Finite Group Databasebeta

← C67×#20181 #30699432

Subgroups — C67×#20181 #30699432

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3385C211yesmaximal, characteristic, complemented
2C67×#3385C211yesmaximal, characteristic, complemented
3C67×#3335C211yesmaximal, characteristic
4C67×#3331C211yesmaximal, characteristic
5C67×#1723C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1695C211yesmaximal, characteristic, complemented
8C67×#1692C211yesmaximal, characteristic, complemented
9C67×#1675C211yesmaximal, characteristic, complemented
10C67×#1645C211yesmaximal, characteristic, complemented
11C67×#1635C211yesmaximal, characteristic
12C67×#1632C211yesmaximal, characteristic, complemented
13C67×#1632C211yesmaximal, characteristic, complemented
14C67×#1592C211yesmaximal, characteristic, complemented
15C67×#1582C211yesmaximal, characteristic, complemented
16Q82×C67C2211yescharacteristic
17#554×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19#547×C67C2211yescharacteristic
20#501×C67—21no—
21#501×C67—21no—
22#500×C67C2211yescharacteristic
23#500×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#494×C67C2211yescharacteristic
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#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#476×C67C2211yescharacteristic
36#476×C67C2211yescharacteristic
37#476×C67C2211yescharacteristic
38#476×C67C2211yescharacteristic
39#475×C67C2211yescharacteristic
40#474×C67—21no—
41#474×C67—21no—
42#470×C67—21no—
43#465×C67—21no—
44#462×C67C2211yescharacteristic
45#452×C67—21no—
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48#440×C67—21no—
49#440×C67—21no—
50C4×SD16×C67C2211yescharacteristic
51C4×SD16×C67C2211yescharacteristic
52#436×C67C2211yescharacteristic
53C8×D4×C67C2211yescharacteristic
54C2×#106×C67—21no—
55#417×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#333×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×#41×C67—21no—
68C2×Q16×C67—21no—
69C2×Q16×C67—41no—
70C2×SD16×C67D411yescharacteristic
71C2×SD16×C67D411yescharacteristic
72C2×M16×C67—21no—
73C22×C8×C67—21no—
74C4⋊Q8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67—41no—
77C4⋊Q8×C67C2311yescharacteristic
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81#125×C67—21no—
82#125×C67—21no—
83#123×C67—41no—
84C22⋊Q8×C67—21no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—41no—
88C22⋊Q8×C67—41no—
89#121×C67—21no—
90C4×Q8×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67C2311yescharacteristic
93C4×Q8×C67C2311yescharacteristic
94C4×D4×C67—21no—
95C4×D4×C67—21no—
96C4×D4×C67C2311yescharacteristic
97C4×D4×C67C2311yescharacteristic
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67—21no—
101#107×C67—21no—
102#107×C67—21no—
103#107×C67—21no—
104#107×C67—21no—
105#107×C67—21no—
106#107×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×C67—21no—
114#106×C67D411yescharacteristic
115#106×C67D411yescharacteristic
116C4⋊C8×C67—21no—
117C4⋊C8×C67—21no—
118C4⋊C8×C67C2311yescharacteristic
119C4⋊C8×C67C2311yescharacteristic
120C4⋊C8×C67C2311yescharacteristic
121C4⋊C8×C67C2311yescharacteristic
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×C67D411yescharacteristic
130#103×C67D411yescharacteristic
131#102×C67—21no—
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—
140#97×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×C67C2×D411yescharacteristic
152C2×Q8×C67C2×D411yescharacteristic
153C2×D4×C67—41no—
154C2×D4×C67C2×D411yescharacteristic
155C2×D4×C67C2×D411yescharacteristic
156C22×C4×C67—21no—
157C22×C4×C67—21no—
158C22×C4×C67—41no—
159C22×C4×C67—41no—
160C22×C4×C67—41no—
161Q16×C67—81no—
162Q16×C67—82no—
163SD16×C67—22no—
164SD16×C67—22no—
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×C67C2×D411yescharacteristic
