Finite Group Databasebeta

← C67×#19917 #30699168

Subgroups — C67×#19917 #30699168

Order 17152.

588 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3381C211yesmaximal, characteristic, complemented
2C67×#3372C211yesmaximal, characteristic, complemented
3C67×#3335C211yesmaximal, characteristic
4C67×#3331C211yesmaximal, characteristic
5C67×#1707C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1703C211yesmaximal, characteristic, complemented
8C67×#1672C211yesmaximal, characteristic, complemented
9C67×#1671C211yesmaximal, characteristic, complemented
10C67×#1648C211yesmaximal, characteristic, complemented
11C67×#1647C211yesmaximal, characteristic
12C67×#1632C211yesmaximal, characteristic, complemented
13C67×#1632C211yesmaximal, characteristic, complemented
14C67×#1590C211yesmaximal, characteristic, complemented
15C8×SD16×C67C211yesmaximal, characteristic, complemented
16#557×C67C2211yescharacteristic
17#554×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#499×C67—21no—
22#498×C67C2211yescharacteristic
23#498×C67C2211yescharacteristic
24#494×C67C2211yescharacteristic
25#491×C67—21no—
26#487×C67C2211yescharacteristic
27#487×C67C2211yescharacteristic
28#483×C67—21no—
29#483×C67—21no—
30#481×C67—21no—
31#479×C67—21no—
32#479×C67—21no—
33#478×C67—21no—
34#477×C67C2211yescharacteristic
35#477×C67C2211yescharacteristic
36#476×C67C2211yescharacteristic
37#476×C67C2211yescharacteristic
38#475×C67C2211yescharacteristic
39#474×C67C2211yescharacteristic
40#470×C67—21no—
41#465×C67—21no—
42#462×C67C2211yescharacteristic
43#461×C67—21no—
44#461×C67C2211yescharacteristic
45#452×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
54C2×#106×C67—21no—
55#417×C67—21no—
56#334×C67C2211yescharacteristic
57#334×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
66C82×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×Q16×C67—21no—
69C2×Q16×C67—41no—
70C2×SD16×C67—21no—
71C2×SD16×C67D411yescharacteristic
72C2×SD16×C67D411yescharacteristic
73C2×M16×C67—21no—
74C22×C8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67C2311yescharacteristic
77C4⋊Q8×C67C2311yescharacteristic
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80#125×C67—21no—
81#125×C67—21no—
82#125×C67—21no—
83#125×C67—41no—
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—
101C2×C4⋊C4×C67—21no—
102#107×C67—21no—
103#107×C67—21no—
104#107×C67—21no—
105#107×C67—41no—
106#106×C67—21no—
107#106×C67—21no—
108#106×C67—21no—
109#106×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×C67—21no—
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#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×C67C2×D411yescharacteristic
151C2×Q8×C67C2×D411yescharacteristic
152C2×D4×C67—41no—
153C2×D4×C67C2×D411yescharacteristic
154C2×D4×C67C2×D411yescharacteristic
155C22×C4×C67—21no—
156C22×C4×C67—21no—
157C22×C4×C67—41no—
158C22×C4×C67—41no—
159C22×C4×C67—41no—
160Q16×C67—82no—
161Q16×C67—81no—
162SD16×C67—22no—
163SD16×C67—22no—
164SD16×C67—42no—
165M16×C67—81no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C2×C8×C67—21no—
170C2×C8×C67—21no—
171C2×C8×C67—42no—
172C2×C8×C67C2×D411yescharacteristic
173C2×C8×C67C2×D411yescharacteristic
174C2×C8×C67C2×D411yescharacteristic
175C2×C8×C67C2×D411yescharacteristic
176C4⋊C4×C67—21no—
177C4⋊C4×C67—21no—
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—41no—
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×C67—41no—
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C4⋊C4×C67C2×D411yescharacteristic
198C22⋊C4×C67—21no—
199C22⋊C4×C67—21no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C22⋊C4×C67—41no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—41no—
206C42×C67C2411yescharacteristic
207C23×C67—21no—
208C23×C67—21no—
209Q8×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—81no—
