented
289#1692C2×C6711yescharacteristic, complemented
290#1685C2×C6711yescharacteristic
291#1672C2×C6711yescharacteristic, complemented
292#1666C2×C6711yescharacteristic, complemented
293#1653C2×C6711yescharacteristic, complemented
294#1652C2×C6711yescharacteristic, complemented
295#1630C2×C6711yescharacteristic, complemented
296#1622C2×C6711yescharacteristic
297#1590C2×C6711yescharacteristic, complemented
298C8×Q16C2×C6711yescharacteristic, complemented
299C67#1995211yescharacteristic, direct product factor, Sylow
300#559C22×C6711yescharacteristic
301#557C22×C6711yescharacteristic
302#556C22×C6711yescharacteristic
303#554C22×C6711yescharacteristic
304#499—21no—
305#498C22×C6711yescharacteristic
306#498C22×C6711yescharacteristic
307#498C22×C6711yescharacteristic
308#497—21no—
309#495—21no—
310#492C22×C6711yescharacteristic
311#492C22×C6711yescharacteristic
312#492C22×C6711yescharacteristic
313#485—21no—
314#483—21no—
315#479—21no—
316#478—21no—
317#477C22×C6711yescharacteristic
318#477C22×C6711yescharacteristic
319#475C22×C6711yescharacteristic
320#474C22×C6711yescharacteristic
321#470—21no—
322#464—21no—
323#464—21no—
324#463—21no—
325#462C22×C6711yescharacteristic
326#462C22×C6711yescharacteristic
327#461—21no—
328#460C22×C6711yescharacteristic
329#459C22×C6711yescharacteristic
330#446C22×C6711yescharacteristic
331C8×Q8C22×C6711yescharacteristic
332C8×Q8C22×C6711yescharacteristic
333#440—21no—
334C4×Q16C22×C6711yescharacteristic
335C4×SD16—21no—
336C4×SD16—21no—
337C4×SD16C22×C6711yescharacteristic
338#436C22×C6711yescharacteristic
339#428—21no—
340#335C22×C6711yescharacteristic
341#334C22×C6711yescharacteristic
342#333C22×C6711yescharacteristic
343#332C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345#331C22×C6711yescharacteristic
346Q8⋊C8C22×C6711yescharacteristic
347Q8⋊C8C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349D4⋊C8C22×C6711yescharacteristic
350C82C22×C6711yescharacteristic
351C2×Q16—21no—
352C2×Q16—21no—
353C2×Q16D4×C6711yescharacteristic
354C2×SD16—21no—
355C2×SD16—21no—
356C2×SD16—21no—
357C2×SD16—21no—
358C2×SD16—21no—
359C2×SD16—41no—
360C2×SD16D4×C6711yescharacteristic
361C2×D8—21no—
362C2×M16—21no—
363C4⋊Q8—21no—
364C4⋊Q8C23×C6711yescharacteristic
365C4⋊Q8C23×C6711yescharacteristic
366C4⋊Q8C23×C6711yescharacteristic
367#127C23×C6711yescharacteristic
368#126—41no—
369#125—21no—
370#125—21no—
371#125—41no—
372#124—21no—
373#124—21no—
374#124—21no—
375#123—41no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—41no—
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×D4—21no—
388C4×D4C23×C6711yescharacteristic
389#117—21no—
390C2×C4⋊C4—21no—
391#107—21no—
392#107—21no—
393#107—21no—
394#107D4×C6711yescharacteristic
395#106—21no—
396#106—21no—
397#106—21no—
398#106—21no—
399#106—21no—
400#106—41no—
401#106D4×C6711yescharacteristic
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—21no—
405C4⋊C8C23×C6711yescharacteristic
406C4⋊C8C23×C6711yescharacteristic
407C4⋊C8C23×C6711yescharacteristic
408C4⋊C8C23×C6711yescharacteristic
409#103—21no—
410#103—21no—
411#103—21no—
412#103—21no—
413#103—21no—
414#103D4×C6711yescharacteristic
415#103D4×C6711yescharacteristic
416#103D4×C6711yescharacteristic
417#102—21no—
418#102—21no—
419#102—21no—
420#102D4×C6711yescharacteristic
421C22⋊C8—21no—
422#97—21no—
423C4×C8—21no—
424C4×C8—21no—
425C4×C8C23×C6711yescharacteristic
426C4×C8C23×C6711yescharacteristic
427C4×C8C23×C6711yescharacteristic
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×D4×C6711yescharacteristic
432C2×Q8C2×D4×C6711yescharacteristic
433C2×Q8C2×D4×C6711yescharacteristic
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4C2×D4×C6711yescharacteristic
438C22×C4—21no—
439C22×C4—41no—
440C22×C4—41no—
441Q16—22no—
442Q16—81no—
443Q16—81no—
444SD16—22no—
445SD16—42no—
446SD16—42no—
447SD16—81no—
448SD16—81no—
449SD16—82no—
450SD16—81no—
451D8—81no—
452M16—42no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8C2×D4×C6711yescharacteristic
459C2×C8C2×D4×C6711yescharacteristic
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C4⋊C4—21no—
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—41no—
472C4⋊C4—41no—
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—21no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—81no—
507D4—21no—
508D4—21no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—81no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—161no—
527C2×C4—161no—
528C2×C4C22×D4×C6711yescharacteristic
529C2×C4C22×D4×C6711yescharacteristic
530C2×C4H(5,𝔽2)×C6711yescharacteristic
531C8—22no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8#135×C6712yes—
539C8#137×C6712yes—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22D42×C6711yescharacteristic
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4C2×#135×C6711yescharacteristic
559C4C2×#137×C6711yescharacteristic
560C4#577×C6711yescharacteristic
561C4#577×C6711yescharacteristic
562C2—81no—
563C2—161no—
564C2C67×#327811yescharacteristic
565C2C67×#328411yescharacteristic
566C2C67×#329111yescharacteristic
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
1. C67×#3400C2semidirect
3. C67×#3381C2semidirect
4. C67×#3356C2semidirect
5. C67×#1705C2semidirect
6. C67×#1692C2semidirect
8. C67×#1672C2semidirect
9. C67×#1666C2semidirect
10. C67×#1653C2semidirect
11. C67×#1652C2semidirect
12. C67×#1630C2semidirect
14. C67×#1590C2semidirect
15. C8×Q16×C67C2semidirect
278. #19952C67direct (N × H)
284. #3400C2×C67semidirect
286. #3381C2×C67semidirect
287. #3356C2×C67semidirect
288. #1705C2×C67semidirect
289. #1692C2×C67semidirect
291. #1672C2×C67semidirect
292. #1666C2×C67semidirect
293. #1653C2×C67semidirect
294. #1652C2×C67semidirect
295. #1630C2×C67semidirect
297. #1590C2×C67semidirect
298. C8×Q16C2×C67semidirect
299. C67#19952direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#34002
C67×#33942
C67×#33812
C67×#33562
C67×#17052
C67×#16922
C67×#16852
C67×#16722
C67×#16662
C67×#16532
C67×#16522
C67×#16302
C67×#16222
C67×#15902
C8×Q16×C672
#1995267