194C4⋊C4×C67C2×D411yescharacteristic
195C4⋊C4×C67C2×D411yescharacteristic
196C4⋊C4×C67C2×D411yescharacteristic
197C22⋊C4×C67—21no—
198C22⋊C4×C67—21no—
199C22⋊C4×C67—41no—
200C22⋊C4×C67—41no—
201C22⋊C4×C67—41no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67—41no—
205C42×C67C2411yescharacteristic
206C23×C67—21no—
207C23×C67—21no—
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—41no—
213Q8×C67—41no—
214Q8×C67—81no—
215Q8×C67—81no—
216Q8×C67—81no—
217Q8×C67—161no—
218D4×C67—21no—
219D4×C67—21no—
220D4×C67—21no—
221D4×C67—82no—
222D4×C67#13712yes—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—41no—
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—82no—
238C2×C4×C67—161no—
239C2×C4×C67—161no—
240C2×C4×C67C22×D411yescharacteristic
241C2×C4×C67C22×D411yescharacteristic
242C2×C4×C67H(5,𝔽2)11yescharacteristic
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—41no—
247C8×C67—41no—
248C8×C67—41no—
249C8×C67—41no—
250C8×C67—41no—
251C8×C67#13712yes—
252C22×C67—22no—
253C22×C67—41no—
254C22×C67—41no—
255C22×C67—41no—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67D4211yescharacteristic
259C4×C67—21no—
260C4×C67—41no—
261C4×C67—41no—
262C4×C67—41no—
263C4×C67—41no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—81no—
267C4×C67—81no—
268C4×C67—81no—
269C4×C67—81no—
270C4×C67C2×#13711yescharacteristic
271C4×C67C2×#13711yescharacteristic
272C4×C67#57711yescharacteristic
273C4×C67#57811yescharacteristic
274#20181C6711yesmaximal, characteristic, direct product factor, Sylow
275C2×C67—42no—
276C2×C67—81no—
277C2×C67#326611yescharacteristic
278C2×C67#327611yescharacteristic
279C2×C67#328811yescharacteristic
280#3385C2×C6711yescharacteristic, complemented
281#3385C2×C6711yescharacteristic, complemented
282#3335C2×C6711yescharacteristic
283#3331C2×C6711yescharacteristic
284#1723C2×C6711yescharacteristic, complemented
285#1704C2×C6711yescharacteristic, complemented
286#1695C2×C6711yescharacteristic, complemented
287#1692C2×C6711yescharacteristic, complemented
288#1675C2×C6711yescharacteristic, complemented
289#1645C2×C6711yescharacteristic, complemented
290#1635C2×C6711yescharacteristic
291#1632C2×C6711yescharacteristic, complemented
292#1632C2×C6711yescharacteristic, complemented
293#1592C2×C6711yescharacteristic, complemented
294#1582C2×C6711yescharacteristic, complemented
295C67#2018111yescharacteristic, direct product factor, Sylow
296Q82C22×C6711yescharacteristic
297#554C22×C6711yescharacteristic
298#554C22×C6711yescharacteristic
299#547C22×C6711yescharacteristic
300#501—21no—
301#501—21no—
302#500C22×C6711yescharacteristic
303#500C22×C6711yescharacteristic
304#498C22×C6711yescharacteristic
305#494C22×C6711yescharacteristic
306#487C22×C6711yescharacteristic
307#487C22×C6711yescharacteristic
308#483—21no—
309#483—21no—
310#481—21no—
311#479—21no—
312#479—21no—
313#477C22×C6711yescharacteristic
314#477C22×C6711yescharacteristic
315#476C22×C6711yescharacteristic
316#476C22×C6711yescharacteristic
317#476C22×C6711yescharacteristic
318#476C22×C6711yescharacteristic
319#475C22×C6711yescharacteristic
320#474—21no—
321#474—21no—
322#470—21no—
323#465—21no—
324#462C22×C6711yescharacteristic
325#452—21no—
326#446C22×C6711yescharacteristic
327#446C22×C6711yescharacteristic
328#440—21no—
329#440—21no—
330C4×SD16C22×C6711yescharacteristic
331C4×SD16C22×C6711yescharacteristic
332#436C22×C6711yescharacteristic
333C8×D4C22×C6711yescharacteristic
334C2×#106—21no—
335#417—21no—
336#335C22×C6711yescharacteristic
337#334C22×C6711yescharacteristic
338#333C22×C6711yescharacteristic
339#333C22×C6711yescharacteristic
340#333C22×C6711yescharacteristic
341#332C22×C6711yescharacteristic
342Q8⋊C8C22×C6711yescharacteristic
343Q8⋊C8C22×C6711yescharacteristic
