Finite Group Databasebeta

← C67×#20149 #30699400

Subgroups — C67×#20149 #30699400

Order 17152.

576 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3394C211yesmaximal, characteristic
2C67×#3372C211yesmaximal, characteristic, complemented
3C67×#3365C211yesmaximal, characteristic, complemented
4C67×#3356C211yesmaximal, characteristic, complemented
5C67×#1714C211yesmaximal, characteristic, complemented
6C67×#1701C211yesmaximal, characteristic
7C67×#1686C211yesmaximal, characteristic, complemented
8C67×#1684C211yesmaximal, characteristic, complemented
9C67×#1672C211yesmaximal, characteristic, complemented
10C67×#1650C211yesmaximal, characteristic, complemented
11C67×#1648C211yesmaximal, characteristic, complemented
12C67×#1630C211yesmaximal, characteristic, complemented
13C67×#1622C211yesmaximal, characteristic
14C67×#1588C211yesmaximal, characteristic, complemented
15C67×#1587C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19#554×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#495×C67—21no—
23#492×C67C2211yescharacteristic
24#492×C67C2211yescharacteristic
25#491×C67—21no—
26#488×C67—21no—
27#487×C67C2211yescharacteristic
28#486×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#475×C67C2211yescharacteristic
37#470×C67—21no—
38#464×C67—21no—
39#464×C67—21no—
40#463×C67—21no—
41#462×C67C2211yescharacteristic
42#461×C67—21no—
43#461×C67C2211yescharacteristic
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
66#322×C67C2211yescharacteristic
67C2×Q16×C67—21no—
68C2×Q16×C67D411yescharacteristic
69C2×SD16×C67—21no—
70C2×SD16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67—21no—
73C2×SD16×C67—41no—
74C2×SD16×C67D411yescharacteristic
75C2×D8×C67—21no—
76C2×M16×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81#127×C67C2311yescharacteristic
82#126×C67—41no—
83#125×C67—21no—
84#125×C67—21no—
85#125×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—
104C2×C4⋊C4×C67—21no—
105#107×C67—21no—
106#107×C67—21no—
107#107×C67—41no—
108#107×C67D411yescharacteristic
109#106×C67—21no—
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—21no—
113#106×C67D411yescharacteristic
114C4⋊C8×C67—21no—
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×C67D411yescharacteristic
130#103×C67D411yescharacteristic
131#103×C67D411yescharacteristic
132#102×C67—21no—
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—41no—
158Q16×C67—81no—
159SD16×C67—41no—
160SD16×C67—42no—
161SD16×C67—42no—
162SD16×C67—42no—
163SD16×C67—81no—
164SD16×C67—82no—
165D8×C67—81no—
166M16×C67—42no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C2×C8×C67—21no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
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—41no—
186C4⋊C4×C67—41no—
187C4⋊C4×C67—41no—
188C4⋊C4×C67—41no—
189C4⋊C4×C67—41no—
190C4⋊C4×C67—41no—
191C4⋊C4×C67C2×D411yescharacteristic
192C4⋊C4×C67C2×D411yescharacteristic
193C4⋊C4×C67C2×D411yescharacteristic
194C4⋊C4×C67C2×D411yescharacteristic
195C22⋊C4×C67—21no—
196C22⋊C4×C67—41no—
197C22⋊C4×C67—41no—
198C22⋊C4×C67—41no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—41no—
206C42×C67—41no—
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209C23×C67—41no—
210Q8×C67—21no—
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—81no—
221D4×C67—21no—
222D4×C67—21no—
223D4×C67—41no—
224D4×C67—41no—
225D4×C67—41no—
226D4×C67—41no—
227D4×C67—81no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—21no—
234C2×C4×C67—21no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—161no—
241C2×C4×C67—161no—