218Q8×C67—81no—
219D4×C67—21no—
220D4×C67—21no—
221D4×C67—21no—
222D4×C67—82no—
223D4×C67#13512yes—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—41no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—82no—
239C2×C4×C67—161no—
240C2×C4×C67—161no—
241C2×C4×C67C22×D411yescharacteristic
242C2×C4×C67C22×D411yescharacteristic
243C2×C4×C67H(5,𝔽2)11yescharacteristic
244C8×C67—22no—
245C8×C67—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—41no—
249C8×C67—41no—
250C8×C67—41no—
251C8×C67—41no—
252C8×C67#13712yes—
253C22×C67—22no—
254C22×C67—41no—
255C22×C67—41no—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67D4211yescharacteristic
260C4×C67—21no—
261C4×C67—41no—
262C4×C67—41no—
263C4×C67—41no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—81no—
268C4×C67—81no—
269C4×C67—81no—
270C4×C67—81no—
271C4×C67C2×#13511yescharacteristic
272C4×C67C2×#13711yescharacteristic
273C4×C67#57711yescharacteristic
274C4×C67#57811yescharacteristic
275#19917C6711yesmaximal, characteristic, direct product factor, Sylow
276C2×C67—42no—
277C2×C67—81no—
278C2×C67#327611yescharacteristic
279C2×C67#328211yescharacteristic
280C2×C67#329111yescharacteristic
281#3381C2×C6711yescharacteristic, complemented
282#3372C2×C6711yescharacteristic, complemented
283#3335C2×C6711yescharacteristic
284#3331C2×C6711yescharacteristic
285#1707C2×C6711yescharacteristic, complemented
286#1704C2×C6711yescharacteristic, complemented
287#1703C2×C6711yescharacteristic, complemented
288#1672C2×C6711yescharacteristic, complemented
289#1671C2×C6711yescharacteristic, complemented
290#1648C2×C6711yescharacteristic, complemented
291#1647C2×C6711yescharacteristic
292#1632C2×C6711yescharacteristic, complemented
293#1632C2×C6711yescharacteristic, complemented
294#1590C2×C6711yescharacteristic, complemented
295C8×SD16C2×C6711yescharacteristic, complemented
296C67#1991711yescharacteristic, direct product factor, Sylow
297#557C22×C6711yescharacteristic
298#554C22×C6711yescharacteristic
299#554C22×C6711yescharacteristic
300#547C22×C6711yescharacteristic
301#500C22×C6711yescharacteristic
302#499—21no—
303#498C22×C6711yescharacteristic
304#498C22×C6711yescharacteristic
305#494C22×C6711yescharacteristic
306#491—21no—
307#487C22×C6711yescharacteristic
308#487C22×C6711yescharacteristic
309#483—21no—
310#483—21no—
311#481—21no—
312#479—21no—
313#479—21no—
314#478—21no—
315#477C22×C6711yescharacteristic
316#477C22×C6711yescharacteristic
317#476C22×C6711yescharacteristic
318#476C22×C6711yescharacteristic
319#475C22×C6711yescharacteristic
320#474C22×C6711yescharacteristic
321#470—21no—
322#465—21no—
323#462C22×C6711yescharacteristic
324#461—21no—
325#461C22×C6711yescharacteristic
326#452—21no—
327#446C22×C6711yescharacteristic
328C8×Q8C22×C6711yescharacteristic
329#440—21no—
330C4×SD16—21no—
331C4×SD16C22×C6711yescharacteristic
332C4×SD16C22×C6711yescharacteristic
333#436C22×C6711yescharacteristic
334C8×D4C22×C6711yescharacteristic
335C2×#106—21no—
336#417—21no—
337#334C22×C6711yescharacteristic
338#334C22×C6711yescharacteristic
339#333C22×C6711yescharacteristic
340#333C22×C6711yescharacteristic
341#332C22×C6711yescharacteristic
342#332C22×C6711yescharacteristic
343Q8⋊C8C22×C6711yescharacteristic
344Q8⋊C8C22×C6711yescharacteristic
345D4⋊C8C22×C6711yescharacteristic
346D4⋊C8C22×C6711yescharacteristic
347C82C22×C6711yescharacteristic
348C2×#41—21no—
349C2×Q16—21no—
350C2×Q16—41no—
351C2×SD16—21no—
352C2×SD16D4×C6711yescharacteristic
353C2×SD16D4×C6711yescharacteristic
354C2×M16—21no—
355C22×C8—21no—
356C4⋊Q8—21no—
357C4⋊Q8C23×C6711yescharacteristic
358C4⋊Q8C23×C6711yescharacteristic
359C4⋊Q8C23×C6711yescharacteristic
360C4⋊Q8C23×C6711yescharacteristic
361#125—21no—
362#125—21no—
363#125—21no—
364#125—41no—
