Finite Group Databasebeta

← C67×#20123 #30699374

Subgroups — C67×#20123 #30699374

Order 17152.

584 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3367C211yesmaximal, characteristic, complemented
2C67×#3341C211yesmaximal, characteristic
3C67×#3340C211yesmaximal, characteristic, complemented
4C67×#3329C211yesmaximal, characteristic, complemented
5C67×#1718C211yesmaximal, characteristic, complemented
6C67×#1696C211yesmaximal, characteristic, complemented
7C67×#1691C211yesmaximal, characteristic, complemented
8C67×#1683C211yesmaximal, characteristic, complemented
9C67×#1679C211yesmaximal, characteristic, complemented
10C67×#1646C211yesmaximal, characteristic, complemented
11C67×#1646C211yesmaximal, characteristic, complemented
12C67×#1638C211yesmaximal, characteristic, complemented
13C67×#1637C211yesmaximal, characteristic, complemented
14C67×#1586C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic, complemented
16#556×C67C2211yescharacteristic
17#555×C67C2211yescharacteristic
18#555×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#500×C67C2211yescharacteristic
21#498×C67C2211yescharacteristic
22#489×C67—21no—
23#487×C67C2211yescharacteristic
24#487×C67C2211yescharacteristic
25#487×C67C2211yescharacteristic
26#487×C67C2211yescharacteristic
27#485×C67—21no—
28#485×C67—21no—
29#482×C67—21no—
30#482×C67—21no—
31#482×C67—21no—
32#477×C67—21no—
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#476×C67C2211yescharacteristic
36#476×C67C2211yescharacteristic
37#474×C67C2211yescharacteristic
38#474×C67C2211yescharacteristic
39#468×C67—21no—
40#466×C67—21no—
41#465×C67—21no—
42#463×C67—21no—
43#461×C67C2211yescharacteristic
44#461×C67C2211yescharacteristic
45#448×C67—21no—
46#446×C67C2211yescharacteristic
47#440×C67—21no—
48C4×SD16×C67C2211yescharacteristic
49C4×D8×C67C2211yescharacteristic
50#436×C67C2211yescharacteristic
51C8×D4×C67C2211yescharacteristic
52C8×D4×C67C2211yescharacteristic
53#427×C67—21no—
54#427×C67—21no—
55#419×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63D4⋊C8×C67C2211yescharacteristic
64D4⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C22×Q8×C67—21no—
68C2×SD16×C67—21no—
69C2×SD16×C67—21no—
70C2×SD16×C67D411yescharacteristic
71C2×D8×C67D411yescharacteristic
72C2×M16×C67—21no—
73C22×C8×C67—21no—
74C22×C8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67C2311yescharacteristic
77C4⋊Q8×C67C2311yescharacteristic
78C4⋊Q8×C67C2311yescharacteristic
79C4⋊Q8×C67C2311yescharacteristic
80#126×C67—41no—
81#124×C67—21no—
82#124×C67—41no—
83#124×C67—41no—
84C22⋊Q8×C67—21no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—41no—
87#121×C67—21no—
88#121×C67—21no—
89#121×C67—21no—
90#121×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67C2311yescharacteristic
93C4×D4×C67—21no—
94C4×D4×C67C2311yescharacteristic
95C4×D4×C67C2311yescharacteristic
96C4×D4×C67C2311yescharacteristic
97#117×C67—21no—
98#117×C67—21no—
99#117×C67—21no—
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#107×C67—21no—
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×C67C2311yescharacteristic
116C4⋊C8×C67C2311yescharacteristic
117C4⋊C8×C67C2311yescharacteristic
118C4⋊C8×C67C2311yescharacteristic
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—41no—
128#103×C67—41no—
129#103×C67D411yescharacteristic
130#102×C67—21no—
131#102×C67—21no—
132#102×C67—21no—
133#102×C67—21no—
134#102×C67D411yescharacteristic
135#102×C67D411yescharacteristic
136#102×C67D411yescharacteristic
137C22⋊C8×C67—21no—
138C22⋊C8×C67—21no—
139C22⋊C8×C67—21no—
140#97×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—42no—
150C2×Q8×C67—81no—
151C2×Q8×C67C2×D411yescharacteristic
152C2×D4×C67—41no—
153C2×D4×C67—41no—
154C2×D4×C67C2×D411yescharacteristic
155C2×D4×C67C2×D411yescharacteristic
156C2×D4×C67C2×D411yescharacteristic
157C22×C4×C67—21no—
158C22×C4×C67—21no—
159C22×C4×C67—21no—
160C22×C4×C67—41no—
161SD16×C67—41no—
162SD16×C67—42no—
163SD16×C67—81no—
164D8×C67—41no—
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×C67—42no—
173C2×C8×C67C2×D411yescharacteristic
174C2×C8×C67C2×D411yescharacteristic
