Finite Group Databasebeta

← C67×#20074 #30699325

Subgroups — C67×#20074 #30699325

Order 17152.

612 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3386C211yesmaximal, characteristic
2C67×#3374C211yesmaximal, characteristic, complemented
3C67×#3359C211yesmaximal, characteristic
4D4×Q16×C67C211yesmaximal, characteristic
5C67×#1723C211yesmaximal, characteristic, complemented
6C67×#1700C211yesmaximal, characteristic, complemented
7C67×#1690C211yesmaximal, characteristic, complemented
8C67×#1679C211yesmaximal, characteristic
9C67×#1674C211yesmaximal, characteristic, complemented
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1651C211yesmaximal, characteristic
12C67×#1649C211yesmaximal, characteristic
13C67×#1648C211yesmaximal, characteristic, complemented
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic
16C22×Q16×C67—21no—
17#557×C67C2211yescharacteristic
18#557×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20D4×Q8×C67C2211yescharacteristic
21#501×C67—21no—
22#500×C67C2211yescharacteristic
23#500×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#494×C67C2211yescharacteristic
26#491×C67—21no—
27#487×C67—21no—
28#487×C67C2211yescharacteristic
29#487×C67C2211yescharacteristic
30#479×C67—21no—
31#478×C67—21no—
32#477×C67—21no—
33#475×C67—21no—
34#475×C67C2211yescharacteristic
35#475×C67C2211yescharacteristic
36#475×C67C2211yescharacteristic
37#474×C67C2211yescharacteristic
38#467×C67—21no—
39#462×C67—21no—
40#462×C67C2211yescharacteristic
41#462×C67C2211yescharacteristic
42#462×C67C2211yescharacteristic
43#461×C67—21no—
44#461×C67C2211yescharacteristic
45#451×C67—21no—
46#451×C67—21no—
47#446×C67C2211yescharacteristic
48#446×C67C2211yescharacteristic
49C8×Q8×C67C2211yescharacteristic
50#441×C67—21no—
51#440×C67—21no—
52C4×Q16×C67C2211yescharacteristic
53C4×SD16×C67—21no—
54C4×SD16×C67C2211yescharacteristic
55C8×D4×C67C2211yescharacteristic
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#332×C67C2211yescharacteristic
59#332×C67C2211yescharacteristic
60#332×C67C2211yescharacteristic
61#332×C67C2211yescharacteristic
62Q8⋊C8×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65D4⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C22×Q8×C67—21no—
68C22×Q8×C67—21no—
69C2×Q16×C67—21no—
70C2×Q16×C67—21no—
71C2×Q16×C67—21no—
72C2×Q16×C67—21no—
73C2×Q16×C67—21no—
74C2×Q16×C67—44no—
75C2×Q16×C67D411yescharacteristic
76C2×SD16×C67—21no—
77C2×SD16×C67D411yescharacteristic
78C22×C8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67—21no—
81C4⋊Q8×C67—21no—
82C4⋊Q8×C67—21no—
83C4⋊Q8×C67C2311yescharacteristic
84C4⋊Q8×C67C2311yescharacteristic
85C4⋊Q8×C67C2311yescharacteristic
86C4⋊Q8×C67C2311yescharacteristic
87#125×C67—21no—
88#125×C67—21no—
89#125×C67—41no—
90#125×C67—41no—
91C22⋊Q8×C67—21no—
92C22⋊Q8×C67—21no—
93C22⋊Q8×C67—41no—
94C22⋊Q8×C67—41no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67—21no—
98C4×Q8×C67—21no—
99C4×Q8×C67C2311yescharacteristic
100C4×Q8×C67C2311yescharacteristic
101C4×Q8×C67C2311yescharacteristic
102C4×D4×C67—21no—
103C4×D4×C67—21no—
104C4×D4×C67C2311yescharacteristic
