Finite Group Databasebeta

← C67×#20048 #30699299

Subgroups — C67×#20048 #30699299

Order 17152.

576 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3387C211yesmaximal, characteristic, complemented
2Q8×SD16×C67C211yesmaximal, characteristic
3C67×#3372C211yesmaximal, characteristic
4C67×#3299C211yesmaximal, characteristic
5C67×#1716C211yesmaximal, characteristic, complemented
6C67×#1704C211yesmaximal, characteristic, complemented
7C67×#1692C211yesmaximal, characteristic, complemented
8C67×#1684C211yesmaximal, characteristic, complemented
9C67×#1671C211yesmaximal, characteristic
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1651C211yesmaximal, characteristic
12C67×#1633C211yesmaximal, characteristic, complemented
13C67×#1632C211yesmaximal, characteristic
14C67×#1587C211yesmaximal, characteristic, complemented
15C67×#1585C211yesmaximal, characteristic
16Q82×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#554×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#501×C67—21no—
21#498×C67—21no—
22#498×C67C2211yescharacteristic
23#498×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#494×C67C2211yescharacteristic
26#491×C67—21no—
27#487×C67C2211yescharacteristic
28#487×C67C2211yescharacteristic
29#484×C67—21no—
30#479×C67—21no—
31#479×C67—21no—
32#477×C67C2211yescharacteristic
33#477×C67C2211yescharacteristic
34#477×C67C2211yescharacteristic
35#476×C67—21no—
36#476×C67C2211yescharacteristic
37#475×C67—21no—
38#475×C67—21no—
39#475×C67C2211yescharacteristic
40#475×C67C2211yescharacteristic
41#467×C67—21no—
42#462×C67C2211yescharacteristic
43#461×C67—21no—
44#461×C67C2211yescharacteristic
45#451×C67—21no—
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#441×C67—21no—
50#440×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67—21no—
53C4×SD16×C67C2211yescharacteristic
54C8×D4×C67C2211yescharacteristic
55C2×#103×C67—21no—
56#335×C67C2211yescharacteristic
57#334×C67C2211yescharacteristic
58#333×C67C2211yescharacteristic
59#333×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—
68C2×Q16×C67—21no—
69C2×Q16×C67—41no—
70C2×Q16×C67D411yescharacteristic
71C2×SD16×C67—21no—
72C2×SD16×C67D411yescharacteristic
73C22×C8×C67—21no—
74C4⋊Q8×C67—21no—
75C4⋊Q8×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—41no—
79C4⋊Q8×C67C2311yescharacteristic
80C4⋊Q8×C67C2311yescharacteristic
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83#125×C67—21no—
84#125×C67—21no—
85#125×C67—41no—
86C22⋊Q8×C67—21no—
87C22⋊Q8×C67—21no—
88C22⋊Q8×C67—41no—
89C22⋊Q8×C67—41no—
90C4×Q8×C67—21no—
91C4×Q8×C67—21no—
92C4×Q8×C67—21no—
93C4×Q8×C67—21no—
94C4×Q8×C67C2311yescharacteristic
95C4×Q8×C67C2311yescharacteristic
96C4×Q8×C67C2311yescharacteristic
97C4×D4×C67—21no—
98C4×D4×C67—21no—
99C4×D4×C67C2311yescharacteristic
100C2×C4⋊C4×C67—21no—
101#107×C67—21no—
102#107×C67—21no—
103#107×C67—21no—
104#107×C67D411yescharacteristic
105#106×C67—21no—
106#106×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—41no—
113#106×C67D411yescharacteristic
114C4⋊C8×C67—21no—
115C4⋊C8×C67—21no—
116C4⋊C8×C67—21no—
117C4⋊C8×C67C2311yescharacteristic
118C4⋊C8×C67C2311yescharacteristic
119C4⋊C8×C67C2311yescharacteristic
120C4⋊C8×C67C2311yescharacteristic
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—21no—
132#103×C67D411yescharacteristic
133#103×C67D411yescharacteristic
134#103×C67D411yescharacteristic
135#102×C67—21no—
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—21no—
148C2×Q8×C67—41no—
149C2×Q8×C67—41no—
150C2×Q8×C67—42no—
151C2×Q8×C67—81no—
152C2×Q8×C67C2×D411yescharacteristic
153C2×Q8×C67C2×D411yescharacteristic
154C2×Q8×C67C2×D411yescharacteristic
155C2×D4×C67C2×D411yescharacteristic
156C22×C4×C67—21no—
157C22×C4×C67—41no—
158C22×C4×C67—41no—
159Q16×C67—22no—
160Q16×C67—82no—
161Q16×C67—81no—
162SD16×C67—22no—
163SD16×C67—42no—
164C2×C8×C67—21no—