175C2×C8×C67C2×D411yescharacteristic
176C2×C8×C67C2×D411yescharacteristic
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×C67C2×D411yescharacteristic
189C4⋊C4×C67C2×D411yescharacteristic
190C4⋊C4×C67C2×D411yescharacteristic
191C4⋊C4×C67C2×D411yescharacteristic
192C22⋊C4×C67—21no—
193C22⋊C4×C67—21no—
194C22⋊C4×C67—21no—
195C22⋊C4×C67—41no—
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—41no—
204C42×C67—41no—
205C42×C67—41no—
206C42×C67C2411yescharacteristic
207C23×C67—21no—
208C23×C67—21no—
209C23×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—41no—
213Q8×C67—41no—
214Q8×C67—41no—
215Q8×C67—41no—
216Q8×C67—81no—
217Q8×C67—81no—
218Q8×C67—82no—
219D4×C67—21no—
220D4×C67—21no—
221D4×C67—21no—
222D4×C67—21no—
223D4×C67—21no—
224D4×C67—21no—
225D4×C67—161no—
226D4×C67—161no—
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—41no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—82no—
243C2×C4×C67C22×D411yescharacteristic
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67H(5,𝔽2)11yescharacteristic
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—41no—
251C8×C67—41no—
252C8×C67—41no—
253C8×C67—41no—
254C8×C67—41no—
255C22×C67—41no—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67—41no—
264C22×C67D4211yescharacteristic
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×C67#57511yescharacteristic
276C4×C67#57511yescharacteristic
277C4×C67#57711yescharacteristic
278C4×C67#57811yescharacteristic
279#20123C6711yesmaximal, characteristic, direct product factor, Sylow
280C2×C67—81no—
281C2×C67—81no—
282C2×C67—81no—
283C2×C67#326711yescharacteristic
284C2×C67#326911yescharacteristic
285C2×C67#329011yescharacteristic
286#3367C2×C6711yescharacteristic, complemented
287#3341C2×C6711yescharacteristic
288#3340C2×C6711yescharacteristic, complemented
289#3329C2×C6711yescharacteristic, complemented
290#1718C2×C6711yescharacteristic, complemented
291#1696C2×C6711yescharacteristic, complemented
292#1691C2×C6711yescharacteristic, complemented
293#1683C2×C6711yescharacteristic, complemented
294#1679C2×C6711yescharacteristic, complemented
295#1646C2×C6711yescharacteristic, complemented
296#1646C2×C6711yescharacteristic, complemented
297#1638C2×C6711yescharacteristic, complemented
298#1637C2×C6711yescharacteristic, complemented
299#1586C2×C6711yescharacteristic, complemented
300#1585C2×C6711yescharacteristic, complemented
301C67#2012311yescharacteristic, direct product factor, Sylow
302#556C22×C6711yescharacteristic
303#555C22×C6711yescharacteristic
304#555C22×C6711yescharacteristic
305D4×Q8C22×C6711yescharacteristic
306#500C22×C6711yescharacteristic
307#498C22×C6711yescharacteristic
308#489—21no—
309#487C22×C6711yescharacteristic
310#487C22×C6711yescharacteristic
311#487C22×C6711yescharacteristic
312#487C22×C6711yescharacteristic
313#485—21no—
314#485—21no—
315#482—21no—
316#482—21no—
317#482—21no—
318#477—21no—
319#477C22×C6711yescharacteristic
320#477C22×C6711yescharacteristic
321#476C22×C6711yescharacteristic
322#476C22×C6711yescharacteristic
323#474C22×C6711yescharacteristic
324#474C22×C6711yescharacteristic
325#468—21no—
326#466—21no—
327#465—21no—
328#463—21no—
329#461C22×C6711yescharacteristic
330#461C22×C6711yescharacteristic
331#448—21no—
332#446C22×C6711yescharacteristic
333#440—21no—
334C4×SD16C22×C6711yescharacteristic
335C4×D8C22×C6711yescharacteristic
336#436C22×C6711yescharacteristic
337C8×D4C22×C6711yescharacteristic
338C8×D4C22×C6711yescharacteristic
339#427—21no—
340#427—21no—
341#419—21no—
342#335C22×C6711yescharacteristic
343#334C22×C6711yescharacteristic
344#333C22×C6711yescharacteristic
345#333C22×C6711yescharacteristic
346#332C22×C6711yescharacteristic
347#332C22×C6711yescharacteristic
348Q8⋊C8C22×C6711yescharacteristic
349D4⋊C8C22×C6711yescharacteristic
350D4⋊C8C22×C6711yescharacteristic
351D4⋊C8C22×C6711yescharacteristic
352#322C22×C6711yescharacteristic
353C22×Q8—21no—
354C2×SD16—21no—
355C2×SD16—21no—
356C2×SD16D4×C6711yescharacteristic
357C2×D8D4×C6711yescharacteristic
358C2×M16—21no—
359C22×C8—21no—
360C22×C8—21no—
361C4⋊Q8—21no—
362C4⋊Q8C23×C6711yescharacteristic