105#107×C67—21no—
106#107×C67—21no—
107#107×C67—21no—
108#107×C67—41no—
109#107×C67D411yescharacteristic
110#106×C67—21no—
111#106×C67—21no—
112#106×C67—21no—
113#106×C67—41no—
114#106×C67D411yescharacteristic
115C4⋊C8×C67—21no—
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—21no—
129#103×C67—21no—
130#103×C67—21no—
131#103×C67—21no—
132#103×C67—21no—
133#103×C67D411yescharacteristic
134#103×C67D411yescharacteristic
135#103×C67D411yescharacteristic
136#102×C67—21no—
137#102×C67D411yescharacteristic
138C22⋊C8×C67—21no—
139#97×C67—21no—
140#97×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C4×C8×C67C2311yescharacteristic
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—21no—
148C2×Q8×C67—21no—
149C2×Q8×C67—41no—
150C2×Q8×C67—41no—
151C2×Q8×C67—42no—
152C2×Q8×C67—42no—
153C2×Q8×C67—81no—
154C2×Q8×C67—81no—
155C2×Q8×C67C2×D411yescharacteristic
156C2×Q8×C67C2×D411yescharacteristic
157C2×Q8×C67C2×D411yescharacteristic
158C2×D4×C67C2×D411yescharacteristic
159C22×C4×C67—21no—
160C22×C4×C67—41no—
161C22×C4×C67—41no—
162Q16×C67—22no—
163Q16×C67—42no—
164Q16×C67—42no—
165Q16×C67—42no—
166Q16×C67—42no—
167Q16×C67—81no—
168SD16×C67—22no—
169SD16×C67—42no—
170C2×C8×C67—21no—
171C2×C8×C67—21no—
172C2×C8×C67—21no—
173C2×C8×C67—21no—
174C2×C8×C67—21no—
175C2×C8×C67—42no—
176C2×C8×C67C2×D411yescharacteristic
177C2×C8×C67C2×D411yescharacteristic
178C2×C8×C67C2×D411yescharacteristic
179C2×C8×C67C2×D411yescharacteristic
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—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—41no—
192C4⋊C4×C67—41no—
193C4⋊C4×C67—41no—
194C4⋊C4×C67—41no—
195C4⋊C4×C67—41no—
196C4⋊C4×C67—41no—
197C4⋊C4×C67—41no—
198C4⋊C4×C67—41no—
199C4⋊C4×C67C2×D411yescharacteristic
200C4⋊C4×C67C2×D411yescharacteristic
201C4⋊C4×C67C2×D411yescharacteristic
202C4⋊C4×C67C2×D411yescharacteristic
203C22⋊C4×C67—21no—
204C22⋊C4×C67—41no—
205C22⋊C4×C67—41no—
206C42×C67—21no—
207C42×C67—21no—
208C42×C67—21no—
209C42×C67—41no—
210C42×C67—41no—
211C42×C67C2411yescharacteristic
212C23×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—21no—
218Q8×C67—41no—
219Q8×C67—41no—
220Q8×C67—41no—
221Q8×C67—41no—
222Q8×C67—41no—
223Q8×C67—41no—
224Q8×C67—41no—
225Q8×C67—41no—
226Q8×C67—82no—
227Q8×C67—82no—
228Q8×C67AGL(1,ℤ/8)12yes—
229D4×C67—21no—
230D4×C67#13712yes—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—21no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
236C2×C4×C67—21no—
237C2×C4×C67—21no—
238C2×C4×C67—41no—
239C2×C4×C67—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—82no—
245C2×C4×C67—82no—
246C2×C4×C67C22×D411yescharacteristic
247C2×C4×C67C22×D411yescharacteristic
248C2×C4×C67H(5,𝔽2)11yescharacteristic
249C8×C67—22no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—21no—
254C8×C67—41no—
255C8×C67—41no—
256C8×C67—41no—
257C8×C67—41no—
258C22×C67—22no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67D4211yescharacteristic
262C4×C67—22no—
263C4×C67—21no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—81no—
271C4×C67—81no—
272C4×C67—81no—
273C4×C67—81no—