165C2×C8×C67—21no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C2×C8×C67—81no—
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—21no—
184C4⋊C4×C67—21no—
185C4⋊C4×C67—41no—
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—41no—
199C22⋊C4×C67—41no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67—41no—
204C42×C67—41no—
205C42×C67C2411yescharacteristic
206C23×C67—21no—
207Q8×C67—21no—
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—41no—
213Q8×C67—41no—
214Q8×C67—41no—
215Q8×C67—41no—
216Q8×C67—82no—
217Q8×C67—81no—
218Q8×C67—81no—
219Q8×C67—161no—
220Q8×C67#13712yes—
221D4×C67—21no—
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—21no—
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—82no—
237C2×C4×C67—161no—
238C2×C4×C67C22×D411yescharacteristic
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67H(5,𝔽2)11yescharacteristic
241C8×C67—22no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—41no—
247C8×C67—41no—
248C8×C67—41no—
249C8×C67—41no—
250C22×C67—22no—
251C22×C67—41no—
252C22×C67—41no—
253C22×C67D4211yescharacteristic
254C4×C67—22no—
255C4×C67—21no—
256C4×C67—41no—
257C4×C67—41no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67—41no—
261C4×C67—41no—
262C4×C67—81no—
263C4×C67—81no—
264C4×C67—81no—
265C4×C67—81no—
266C4×C67C2×#13711yescharacteristic
267C4×C67C2×#13711yescharacteristic
268C4×C67#57711yescharacteristic
269C4×C67#57811yescharacteristic
270#20048C6711yesmaximal, characteristic, direct product factor, Sylow
271C2×C67—42no—
272C2×C67#326611yescharacteristic
273C2×C67#327011yescharacteristic
274C2×C67#329111yescharacteristic
275#3387C2×C6711yescharacteristic, complemented
276Q8×SD16C2×C6711yescharacteristic
277#3372C2×C6711yescharacteristic
278#3299C2×C6711yescharacteristic
279#1716C2×C6711yescharacteristic, complemented
280#1704C2×C6711yescharacteristic, complemented
281#1692C2×C6711yescharacteristic, complemented
282#1684C2×C6711yescharacteristic, complemented
283#1671C2×C6711yescharacteristic
284#1653C2×C6711yescharacteristic, complemented
285#1651C2×C6711yescharacteristic
286#1633C2×C6711yescharacteristic, complemented
287#1632C2×C6711yescharacteristic
288#1587C2×C6711yescharacteristic, complemented
289#1585C2×C6711yescharacteristic
290C67#2004811yescharacteristic, direct product factor, Sylow
291Q82C22×C6711yescharacteristic
292#557C22×C6711yescharacteristic
293#554C22×C6711yescharacteristic
294D4×Q8C22×C6711yescharacteristic
295#501—21no—
296#498—21no—
297#498C22×C6711yescharacteristic
298#498C22×C6711yescharacteristic
299#498C22×C6711yescharacteristic
300#494C22×C6711yescharacteristic
301#491—21no—
302#487C22×C6711yescharacteristic
303#487C22×C6711yescharacteristic
304#484—21no—
305#479—21no—
306#479—21no—
307#477C22×C6711yescharacteristic
308#477C22×C6711yescharacteristic
309#477C22×C6711yescharacteristic
310#476—21no—
311#476C22×C6711yescharacteristic
312#475—21no—
313#475—21no—
314#475C22×C6711yescharacteristic
315#475C22×C6711yescharacteristic
316#467—21no—
317#462C22×C6711yescharacteristic
318#461—21no—
319#461C22×C6711yescharacteristic
320#451—21no—
321#446C22×C6711yescharacteristic
322#446C22×C6711yescharacteristic
323C8×Q8C22×C6711yescharacteristic
324#441—21no—
325#440—21no—
326C4×Q16C22×C6711yescharacteristic
327C4×SD16—21no—
328C4×SD16C22×C6711yescharacteristic
329C8×D4C22×C6711yescharacteristic
330C2×#103—21no—
331#335C22×C6711yescharacteristic
332#334C22×C6711yescharacteristic
333#333C22×C6711yescharacteristic
334#333C22×C6711yescharacteristic
335#332C22×C6711yescharacteristic
336#332C22×C6711yescharacteristic
337Q8⋊C8C22×C6711yescharacteristic
338Q8⋊C8C22×C6711yescharacteristic
339Q8⋊C8C22×C6711yescharacteristic
340D4⋊C8C22×C6711yescharacteristic
341#322C22×C6711yescharacteristic
342C22×Q8—21no—
343C2×Q16—21no—
344C2×Q16—41no—
345C2×Q16D4×C6711yescharacteristic
346C2×SD16—21no—
347C2×SD16D4×C6711yescharacteristic