242C2×C4×C67C22×D411yescharacteristic
243C2×C4×C67C22×D411yescharacteristic
244C2×C4×C67H(5,𝔽2)11yescharacteristic
245C8×C67—22no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—41no—
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C22×C67—41no—
255C22×C67—41no—
256C22×C67—41no—
257C22×C67—81no—
258C22×C67—81no—
259C22×C67—81no—
260C22×C67D4211yescharacteristic
261C4×C67—21no—
262C4×C67—41no—
263C4×C67—41no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—81no—
270C4×C67—81no—
271C4×C67—81no—
272C4×C67#57511yescharacteristic
273C4×C67#57511yescharacteristic
274C4×C67#57711yescharacteristic
275C4×C67#57811yescharacteristic
276#20149C6711yesmaximal, characteristic, direct product factor, Sylow
277C2×C67—81no—
278C2×C67—161no—
279C2×C67#326911yescharacteristic
280C2×C67#327111yescharacteristic
281C2×C67#329111yescharacteristic
282#3394C2×C6711yescharacteristic
283#3372C2×C6711yescharacteristic, complemented
284#3365C2×C6711yescharacteristic, complemented
285#3356C2×C6711yescharacteristic, complemented
286#1714C2×C6711yescharacteristic, complemented
287#1701C2×C6711yescharacteristic
288#1686C2×C6711yescharacteristic, complemented
289#1684C2×C6711yescharacteristic, complemented
290#1672C2×C6711yescharacteristic, complemented
291#1650C2×C6711yescharacteristic, complemented
292#1648C2×C6711yescharacteristic, complemented
293#1630C2×C6711yescharacteristic, complemented
294#1622C2×C6711yescharacteristic
295#1588C2×C6711yescharacteristic, complemented
296#1587C2×C6711yescharacteristic, complemented
297C67#2014911yescharacteristic, direct product factor, Sylow
298#559C22×C6711yescharacteristic
299#557C22×C6711yescharacteristic
300#556C22×C6711yescharacteristic
301#554C22×C6711yescharacteristic
302#500C22×C6711yescharacteristic
303#498C22×C6711yescharacteristic
304#495—21no—
305#492C22×C6711yescharacteristic
306#492C22×C6711yescharacteristic
307#491—21no—
308#488—21no—
309#487C22×C6711yescharacteristic
310#486C22×C6711yescharacteristic
311#485—21no—
312#483—21no—
313#479—21no—
314#478—21no—
315#477C22×C6711yescharacteristic
316#477C22×C6711yescharacteristic
317#475C22×C6711yescharacteristic
318#475C22×C6711yescharacteristic
319#470—21no—
320#464—21no—
321#464—21no—
322#463—21no—
323#462C22×C6711yescharacteristic
324#461—21no—
325#461C22×C6711yescharacteristic
326#460C22×C6711yescharacteristic
327#459C22×C6711yescharacteristic
328#446C22×C6711yescharacteristic
329C8×Q8C22×C6711yescharacteristic
330C8×Q8C22×C6711yescharacteristic
331#440—21no—
332C4×Q16C22×C6711yescharacteristic
333C4×SD16—21no—
334C4×SD16—21no—
335C4×SD16C22×C6711yescharacteristic
336#436C22×C6711yescharacteristic
337#428—21no—
338#335C22×C6711yescharacteristic
339#334C22×C6711yescharacteristic
340#333C22×C6711yescharacteristic
341#332C22×C6711yescharacteristic
342#332C22×C6711yescharacteristic
343#331C22×C6711yescharacteristic
344Q8⋊C8C22×C6711yescharacteristic
345Q8⋊C8C22×C6711yescharacteristic
346Q8⋊C8C22×C6711yescharacteristic
347D4⋊C8C22×C6711yescharacteristic
348#322C22×C6711yescharacteristic
349C2×Q16—21no—
350C2×Q16D4×C6711yescharacteristic
351C2×SD16—21no—
352C2×SD16—21no—
353C2×SD16—21no—
354C2×SD16—21no—
355C2×SD16—41no—
356C2×SD16D4×C6711yescharacteristic
357C2×D8—21no—
358C2×M16—21no—
359C4⋊Q8—21no—
360C4⋊Q8C23×C6711yescharacteristic
361C4⋊Q8C23×C6711yescharacteristic
362C4⋊Q8C23×C6711yescharacteristic
363#127C23×C6711yescharacteristic
364#126—41no—
365#125—21no—
366#125—21no—
367#125—41no—
368#124—21no—
369#124—21no—
370#124—21no—
