Finite Group Databasebeta

← C67×#20185 #30699436

Subgroups — C67×#20185 #30699436

Order 17152.

572 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3398C211yesmaximal, characteristic
2C67×#3387C211yesmaximal, characteristic, complemented
3Q8×SD16×C67C211yesmaximal, characteristic
4C67×#3328C211yesmaximal, characteristic
5C67×#1716C211yesmaximal, characteristic, complemented
6C67×#1700C211yesmaximal, characteristic, complemented
7C67×#1692C211yesmaximal, characteristic, complemented
8C67×#1691C211yesmaximal, characteristic
9C67×#1670C211yesmaximal, characteristic, complemented
10C67×#1653C211yesmaximal, characteristic, complemented
11C67×#1651C211yesmaximal, characteristic
12C67×#1645C211yesmaximal, characteristic, complemented
13C67×#1638C211yesmaximal, characteristic
14C67×#1592C211yesmaximal, characteristic
15C67×#1584C211yesmaximal, characteristic, complemented
16Q82×C67C2211yescharacteristic
17#557×C67C2211yescharacteristic
18#556×C67C2211yescharacteristic
19D4×Q8×C67C2211yescharacteristic
20#501×C67—21no—
21#498×C67—21no—
22#498×C67C2211yescharacteristic
23#498×C67C2211yescharacteristic
24#498×C67C2211yescharacteristic
25#497×C67—21no—
26#494×C67C2211yescharacteristic
27#487×C67C2211yescharacteristic
28#487×C67C2211yescharacteristic
29#485×C67—21no—
30#479×C67—21no—
31#477×C67C2211yescharacteristic
32#476×C67—21no—
33#476×C67C2211yescharacteristic
34#475×C67—21no—
35#475×C67—21no—
36#475×C67C2211yescharacteristic
37#475×C67C2211yescharacteristic
38#475×C67C2211yescharacteristic
39#470×C67—21no—
40#464×C67—21no—
41#462×C67—21no—
42#462×C67C2211yescharacteristic
43#462×C67C2211yescharacteristic
44#461×C67C2211yescharacteristic
45#451×C67—21no—
46#446×C67C2211yescharacteristic
47#446×C67C2211yescharacteristic
48C8×Q8×C67C2211yescharacteristic
49#441×C67—21no—
50#441×C67—21no—
51C4×Q16×C67C2211yescharacteristic
52C4×SD16×C67—21no—
53C4×SD16×C67C2211yescharacteristic
54#436×C67C2211yescharacteristic
55#419×C67—21no—
56#335×C67C2211yescharacteristic
57#335×C67C2211yescharacteristic
58#333×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—
68C2×Q16×C67—21no—
69C2×Q16×C67—21no—
70C2×Q16×C67—21no—
71C2×Q16×C67—21no—
72C2×Q16×C67D411yescharacteristic
73C2×SD16×C67—21no—
74C2×SD16×C67D411yescharacteristic
75C2×M16×C67—21no—
76C4⋊Q8×C67—21no—
77C4⋊Q8×C67—21no—
78C4⋊Q8×C67—21no—
79C4⋊Q8×C67—21no—
80C4⋊Q8×C67—41no—
81C4⋊Q8×C67C2311yescharacteristic
82C4⋊Q8×C67C2311yescharacteristic
83C4⋊Q8×C67C2311yescharacteristic
84C4⋊Q8×C67C2311yescharacteristic
85#126×C67—41no—
86#125×C67—21no—
87#125×C67—41no—
88#124×C67—21no—
89C22⋊Q8×C67—21no—
90C22⋊Q8×C67—21no—
91C22⋊Q8×C67—41no—
92C4×Q8×C67—21no—
93C4×Q8×C67—21no—
94C4×Q8×C67—21no—
95C4×Q8×C67—21no—
96C4×Q8×C67—21no—
97C4×Q8×C67C2311yescharacteristic
98C4×Q8×C67C2311yescharacteristic
99C4×Q8×C67C2311yescharacteristic
100C4×D4×C67—21no—
101C4×D4×C67C2311yescharacteristic
102#117×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—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—41no—
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—
158Q16×C67—22no—
159Q16×C67—42no—
160Q16×C67—81no—
161Q16×C67—81no—
162Q16×C67—81no—
163SD16×C67—22no—
164SD16×C67—81no—
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×C67C2×D411yescharacteristic
172C2×C8×C67C2×D411yescharacteristic
173C2×C8×C67C2×D411yescharacteristic
174C2×C8×C67C2×D411yescharacteristic
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—21no—
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—
200C22⋊C4×C67—41no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67—41no—
205C42×C67—41no—
