Finite Group Databasebeta

← C67×#20182 #30699433

Subgroups — C67×#20182 #30699433

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3386C211yesmaximal, characteristic, complemented
2C67×#3359C211yesmaximal, characteristic, complemented
3C67×#3329C211yesmaximal, characteristic
4C67×#3329C211yesmaximal, characteristic
5C67×#1718C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1699C211yesmaximal, characteristic, complemented
8C67×#1693C211yesmaximal, characteristic, complemented
9C67×#1684C211yesmaximal, characteristic, complemented
10C67×#1651C211yesmaximal, characteristic, complemented
11C67×#1651C211yesmaximal, characteristic, complemented
12C67×#1646C211yesmaximal, characteristic
13C67×#1636C211yesmaximal, characteristic, complemented
14C67×#1592C211yesmaximal, characteristic, complemented
15C67×#1583C211yesmaximal, characteristic, complemented
16#557×C67C2211yescharacteristic
17#555×C67C2211yescharacteristic
18D4×Q8×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#501×C67—21no—
21#500×C67C2211yescharacteristic
22#498×C67C2211yescharacteristic
23#487×C67—21no—
24#487×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#487×C67C2211yescharacteristic
28#485×C67—21no—
29#485×C67—21no—
30#478×C67—21no—
31#477×C67—21no—
32#477×C67C2211yescharacteristic
33#477×C67C2211yescharacteristic
34#475×C67—21no—
35#475×C67C2211yescharacteristic
36#475×C67C2211yescharacteristic
37#474×C67C2211yescharacteristic
38#468×C67—21no—
39#468×C67—21no—
40#461×C67—21no—
41#461×C67C2211yescharacteristic
42#461×C67C2211yescharacteristic
43#461×C67C2211yescharacteristic
44#448×C67—21no—
45#448×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
53#436×C67C2211yescharacteristic
54#419×C67—21no—
55#419×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C22×Q8×C67—21no—
68C22×Q8×C67—21no—
69C2×SD16×C67—21no—
70C2×SD16×C67—21no—
71C2×SD16×C67—21no—
72C2×SD16×C67D411yescharacteristic
73C2×SD16×C67D411yescharacteristic
74C2×M16×C67—21no—
75C2×M16×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#125×C67—21no—
84#125×C67—41no—
85#124×C67—41no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—21no—
88C22⋊Q8×C67—41no—
89C22⋊Q8×C67—41no—
90#121×C67—21no—
91#121×C67—21no—
92C4×Q8×C67—21no—
93C4×Q8×C67—21no—
94C4×Q8×C67C2311yescharacteristic
95C4×Q8×C67C2311yescharacteristic
96C4×D4×C67—21no—
97C4×D4×C67—21no—
98C4×D4×C67C2311yescharacteristic
99C4×D4×C67C2311yescharacteristic
100#117×C67—21no—
101#117×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—41no—
110#106×C67D411yescharacteristic
111#106×C67D411yescharacteristic
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×C67—21no—
126#103×C67—21no—
127#103×C67—21no—
128#103×C67—21no—
129#103×C67—21no—
130#103×C67—21no—
131#103×C67—41no—
132#103×C67D411yescharacteristic
133#103×C67D411yescharacteristic
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
144C2×Q8×C67—21no—
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67—41no—
149C2×Q8×C67—41no—
150C2×Q8×C67—42no—
151C2×Q8×C67—42no—
152C2×Q8×C67—81no—
153C2×Q8×C67—81no—
154C2×Q8×C67C2×D411yescharacteristic
155C2×Q8×C67C2×D411yescharacteristic
156C2×D4×C67—41no—
157C2×D4×C67C2×D411yescharacteristic
158C2×D4×C67C2×D411yescharacteristic
159C22×C4×C67—21no—
160C22×C4×C67—21no—
161C22×C4×C67—41no—
162C22×C4×C67—41no—
163SD16×C67—22no—
164SD16×C67—41no—
165SD16×C67—42no—
166SD16×C67—81no—
167SD16×C67—81no—
168M16×C67—81no—
169M16×C67—81no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67C2×D411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
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—21no—
188C4⋊C4×C67—21no—