371#123—41no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—41no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8C23×C6711yescharacteristic
379C4×Q8C23×C6711yescharacteristic
380C4×Q8C23×C6711yescharacteristic
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4C23×C6711yescharacteristic
385#117—21no—
386C2×C4⋊C4—21no—
387#107—21no—
388#107—21no—
389#107—41no—
390#107D4×C6711yescharacteristic
391#106—21no—
392#106—21no—
393#106—21no—
394#106—21no—
395#106D4×C6711yescharacteristic
396C4⋊C8—21no—
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#103D4×C6711yescharacteristic
412#103D4×C6711yescharacteristic
413#103D4×C6711yescharacteristic
414#102—21no—
415#102—21no—
416#102—21no—
417#102—21no—
418#102D4×C6711yescharacteristic
419C22⋊C8—21no—
420#97—21no—
421C4×C8—21no—
422C4×C8—21no—
423C4×C8C23×C6711yescharacteristic
424C4×C8C23×C6711yescharacteristic
425C4×C8C23×C6711yescharacteristic
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8C2×D4×C6711yescharacteristic
430C2×Q8C2×D4×C6711yescharacteristic
431C2×Q8C2×D4×C6711yescharacteristic
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4C2×D4×C6711yescharacteristic
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—41no—
439Q16—41no—
440Q16—81no—
441SD16—41no—
442SD16—42no—
443SD16—42no—
444SD16—42no—
445SD16—81no—
446SD16—82no—
447D8—81no—
448M16—42no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8C2×D4×C6711yescharacteristic
455C2×C8C2×D4×C6711yescharacteristic
456C2×C8C2×D4×C6711yescharacteristic
457C2×C8C2×D4×C6711yescharacteristic
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⋊C4C2×D4×C6711yescharacteristic
474C4⋊C4C2×D4×C6711yescharacteristic
475C4⋊C4C2×D4×C6711yescharacteristic
476C4⋊C4C2×D4×C6711yescharacteristic
477C22⋊C4—21no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—41no—
488C42—41no—
489C42C24×C6711yescharacteristic
490C23—21no—
491C23—41no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—81no—
503D4—21no—
504D4—21no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—81no—
510C2×C4—21no—
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—161no—
523C2×C4—161no—
524C2×C4C22×D4×C6711yescharacteristic
525C2×C4C22×D4×C6711yescharacteristic
526C2×C4H(5,𝔽2)×C6711yescharacteristic
527C8—22no—
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22D42×C6711yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4#575×C6711yescharacteristic
555C4#575×C6711yescharacteristic
556C4#577×C6711yescharacteristic
557C4#578×C6711yescharacteristic
558C2—81no—
559C2—161no—
560C2C67×#326911yescharacteristic
561C2C67×#327111yescharacteristic
562C2C67×#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
2. C67×#3372C2semidirect
3. C67×#3365C2semidirect
4. C67×#3356C2semidirect
5. C67×#1714C2semidirect
7. C67×#1686C2semidirect
8. C67×#1684C2semidirect
9. C67×#1672C2semidirect
10. C67×#1650C2semidirect
11. C67×#1648C2semidirect
12. C67×#1630C2semidirect
14. C67×#1588C2semidirect
15. C67×#1587C2semidirect
276. #20149C67direct (N × H)
283. #3372C2×C67semidirect
284. #3365C2×C67semidirect
285. #3356C2×C67semidirect
286. #1714C2×C67semidirect
288. #1686C2×C67semidirect
289. #1684C2×C67semidirect
290. #1672C2×C67semidirect
291. #1650C2×C67semidirect
292. #1648C2×C67semidirect
293. #1630C2×C67semidirect
295. #1588C2×C67semidirect
296. #1587C2×C67semidirect
297. C67#20149direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33942
C67×#33722
C67×#33652
C67×#33562
C67×#17142
C67×#17012
C67×#16862
C67×#16842
C67×#16722
C67×#16502
C67×#16482
C67×#16302
C67×#16222
C67×#15882
C67×#15872
#2014967