348C22×C8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—21no—
352C4⋊Q8—21no—
353C4⋊Q8—41no—
354C4⋊Q8C23×C6711yescharacteristic
355C4⋊Q8C23×C6711yescharacteristic
356C4⋊Q8C23×C6711yescharacteristic
357C4⋊Q8C23×C6711yescharacteristic
358#125—21no—
359#125—21no—
360#125—41no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—41no—
364C22⋊Q8—41no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8C23×C6711yescharacteristic
370C4×Q8C23×C6711yescharacteristic
371C4×Q8C23×C6711yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C23×C6711yescharacteristic
375C2×C4⋊C4—21no—
376#107—21no—
377#107—21no—
378#107—21no—
379#107D4×C6711yescharacteristic
380#106—21no—
381#106—21no—
382#106—21no—
383#106—21no—
384#106—21no—
385#106—21no—
386#106—21no—
387#106—41no—
388#106D4×C6711yescharacteristic
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8C23×C6711yescharacteristic
393C4⋊C8C23×C6711yescharacteristic
394C4⋊C8C23×C6711yescharacteristic
395C4⋊C8C23×C6711yescharacteristic
396#103—21no—
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103—21no—
404#103—21no—
405#103—21no—
406#103—21no—
407#103D4×C6711yescharacteristic
408#103D4×C6711yescharacteristic
409#103D4×C6711yescharacteristic
410#102—21no—
411#102D4×C6711yescharacteristic
412C22⋊C8—21no—
413#97—21no—
414#97—21no—
415C4×C8—21no—
416C4×C8C23×C6711yescharacteristic
417C4×C8C23×C6711yescharacteristic
418C4×C8C23×C6711yescharacteristic
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—41no—
424C2×Q8—41no—
425C2×Q8—42no—
426C2×Q8—81no—
427C2×Q8C2×D4×C6711yescharacteristic
428C2×Q8C2×D4×C6711yescharacteristic
429C2×Q8C2×D4×C6711yescharacteristic
430C2×D4C2×D4×C6711yescharacteristic
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—41no—
434Q16—22no—
435Q16—82no—
436Q16—81no—
437SD16—22no—
438SD16—42no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—81no—
445C2×C8C2×D4×C6711yescharacteristic
446C2×C8C2×D4×C6711yescharacteristic
447C2×C8C2×D4×C6711yescharacteristic
448C2×C8C2×D4×C6711yescharacteristic
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4C2×D4×C6711yescharacteristic
469C4⋊C4C2×D4×C6711yescharacteristic
470C4⋊C4C2×D4×C6711yescharacteristic
471C4⋊C4C2×D4×C6711yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42—41no—
480C42C24×C6711yescharacteristic
481C23—21no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—41no—
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—82no—
492Q8—81no—
493Q8—81no—
494Q8—161no—
495Q8#137×C6712yes—
496D4—21no—
497D4#137×C6712yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—161no—
513C2×C4C22×D4×C6711yescharacteristic
514C2×C4C22×D4×C6711yescharacteristic
515C2×C4H(5,𝔽2)×C6711yescharacteristic
516C8—22no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C22—22no—
526C22—41no—
527C22—41no—
528C22D42×C6711yescharacteristic
529C4—22no—
530C4—21no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4C2×#137×C6711yescharacteristic
542C4C2×#137×C6711yescharacteristic
543C4#577×C6711yescharacteristic
544C4#578×C6711yescharacteristic
545C2—42no—
546C2C67×#326611yescharacteristic
547C2C67×#327011yescharacteristic
548C2C67×#329111yescharacteristic
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
1. C67×#3387C2semidirect
5. C67×#1716C2semidirect
6. C67×#1704C2semidirect
7. C67×#1692C2semidirect
8. C67×#1684C2semidirect
10. C67×#1653C2semidirect
12. C67×#1633C2semidirect
14. C67×#1587C2semidirect
270. #20048C67direct (N × H)
275. #3387C2×C67semidirect
279. #1716C2×C67semidirect
280. #1704C2×C67semidirect
281. #1692C2×C67semidirect
282. #1684C2×C67semidirect
284. #1653C2×C67semidirect
286. #1633C2×C67semidirect
288. #1587C2×C67semidirect
290. C67#20048direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33872
Q8×SD16×C672
C67×#33722
C67×#32992
C67×#17162
C67×#17042
C67×#16922
C67×#16842
C67×#16712
C67×#16532
C67×#16512
C67×#16332
C67×#16322
C67×#15872
C67×#15852
#2004867