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—
203C22⋊C4×C67—41no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67—41no—
207C42×C67—41no—
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—41no—
215Q8×C67—41no—
216Q8×C67—41no—
217Q8×C67—41no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—82no—
221Q8×C67—81no—
222Q8×C67—82no—
223Q8×C67#13712yes—
224D4×C67—21no—
225D4×C67—21no—
226D4×C67—21no—
227D4×C67—21no—
228D4×C67—161no—
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—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—82no—
244C2×C4×C67—82no—
245C2×C4×C67C22×D411yescharacteristic
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67H(5,𝔽2)11yescharacteristic
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67D4211yescharacteristic
264C4×C67—22no—
265C4×C67—21no—
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×C67—81no—
275C4×C67C2×#13711yescharacteristic
276C4×C67#57511yescharacteristic
277C4×C67#57711yescharacteristic
278C4×C67#57711yescharacteristic
279#20182C6711yesmaximal, characteristic, direct product factor, Sylow
280C2×C67—81no—
281C2×C67—81no—
282C2×C67#326611yescharacteristic
283C2×C67#327511yescharacteristic
284C2×C67#328911yescharacteristic
285#3386C2×C6711yescharacteristic, complemented
286#3359C2×C6711yescharacteristic, complemented
287#3329C2×C6711yescharacteristic
288#3329C2×C6711yescharacteristic
289#1718C2×C6711yescharacteristic, complemented
290#1704C2×C6711yescharacteristic, complemented
291#1699C2×C6711yescharacteristic, complemented
292#1693C2×C6711yescharacteristic, complemented
293#1684C2×C6711yescharacteristic, complemented
294#1651C2×C6711yescharacteristic, complemented
295#1651C2×C6711yescharacteristic, complemented
296#1646C2×C6711yescharacteristic
297#1636C2×C6711yescharacteristic, complemented
298#1592C2×C6711yescharacteristic, complemented
299#1583C2×C6711yescharacteristic, complemented
300C67#2018211yescharacteristic, direct product factor, Sylow
301#557C22×C6711yescharacteristic
302#555C22×C6711yescharacteristic
303D4×Q8C22×C6711yescharacteristic
304D4×Q8C22×C6711yescharacteristic
305#501—21no—
306#500C22×C6711yescharacteristic
307#498C22×C6711yescharacteristic
308#487—21no—
309#487C22×C6711yescharacteristic
310#487C22×C6711yescharacteristic
311#487C22×C6711yescharacteristic
312#487C22×C6711yescharacteristic
313#485—21no—
314#485—21no—
315#478—21no—
316#477—21no—
317#477C22×C6711yescharacteristic
318#477C22×C6711yescharacteristic
319#475—21no—
320#475C22×C6711yescharacteristic
321#475C22×C6711yescharacteristic
322#474C22×C6711yescharacteristic
323#468—21no—
324#468—21no—
325#461—21no—
326#461C22×C6711yescharacteristic
327#461C22×C6711yescharacteristic
328#461C22×C6711yescharacteristic
329#448—21no—
330#448—21no—
331#446C22×C6711yescharacteristic
332C8×Q8C22×C6711yescharacteristic
333#440—21no—
334C4×SD16—21no—
335C4×SD16C22×C6711yescharacteristic
336C4×SD16C22×C6711yescharacteristic
337#436C22×C6711yescharacteristic
338#436C22×C6711yescharacteristic
339#419—21no—
340#419—21no—
341#335C22×C6711yescharacteristic
342#334C22×C6711yescharacteristic
343#333C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345#332C22×C6711yescharacteristic
346#332C22×C6711yescharacteristic
347Q8⋊C8C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349D4⋊C8C22×C6711yescharacteristic
350D4⋊C8C22×C6711yescharacteristic
351#322C22×C6711yescharacteristic
352C22×Q8—21no—
353C22×Q8—21no—
354C2×SD16—21no—
355C2×SD16—21no—
356C2×SD16—21no—
357C2×SD16D4×C6711yescharacteristic
358C2×SD16D4×C6711yescharacteristic
359C2×M16—21no—
360C2×M16—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8C23×C6711yescharacteristic