274C4×C67C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C67C2×#13711yescharacteristic
276C4×C67#57711yescharacteristic
277C4×C67#57811yescharacteristic
278#20074C6711yesmaximal, characteristic, direct product factor, Sylow
279C2×C67—42no—
280C2×C67#326511yescharacteristic
281C2×C67#327011yescharacteristic
282C2×C67#329311yescharacteristic
283#3386C2×C6711yescharacteristic
284#3374C2×C6711yescharacteristic, complemented
285#3359C2×C6711yescharacteristic
286D4×Q16C2×C6711yescharacteristic
287#1723C2×C6711yescharacteristic, complemented
288#1700C2×C6711yescharacteristic, complemented
289#1690C2×C6711yescharacteristic, complemented
290#1679C2×C6711yescharacteristic
291#1674C2×C6711yescharacteristic, complemented
292#1653C2×C6711yescharacteristic, complemented
293#1651C2×C6711yescharacteristic
294#1649C2×C6711yescharacteristic
295#1648C2×C6711yescharacteristic, complemented
296#1587C2×C6711yescharacteristic, complemented
297#1585C2×C6711yescharacteristic
298C67#2007411yescharacteristic, direct product factor, Sylow
299C22×Q16—21no—
300#557C22×C6711yescharacteristic
301#557C22×C6711yescharacteristic
302D4×Q8C22×C6711yescharacteristic
303D4×Q8C22×C6711yescharacteristic
304#501—21no—
305#500C22×C6711yescharacteristic
306#500C22×C6711yescharacteristic
307#498C22×C6711yescharacteristic
308#494C22×C6711yescharacteristic
309#491—21no—
310#487—21no—
311#487C22×C6711yescharacteristic
312#487C22×C6711yescharacteristic
313#479—21no—
314#478—21no—
315#477—21no—
316#475—21no—
317#475C22×C6711yescharacteristic
318#475C22×C6711yescharacteristic
319#475C22×C6711yescharacteristic
320#474C22×C6711yescharacteristic
321#467—21no—
322#462—21no—
323#462C22×C6711yescharacteristic
324#462C22×C6711yescharacteristic
325#462C22×C6711yescharacteristic
326#461—21no—
327#461C22×C6711yescharacteristic
328#451—21no—
329#451—21no—
330#446C22×C6711yescharacteristic
331#446C22×C6711yescharacteristic
332C8×Q8C22×C6711yescharacteristic
333#441—21no—
334#440—21no—
335C4×Q16C22×C6711yescharacteristic
336C4×SD16—21no—
337C4×SD16C22×C6711yescharacteristic
338C8×D4C22×C6711yescharacteristic
339#335C22×C6711yescharacteristic
340#334C22×C6711yescharacteristic
341#332C22×C6711yescharacteristic
342#332C22×C6711yescharacteristic
343#332C22×C6711yescharacteristic
344#332C22×C6711yescharacteristic
345Q8⋊C8C22×C6711yescharacteristic
346Q8⋊C8C22×C6711yescharacteristic
347Q8⋊C8C22×C6711yescharacteristic
348D4⋊C8C22×C6711yescharacteristic
349#322C22×C6711yescharacteristic
350C22×Q8—21no—
351C22×Q8—21no—
352C2×Q16—21no—
353C2×Q16—21no—
354C2×Q16—21no—
355C2×Q16—21no—
356C2×Q16—21no—
357C2×Q16—44no—
358C2×Q16D4×C6711yescharacteristic
359C2×SD16—21no—
360C2×SD16D4×C6711yescharacteristic
361C22×C8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8C23×C6711yescharacteristic
367C4⋊Q8C23×C6711yescharacteristic
368C4⋊Q8C23×C6711yescharacteristic
369C4⋊Q8C23×C6711yescharacteristic
370#125—21no—
371#125—21no—
372#125—41no—
373#125—41no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—41no—
377C22⋊Q8—41no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C23×C6711yescharacteristic
383C4×Q8C23×C6711yescharacteristic