344D4⋊C8C22×C6711yescharacteristic
345D4⋊C8C22×C6711yescharacteristic
346#322C22×C6711yescharacteristic
347C2×#41—21no—
348C2×Q16—21no—
349C2×Q16—41no—
350C2×SD16D4×C6711yescharacteristic
351C2×SD16D4×C6711yescharacteristic
352C2×M16—21no—
353C22×C8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—41no—
357C4⋊Q8C23×C6711yescharacteristic
358C4⋊Q8C23×C6711yescharacteristic
359C4⋊Q8C23×C6711yescharacteristic
360C4⋊Q8C23×C6711yescharacteristic
361#125—21no—
362#125—21no—
363#123—41no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—41no—
368C22⋊Q8—41no—
369#121—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8C23×C6711yescharacteristic
373C4×Q8C23×C6711yescharacteristic
374C4×D4—21no—
375C4×D4—21no—
376C4×D4C23×C6711yescharacteristic
377C4×D4C23×C6711yescharacteristic
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381#107—21no—
382#107—21no—
383#107—21no—
384#107—21no—
385#107—21no—
386#107—21no—
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—41no—
409#103D4×C6711yescharacteristic
410#103D4×C6711yescharacteristic
411#102—21no—
412#102—21no—
413#102—21no—
414#102—21no—
415#102D4×C6711yescharacteristic
416#102D4×C6711yescharacteristic
417C22⋊C8—21no—
418C22⋊C8—21no—
419#97—21no—
420#97—21no—
421C4×C8C23×C6711yescharacteristic
422C4×C8C23×C6711yescharacteristic
423C4×C8C23×C6711yescharacteristic
424#41—42no—
425#41—81no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—41no—
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—81no—
442Q16—82no—
443SD16—22no—
444SD16—22no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—42no—
452C2×C8C2×D4×C6711yescharacteristic
453C2×C8C2×D4×C6711yescharacteristic
454C2×C8C2×D4×C6711yescharacteristic
455C2×C8C2×D4×C6711yescharacteristic
456C4⋊C4—21no—
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—41no—
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—21no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42C24×C6711yescharacteristic
486C23—21no—
487C23—21no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—41no—
493Q8—41no—
494Q8—81no—
495Q8—81no—
496Q8—81no—
497Q8—161no—
498D4—21no—
499D4—21no—
500D4—21no—
501D4—82no—
502D4#137×C6712yes—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—161no—
519C2×C4—161no—
520C2×C4C22×D4×C6711yescharacteristic
521C2×C4C22×D4×C6711yescharacteristic
522C2×C4H(5,𝔽2)×C6711yescharacteristic
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C8#137×C6712yes—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22D42×C6711yescharacteristic
539C4—21no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4C2×#137×C6711yescharacteristic
551C4C2×#137×C6711yescharacteristic
552C4#577×C6711yescharacteristic
553C4#578×C6711yescharacteristic
554C2—42no—
555C2—81no—
556C2C67×#326611yescharacteristic
557C2C67×#327611yescharacteristic
558C2C67×#328811yescharacteristic
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. C67×#3385C2semidirect
5. C67×#1723C2semidirect
6. C67×#1704C2semidirect
7. C67×#1695C2semidirect
8. C67×#1692C2semidirect
9. C67×#1675C2semidirect
10. C67×#1645C2semidirect
12. C67×#1632C2semidirect
13. C67×#1632C2semidirect
14. C67×#1592C2semidirect
15. C67×#1582C2semidirect
274. #20181C67direct (N × H)
280. #3385C2×C67semidirect
281. #3385C2×C67semidirect
284. #1723C2×C67semidirect
285. #1704C2×C67semidirect
286. #1695C2×C67semidirect
287. #1692C2×C67semidirect
288. #1675C2×C67semidirect
289. #1645C2×C67semidirect
291. #1632C2×C67semidirect
292. #1632C2×C67semidirect
293. #1592C2×C67semidirect
294. #1582C2×C67semidirect
295. C67#20181direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33852
C67×#33352
C67×#33312
C67×#17232
C67×#17042
C67×#16952
C67×#16922
C67×#16752
C67×#16452
C67×#16352
C67×#16322
C67×#15922
C67×#15822
#2018167