363C4⋊Q8C23×C6711yescharacteristic
364C4⋊Q8C23×C6711yescharacteristic
365C4⋊Q8C23×C6711yescharacteristic
366#126—41no—
367#124—21no—
368#124—41no—
369#124—41no—
370C22⋊Q8—21no—
371C22⋊Q8—21no—
372C22⋊Q8—41no—
373#121—21no—
374#121—21no—
375#121—21no—
376#121—21no—
377C4×Q8—21no—
378C4×Q8C23×C6711yescharacteristic
379C4×D4—21no—
380C4×D4C23×C6711yescharacteristic
381C4×D4C23×C6711yescharacteristic
382C4×D4C23×C6711yescharacteristic
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#107—21no—
389#107—21no—
390#107—21no—
391#107—21no—
392#107—21no—
393#107—21no—
394#107D4×C6711yescharacteristic
395#106—21no—
396#106—21no—
397#106—21no—
398#106—21no—
399#106D4×C6711yescharacteristic
400C4⋊C8—21no—
401C4⋊C8C23×C6711yescharacteristic
402C4⋊C8C23×C6711yescharacteristic
403C4⋊C8C23×C6711yescharacteristic
404C4⋊C8C23×C6711yescharacteristic
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—41no—
414#103—41no—
415#103D4×C6711yescharacteristic
416#102—21no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102D4×C6711yescharacteristic
421#102D4×C6711yescharacteristic
422#102D4×C6711yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426#97—21no—
427C4×C8C23×C6711yescharacteristic
428C4×C8C23×C6711yescharacteristic
429C4×C8C23×C6711yescharacteristic
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—41no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8C2×D4×C6711yescharacteristic
438C2×D4—41no—
439C2×D4—41no—
440C2×D4C2×D4×C6711yescharacteristic
441C2×D4C2×D4×C6711yescharacteristic
442C2×D4C2×D4×C6711yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447SD16—41no—
448SD16—42no—
449SD16—81no—
450D8—41no—
451M16—81no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8C2×D4×C6711yescharacteristic
460C2×C8C2×D4×C6711yescharacteristic
461C2×C8C2×D4×C6711yescharacteristic
462C2×C8C2×D4×C6711yescharacteristic
463C4⋊C4—21no—
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—41no—
473C4⋊C4—41no—
474C4⋊C4C2×D4×C6711yescharacteristic
475C4⋊C4C2×D4×C6711yescharacteristic
476C4⋊C4C2×D4×C6711yescharacteristic
477C4⋊C4C2×D4×C6711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—21no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42C24×C6711yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502Q8—81no—
503Q8—81no—
504Q8—82no—
505D4—21no—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—161no—
512D4—161no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
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×C4C22×D4×C6711yescharacteristic
530C2×C4C22×D4×C6711yescharacteristic
531C2×C4H(5,𝔽2)×C6711yescharacteristic
532C8—21no—
533C8—21no—
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22D42×C6711yescharacteristic
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4#575×C6711yescharacteristic
562C4#575×C6711yescharacteristic
563C4#577×C6711yescharacteristic
564C4#578×C6711yescharacteristic
565C2—81no—
566C2—81no—
567C2—81no—
568C2C67×#326711yescharacteristic
569C2C67×#326911yescharacteristic
570C2C67×#329011yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3341C2
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×#3367C2semidirect
3. C67×#3340C2semidirect
4. C67×#3329C2semidirect
5. C67×#1718C2semidirect
6. C67×#1696C2semidirect
7. C67×#1691C2semidirect
8. C67×#1683C2semidirect
9. C67×#1679C2semidirect
10. C67×#1646C2semidirect
11. C67×#1646C2semidirect
12. C67×#1638C2semidirect
13. C67×#1637C2semidirect
14. C67×#1586C2semidirect
15. C67×#1585C2semidirect
279. #20123C67direct (N × H)
286. #3367C2×C67semidirect
288. #3340C2×C67semidirect
289. #3329C2×C67semidirect
290. #1718C2×C67semidirect
291. #1696C2×C67semidirect
292. #1691C2×C67semidirect
293. #1683C2×C67semidirect
294. #1679C2×C67semidirect
295. #1646C2×C67semidirect
296. #1646C2×C67semidirect
297. #1638C2×C67semidirect
298. #1637C2×C67semidirect
299. #1586C2×C67semidirect
300. #1585C2×C67semidirect
301. C67#20123direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33672
C67×#33412
C67×#33402
C67×#33292
C67×#17182
C67×#16962
C67×#16912
C67×#16832
C67×#16792
C67×#16462
C67×#16382
C67×#16372
C67×#15862
C67×#15852
#2012367