384C4×Q8C23×C6711yescharacteristic
385C4×D4—21no—
386C4×D4—21no—
387C4×D4C23×C6711yescharacteristic
388#107—21no—
389#107—21no—
390#107—21no—
391#107—41no—
392#107D4×C6711yescharacteristic
393#106—21no—
394#106—21no—
395#106—21no—
396#106—41no—
397#106D4×C6711yescharacteristic
398C4⋊C8—21no—
399C4⋊C8—21no—
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—21no—
414#103—21no—
415#103—21no—
416#103D4×C6711yescharacteristic
417#103D4×C6711yescharacteristic
418#103D4×C6711yescharacteristic
419#102—21no—
420#102D4×C6711yescharacteristic
421C22⋊C8—21no—
422#97—21no—
423#97—21no—
424C4×C8—21no—
425C4×C8C23×C6711yescharacteristic
426C4×C8C23×C6711yescharacteristic
427C4×C8C23×C6711yescharacteristic
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8C2×D4×C6711yescharacteristic
439C2×Q8C2×D4×C6711yescharacteristic
440C2×Q8C2×D4×C6711yescharacteristic
441C2×D4C2×D4×C6711yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—41no—
445Q16—22no—
446Q16—42no—
447Q16—42no—
448Q16—42no—
449Q16—42no—
450Q16—81no—
451SD16—22no—
452SD16—42no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
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—21no—
473C4⋊C4—21no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4C2×D4×C6711yescharacteristic
483C4⋊C4C2×D4×C6711yescharacteristic
484C4⋊C4C2×D4×C6711yescharacteristic
485C4⋊C4C2×D4×C6711yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—41no—
493C42—41no—
494C42C24×C6711yescharacteristic
495C23—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—82no—
510Q8—82no—
511Q8AGL(1,ℤ/8)×C6712yes—
512D4—21no—
513D4#137×C6712yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4C22×D4×C6711yescharacteristic
530C2×C4C22×D4×C6711yescharacteristic
531C2×C4H(5,𝔽2)×C6711yescharacteristic
532C8—22no—
533C8—21no—
534C8—21no—
535C8—21no—
536C8—21no—
537C8—41no—
538C8—41no—
539C8—41no—
540C8—41no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22D42×C6711yescharacteristic
545C4—22no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4C2×AGL(1,ℤ/8)×C6711yescharacteristic
558C4C2×#137×C6711yescharacteristic
559C4#577×C6711yescharacteristic
560C4#578×C6711yescharacteristic
561C2—42no—
562C2C67×#326511yescharacteristic
563C2C67×#327011yescharacteristic
564C2C67×#329311yescharacteristic
subgroupG/H
center, socleC22×C67D42
commutator, Frattini, cosocleC42C24×C67
prime radicalC2×D4×C67C2×D4
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×#3374C2semidirect
5. C67×#1723C2semidirect
6. C67×#1700C2semidirect
7. C67×#1690C2semidirect
9. C67×#1674C2semidirect
10. C67×#1653C2semidirect
13. C67×#1648C2semidirect
14. C67×#1587C2semidirect
278. #20074C67direct (N × H)
284. #3374C2×C67semidirect
287. #1723C2×C67semidirect
288. #1700C2×C67semidirect
289. #1690C2×C67semidirect
291. #1674C2×C67semidirect
292. #1653C2×C67semidirect
295. #1648C2×C67semidirect
296. #1587C2×C67semidirect
298. C67#20074direct (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×#33742
C67×#33592
D4×Q16×C672
C67×#17232
C67×#17002
C67×#16902
C67×#16792
C67×#16742
C67×#16532
C67×#16512
C67×#16492
C67×#16482
C67×#15872
C67×#15852
#2007467