Finite Group Databasebeta

← C67×#20232 #30699483

Subgroups — C67×#20232 #30699483

Order 17152.

568 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3375C211yesmaximal, characteristic, complemented
2C67×#3369C211yesmaximal, characteristic, complemented
3C67×#3361C211yesmaximal, characteristic
4C67×#3356C211yesmaximal, characteristic, complemented
5C67×#1720C211yesmaximal, characteristic, complemented
6C67×#1698C211yesmaximal, characteristic, complemented
7C67×#1697C211yesmaximal, characteristic
8C67×#1692C211yesmaximal, characteristic, complemented
9C67×#1684C211yesmaximal, characteristic, complemented
10C67×#1640C211yesmaximal, characteristic, complemented
11C67×#1633C211yesmaximal, characteristic, complemented
12C67×#1632C211yesmaximal, characteristic, complemented
13C67×#1628C211yesmaximal, characteristic
14C67×#1592C211yesmaximal, characteristic, complemented
15C67×#1584C211yesmaximal, characteristic, complemented
16#559×C67C2211yescharacteristic
17Q82×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19#554×C67C2211yescharacteristic
20#498×C67C2211yescharacteristic
21#494×C67C2211yescharacteristic
22#489×C67—21no—
23#489×C67—21no—
24#488×C67—21no—
25#487×C67C2211yescharacteristic
26#486×C67C2211yescharacteristic
27#486×C67C2211yescharacteristic
28#486×C67C2211yescharacteristic
29#485×C67—21no—
30#483×C67—21no—
31#478×C67—21no—
32#477×C67C2211yescharacteristic
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#476×C67C2211yescharacteristic
36#475×C67—21no—
37#475×C67C2211yescharacteristic
38#474×C67—21no—
39#470×C67—21no—
40#464×C67—21no—
41#464×C67—21no—
42#463×C67—21no—
43#462×C67C2211yescharacteristic
44#460×C67C2211yescharacteristic
45#459×C67C2211yescharacteristic
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#441×C67—21no—
50#440×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67C2211yescharacteristic
53C4×D8×C67—21no—
54#436×C67C2211yescharacteristic
55#428×C67—21no—
56#335×C67C2211yescharacteristic
57#335×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×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×C67—41no—
69C2×Q16×C67D411yescharacteristic
70C2×SD16×C67—21no—
71C2×SD16×C67D411yescharacteristic
72C2×D8×C67—21no—
73C2×M16×C67—21no—
74C4⋊Q8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67—41no—
77C4⋊Q8×C67C2311yescharacteristic
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80#127×C67C2311yescharacteristic
81#126×C67—41no—
82#125×C67—21no—
83#124×C67—21no—
84#124×C67—21no—
85#124×C67—21no—
86#123×C67—41no—
87C22⋊Q8×C67—21no—
88C22⋊Q8×C67—21no—
89C22⋊Q8×C67—41no—
90C4×Q8×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67—21no—
93C4×Q8×C67C2311yescharacteristic
94C4×Q8×C67C2311yescharacteristic
95C4×Q8×C67C2311yescharacteristic
96C4×D4×C67—21no—
97C4×D4×C67—21no—
98C4×D4×C67—21no—
99C4×D4×C67C2311yescharacteristic
100#117×C67—21no—
101C2×C4⋊C4×C67—21no—
102#107×C67—21no—
103#107×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×C67D411yescharacteristic
111C4⋊C8×C67—21no—
112C4⋊C8×C67—21no—
113C4⋊C8×C67—21no—
114C4⋊C8×C67C2311yescharacteristic
115C4⋊C8×C67C2311yescharacteristic
116C4⋊C8×C67C2311yescharacteristic
117C4⋊C8×C67C2311yescharacteristic
118#103×C67—21no—
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×C67D411yescharacteristic
126#103×C67D411yescharacteristic
127#103×C67D411yescharacteristic
128#102×C67—21no—
129#102×C67—21no—
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—
139#97×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×C67C2×D411yescharacteristic
149C2×Q8×C67C2×D411yescharacteristic
150C2×Q8×C67C2×D411yescharacteristic
151C2×D4×C67—21no—
152C2×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67C2×D411yescharacteristic
155C22×C4×C67—21no—
156C22×C4×C67—41no—
157C22×C4×C67—41no—
158Q16×C67—41no—
159Q16×C67—42no—
160Q16×C67—82no—
161SD16×C67—41no—
162SD16×C67—42no—
163D8×C67—42no—
164M16×C67—42no—
165C2×C8×C67—21no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C2×C8×C67—21no—
170C2×C8×C67C2×D411yescharacteristic
171C2×C8×C67C2×D411yescharacteristic
172C2×C8×C67C2×D411yescharacteristic