365#123—41no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—41no—
370C22⋊Q8—41no—
371#121—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8C23×C6711yescharacteristic
375C4×Q8C23×C6711yescharacteristic
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C23×C6711yescharacteristic
379C4×D4C23×C6711yescharacteristic
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383#107—21no—
384#107—21no—
385#107—21no—
386#107—41no—
387#106—21no—
388#106—21no—
389#106—21no—
390#106—21no—
391#106—21no—
392#106—21no—
393#106—21no—
394#106D4×C6711yescharacteristic
395#106D4×C6711yescharacteristic
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8C23×C6711yescharacteristic
399C4⋊C8C23×C6711yescharacteristic
400C4⋊C8C23×C6711yescharacteristic
401C4⋊C8C23×C6711yescharacteristic
402#103—21no—
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#103D4×C6711yescharacteristic
412#103D4×C6711yescharacteristic
413#102—21no—
414#102—21no—
415#102—21no—
416#102D4×C6711yescharacteristic
417#102D4×C6711yescharacteristic
418C22⋊C8—21no—
419C22⋊C8—21no—
420#97—21no—
421C4×C8—21no—
422C4×C8C23×C6711yescharacteristic
423C4×C8C23×C6711yescharacteristic
424C4×C8C23×C6711yescharacteristic
425#41—42no—
426#41—81no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×D4×C6711yescharacteristic
432C2×Q8C2×D4×C6711yescharacteristic
433C2×D4—41no—
434C2×D4C2×D4×C6711yescharacteristic
435C2×D4C2×D4×C6711yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441Q16—82no—
442Q16—81no—
443SD16—22no—
444SD16—22no—
445SD16—42no—
446M16—81no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8C2×D4×C6711yescharacteristic
454C2×C8C2×D4×C6711yescharacteristic
455C2×C8C2×D4×C6711yescharacteristic
456C2×C8C2×D4×C6711yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4C2×D4×C6711yescharacteristic
476C4⋊C4C2×D4×C6711yescharacteristic
477C4⋊C4C2×D4×C6711yescharacteristic
478C4⋊C4C2×D4×C6711yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C42—21no—
485C42—21no—
486C42—41no—
487C42C24×C6711yescharacteristic
488C23—21no—
489C23—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—81no—
499Q8—81no—
500D4—21no—
501D4—21no—
502D4—21no—
503D4—82no—
504D4#135×C6712yes—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—161no—
521C2×C4—161no—
522C2×C4C22×D4×C6711yescharacteristic
523C2×C4C22×D4×C6711yescharacteristic
524C2×C4H(5,𝔽2)×C6711yescharacteristic
525C8—22no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8#137×C6712yes—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22D42×C6711yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C4C2×#135×C6711yescharacteristic
553C4C2×#137×C6711yescharacteristic
554C4#577×C6711yescharacteristic
555C4#578×C6711yescharacteristic
556C2—42no—
557C2—81no—
558C2C67×#327611yescharacteristic
559C2C67×#328211yescharacteristic
560C2C67×#329111yescharacteristic
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×#3381C2semidirect
2. C67×#3372C2semidirect
5. C67×#1707C2semidirect
6. C67×#1704C2semidirect
7. C67×#1703C2semidirect
8. C67×#1672C2semidirect
9. C67×#1671C2semidirect
10. C67×#1648C2semidirect
12. C67×#1632C2semidirect
13. C67×#1632C2semidirect
14. C67×#1590C2semidirect
15. C8×SD16×C67C2semidirect
275. #19917C67direct (N × H)
281. #3381C2×C67semidirect
282. #3372C2×C67semidirect
285. #1707C2×C67semidirect
286. #1704C2×C67semidirect
287. #1703C2×C67semidirect
288. #1672C2×C67semidirect
289. #1671C2×C67semidirect
290. #1648C2×C67semidirect
292. #1632C2×C67semidirect
293. #1632C2×C67semidirect
294. #1590C2×C67semidirect
295. C8×SD16C2×C67semidirect
296. C67#19917direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33812
C67×#33722
C67×#33352
C67×#33312
C67×#17072
C67×#17042
C67×#17032
C67×#16722
C67×#16712
C67×#16482
C67×#16472
C67×#16322
C67×#15902
C8×SD16×C672
#1991767