365C4⋊Q8C23×C6711yescharacteristic
366C4⋊Q8C23×C6711yescharacteristic
367C4⋊Q8C23×C6711yescharacteristic
368#125—21no—
369#125—41no—
370#124—41no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—41no—
374C22⋊Q8—41no—
375#121—21no—
376#121—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C23×C6711yescharacteristic
380C4×Q8C23×C6711yescharacteristic
381C4×D4—21no—
382C4×D4—21no—
383C4×D4C23×C6711yescharacteristic
384C4×D4C23×C6711yescharacteristic
385#117—21no—
386#117—21no—
387#107—21no—
388#107—21no—
389#107—21no—
390#107—21no—
391#107—21no—
392#106—21no—
393#106—21no—
394#106—41no—
395#106D4×C6711yescharacteristic
396#106D4×C6711yescharacteristic
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#103—21no—
412#103—21no—
413#103—21no—
414#103—21no—
415#103—21no—
416#103—41no—
417#103D4×C6711yescharacteristic
418#103D4×C6711yescharacteristic
419#102—21no—
420#102D4×C6711yescharacteristic
421#102D4×C6711yescharacteristic
422C22⋊C8—21no—
423C22⋊C8—21no—
424#97—21no—
425C4×C8—21no—
426C4×C8C23×C6711yescharacteristic
427C4×C8C23×C6711yescharacteristic
428C4×C8C23×C6711yescharacteristic
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—81no—
438C2×Q8—81no—
439C2×Q8C2×D4×C6711yescharacteristic
440C2×Q8C2×D4×C6711yescharacteristic
441C2×D4—41no—
442C2×D4C2×D4×C6711yescharacteristic
443C2×D4C2×D4×C6711yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4—41no—
448SD16—22no—
449SD16—41no—
450SD16—42no—
451SD16—81no—
452SD16—81no—
453M16—81no—
454M16—81no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C2×C8C2×D4×C6711yescharacteristic
463C2×C8C2×D4×C6711yescharacteristic
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C2×D4×C6711yescharacteristic
480C4⋊C4C2×D4×C6711yescharacteristic
481C4⋊C4C2×D4×C6711yescharacteristic
482C4⋊C4C2×D4×C6711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—21no—
490C42—21no—
491C42—41no—
492C42—41no—
493C42C24×C6711yescharacteristic
494C23—21no—
495C23—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—82no—
506Q8—81no—
507Q8—82no—
508Q8#137×C6712yes—
509D4—21no—
510D4—21no—
511D4—21no—
512D4—21no—
513D4—161no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4C22×D4×C6711yescharacteristic
531C2×C4C22×D4×C6711yescharacteristic
532C2×C4H(5,𝔽2)×C6711yescharacteristic
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22D42×C6711yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4C2×#137×C6711yescharacteristic
561C4#575×C6711yescharacteristic
562C4#577×C6711yescharacteristic
563C4#577×C6711yescharacteristic
564C2—81no—
565C2—81no—
566C2C67×#326611yescharacteristic
567C2C67×#327511yescharacteristic
568C2C67×#328911yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radical#555×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×#3386C2semidirect
2. C67×#3359C2semidirect
5. C67×#1718C2semidirect
6. C67×#1704C2semidirect
7. C67×#1699C2semidirect
8. C67×#1693C2semidirect
9. C67×#1684C2semidirect
10. C67×#1651C2semidirect
11. C67×#1651C2semidirect
13. C67×#1636C2semidirect
14. C67×#1592C2semidirect
15. C67×#1583C2semidirect
279. #20182C67direct (N × H)
285. #3386C2×C67semidirect
286. #3359C2×C67semidirect
289. #1718C2×C67semidirect
290. #1704C2×C67semidirect
291. #1699C2×C67semidirect
292. #1693C2×C67semidirect
293. #1684C2×C67semidirect
294. #1651C2×C67semidirect
295. #1651C2×C67semidirect
297. #1636C2×C67semidirect
298. #1592C2×C67semidirect
299. #1583C2×C67semidirect
300. C67#20182direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33862
C67×#33592
C67×#33292
C67×#17182
C67×#17042
C67×#16992
C67×#16932
C67×#16842
C67×#16512
C67×#16462
C67×#16362
C67×#15922
C67×#15832
#2018267