173C2×C8×C67C2×D411yescharacteristic
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
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—41no—
184C4⋊C4×C67—41no—
185C4⋊C4×C67—41no—
186C4⋊C4×C67—41no—
187C4⋊C4×C67—41no—
188C4⋊C4×C67C2×D411yescharacteristic
189C4⋊C4×C67C2×D411yescharacteristic
190C4⋊C4×C67C2×D411yescharacteristic
191C4⋊C4×C67C2×D411yescharacteristic
192C22⋊C4×C67—21no—
193C22⋊C4×C67—41no—
194C22⋊C4×C67—41no—
195C22⋊C4×C67—41no—
196C22⋊C4×C67—41no—
197C22⋊C4×C67—41no—
198C22⋊C4×C67—41no—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—41no—
203C42×C67—41no—
204C42×C67C2411yescharacteristic
205C23×C67—21no—
206C23×C67—41no—
207Q8×C67—21no—
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—41no—
214Q8×C67—41no—
215Q8×C67—81no—
216Q8×C67—81no—
217Q8×C67—161no—
218D4×C67—21no—
219D4×C67—21no—
220D4×C67—41no—
221D4×C67—41no—
222D4×C67—41no—
223D4×C67—41no—
224D4×C67—81no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—161no—
238C2×C4×C67—161no—
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C22×D411yescharacteristic
241C2×C4×C67H(5,𝔽2)11yescharacteristic
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—41no—
247C8×C67—41no—
248C8×C67—41no—
249C8×C67—41no—
250C8×C67—41no—
251C22×C67—41no—
252C22×C67—41no—
253C22×C67—41no—
254C22×C67—81no—
255C22×C67—81no—
256C22×C67—81no—
257C22×C67D4211yescharacteristic
258C4×C67—21no—
259C4×C67—41no—
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#57511yescharacteristic
270C4×C67#57511yescharacteristic
271C4×C67#57811yescharacteristic
272C4×C67#57811yescharacteristic
273#20232C6711yesmaximal, characteristic, direct product factor, Sylow
274C2×C67—81no—
275C2×C67—161no—
276C2×C67#326911yescharacteristic
277C2×C67#327711yescharacteristic
278C2×C67#328811yescharacteristic
279#3375C2×C6711yescharacteristic, complemented
280#3369C2×C6711yescharacteristic, complemented
281#3361C2×C6711yescharacteristic
282#3356C2×C6711yescharacteristic, complemented
283#1720C2×C6711yescharacteristic, complemented
284#1698C2×C6711yescharacteristic, complemented
285#1697C2×C6711yescharacteristic
286#1692C2×C6711yescharacteristic, complemented
287#1684C2×C6711yescharacteristic, complemented
288#1640C2×C6711yescharacteristic, complemented
289#1633C2×C6711yescharacteristic, complemented
290#1632C2×C6711yescharacteristic, complemented
291#1628C2×C6711yescharacteristic
292#1592C2×C6711yescharacteristic, complemented
293#1584C2×C6711yescharacteristic, complemented
294C67#2023211yescharacteristic, direct product factor, Sylow
295#559C22×C6711yescharacteristic
296Q82C22×C6711yescharacteristic
297#556C22×C6711yescharacteristic
298#554C22×C6711yescharacteristic
299#498C22×C6711yescharacteristic
300#494C22×C6711yescharacteristic
301#489—21no—
302#489—21no—
303#488—21no—
304#487C22×C6711yescharacteristic
305#486C22×C6711yescharacteristic
306#486C22×C6711yescharacteristic
307#486C22×C6711yescharacteristic
308#485—21no—
309#483—21no—
310#478—21no—
311#477C22×C6711yescharacteristic
312#477C22×C6711yescharacteristic
313#477C22×C6711yescharacteristic
314#476C22×C6711yescharacteristic
315#475—21no—
316#475C22×C6711yescharacteristic
317#474—21no—
318#470—21no—
319#464—21no—
320#464—21no—
321#463—21no—
322#462C22×C6711yescharacteristic
323#460C22×C6711yescharacteristic
324#459C22×C6711yescharacteristic
325#446C22×C6711yescharacteristic
326#446C22×C6711yescharacteristic
327C8×Q8C22×C6711yescharacteristic
328#441—21no—
329#440—21no—
330C4×Q16C22×C6711yescharacteristic
331C4×SD16C22×C6711yescharacteristic
332C4×D8—21no—
333#436C22×C6711yescharacteristic
334#428—21no—
335#335C22×C6711yescharacteristic
336#335C22×C6711yescharacteristic
337#333C22×C6711yescharacteristic
338#333C22×C6711yescharacteristic
339#332C22×C6711yescharacteristic
340#331C22×C6711yescharacteristic
341Q8⋊C8C22×C6711yescharacteristic
342Q8⋊C8C22×C6711yescharacteristic
343Q8⋊C8C22×C6711yescharacteristic
344D4⋊C8C22×C6711yescharacteristic
345#322C22×C6711yescharacteristic
346C2×Q16—21no—
347C2×Q16—41no—
348C2×Q16D4×C6711yescharacteristic
349C2×SD16—21no—
350C2×SD16D4×C6711yescharacteristic
351C2×D8—21no—
352C2×M16—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—41no—
356C4⋊Q8C23×C6711yescharacteristic
357C4⋊Q8C23×C6711yescharacteristic
358C4⋊Q8C23×C6711yescharacteristic
359#127C23×C6711yescharacteristic
360#126—41no—
361#125—21no—
362#124—21no—
363#124—21no—
364#124—21no—
365#123—41no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—41no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8C23×C6711yescharacteristic
373C4×Q8C23×C6711yescharacteristic
374C4×Q8C23×C6711yescharacteristic
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C23×C6711yescharacteristic
379#117—21no—
380C2×C4⋊C4—21no—
381#107—21no—
382#107—21no—
383#107—21no—
384#107D4×C6711yescharacteristic
385#106—21no—
386#106—21no—
387#106—21no—
388#106—21no—
389#106D4×C6711yescharacteristic
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393C4⋊C8C23×C6711yescharacteristic
394C4⋊C8C23×C6711yescharacteristic
395C4⋊C8C23×C6711yescharacteristic
396C4⋊C8C23×C6711yescharacteristic
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103—21no—
404#103D4×C6711yescharacteristic
405#103D4×C6711yescharacteristic
406#103D4×C6711yescharacteristic
407#102—21no—
408#102—21no—
409#102—21no—
410#102—21no—
411#102—21no—
412#102—21no—
413#102—21no—
414#102—41no—
415#102D4×C6711yescharacteristic
416C22⋊C8—21no—
417#97—21no—
418#97—21no—
419C4×C8—21no—
420C4×C8C23×C6711yescharacteristic
421C4×C8C23×C6711yescharacteristic
422C4×C8C23×C6711yescharacteristic
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—41no—
427C2×Q8C2×D4×C6711yescharacteristic
428C2×Q8C2×D4×C6711yescharacteristic
429C2×Q8C2×D4×C6711yescharacteristic
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4C2×D4×C6711yescharacteristic
434C22×C4—21no—
435C22×C4—41no—
436C22×C4—41no—
437Q16—41no—
438Q16—42no—
439Q16—82no—
440SD16—41no—
441SD16—42no—
442D8—42no—
443M16—42no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8C2×D4×C6711yescharacteristic
450C2×C8C2×D4×C6711yescharacteristic
451C2×C8C2×D4×C6711yescharacteristic
452C2×C8C2×D4×C6711yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4C2×D4×C6711yescharacteristic
468C4⋊C4C2×D4×C6711yescharacteristic
469C4⋊C4C2×D4×C6711yescharacteristic
470C4⋊C4C2×D4×C6711yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—41no—
483C42C24×C6711yescharacteristic
484C23—21no—
485C23—41no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—41no—
493Q8—41no—
494Q8—81no—
495Q8—81no—
496Q8—161no—
497D4—21no—
498D4—21no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—81no—
504C2×C4—21no—
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—161no—
517C2×C4—161no—
518C2×C4C22×D4×C6711yescharacteristic
519C2×C4C22×D4×C6711yescharacteristic
520C2×C4H(5,𝔽2)×C6711yescharacteristic
521C8—21no—
522C8—21no—
523C8—21no—
524C8—21no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22D42×C6711yescharacteristic
537C4—21no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4#575×C6711yescharacteristic
549C4#575×C6711yescharacteristic
550C4#578×C6711yescharacteristic
551C4#578×C6711yescharacteristic
552C2—81no—
553C2—161no—
554C2C67×#326911yescharacteristic
555C2C67×#327711yescharacteristic
556C2C67×#328811yescharacteristic
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×#3375C2semidirect
2. C67×#3369C2semidirect
4. C67×#3356C2semidirect
5. C67×#1720C2semidirect
6. C67×#1698C2semidirect
8. C67×#1692C2semidirect
9. C67×#1684C2semidirect
10. C67×#1640C2semidirect
11. C67×#1633C2semidirect
12. C67×#1632C2semidirect
14. C67×#1592C2semidirect
15. C67×#1584C2semidirect
273. #20232C67direct (N × H)
279. #3375C2×C67semidirect
280. #3369C2×C67semidirect
282. #3356C2×C67semidirect
283. #1720C2×C67semidirect
284. #1698C2×C67semidirect
286. #1692C2×C67semidirect
287. #1684C2×C67semidirect
288. #1640C2×C67semidirect
289. #1633C2×C67semidirect
290. #1632C2×C67semidirect
292. #1592C2×C67semidirect
293. #1584C2×C67semidirect
294. C67#20232direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33752
C67×#33692
C67×#33612
C67×#33562
C67×#17202
C67×#16982
C67×#16972
C67×#16922
C67×#16842
C67×#16402
C67×#16332
C67×#16322
C67×#16282
C67×#15922
C67×#15842
#2023267