206C42×C67—41no—
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209Q8×C67—21no—
210Q8×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—161no—
223Q8×C67#13712yes—
224D4×C67—21no—
225D4×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—21no—
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—82no—
240C2×C4×C67C22×D411yescharacteristic
241C2×C4×C67C22×D411yescharacteristic
242C2×C4×C67H(5,𝔽2)11yescharacteristic
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
246C8×C67—41no—
247C8×C67—41no—
248C8×C67—41no—
249C8×C67—41no—
250C8×C67—41no—
251C8×C67#13712yes—
252C22×C67—41no—
253C22×C67—41no—
254C22×C67—41no—
255C22×C67D4211yescharacteristic
256C4×C67—22no—
257C4×C67—21no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67—41no—
261C4×C67—41no—
262C4×C67—41no—
263C4×C67—41no—
264C4×C67—81no—
265C4×C67—81no—
266C4×C67—81no—
267C4×C67—81no—
268C4×C67C2×#13711yescharacteristic
269C4×C67C2×#13711yescharacteristic
270C4×C67#57711yescharacteristic
271C4×C67#57711yescharacteristic
272#20185C6711yesmaximal, characteristic, direct product factor, Sylow
273C2×C67—81no—
274C2×C67#326611yescharacteristic
275C2×C67#327811yescharacteristic
276C2×C67#328911yescharacteristic
277#3398C2×C6711yescharacteristic
278#3387C2×C6711yescharacteristic, complemented
279Q8×SD16C2×C6711yescharacteristic
280#3328C2×C6711yescharacteristic
281#1716C2×C6711yescharacteristic, complemented
282#1700C2×C6711yescharacteristic, complemented
283#1692C2×C6711yescharacteristic, complemented
284#1691C2×C6711yescharacteristic
285#1670C2×C6711yescharacteristic, complemented
286#1653C2×C6711yescharacteristic, complemented
287#1651C2×C6711yescharacteristic
288#1645C2×C6711yescharacteristic, complemented
289#1638C2×C6711yescharacteristic
290#1592C2×C6711yescharacteristic
291#1584C2×C6711yescharacteristic, complemented
292C67#2018511yescharacteristic, direct product factor, Sylow
293Q82C22×C6711yescharacteristic
294#557C22×C6711yescharacteristic
295#556C22×C6711yescharacteristic
296D4×Q8C22×C6711yescharacteristic
297#501—21no—
298#498—21no—
299#498C22×C6711yescharacteristic
300#498C22×C6711yescharacteristic
301#498C22×C6711yescharacteristic
302#497—21no—
303#494C22×C6711yescharacteristic
304#487C22×C6711yescharacteristic
305#487C22×C6711yescharacteristic
306#485—21no—
307#479—21no—
308#477C22×C6711yescharacteristic
309#476—21no—
310#476C22×C6711yescharacteristic
311#475—21no—
312#475—21no—
313#475C22×C6711yescharacteristic
314#475C22×C6711yescharacteristic
315#475C22×C6711yescharacteristic
316#470—21no—
317#464—21no—
318#462—21no—
319#462C22×C6711yescharacteristic
320#462C22×C6711yescharacteristic
321#461C22×C6711yescharacteristic
322#451—21no—
323#446C22×C6711yescharacteristic
324#446C22×C6711yescharacteristic
325C8×Q8C22×C6711yescharacteristic
326#441—21no—
327#441—21no—
328C4×Q16C22×C6711yescharacteristic
329C4×SD16—21no—
330C4×SD16C22×C6711yescharacteristic
331#436C22×C6711yescharacteristic
332#419—21no—
333#335C22×C6711yescharacteristic
334#335C22×C6711yescharacteristic
335#333C22×C6711yescharacteristic
336#332C22×C6711yescharacteristic
337#332C22×C6711yescharacteristic
338#332C22×C6711yescharacteristic
339Q8⋊C8C22×C6711yescharacteristic
340Q8⋊C8C22×C6711yescharacteristic
341Q8⋊C8C22×C6711yescharacteristic
342D4⋊C8C22×C6711yescharacteristic
343#322C22×C6711yescharacteristic
344C22×Q8—21no—
345C2×Q16—21no—
346C2×Q16—21no—
347C2×Q16—21no—
348C2×Q16—21no—
349C2×Q16D4×C6711yescharacteristic
350C2×SD16—21no—
351C2×SD16D4×C6711yescharacteristic
352C2×M16—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—41no—
358C4⋊Q8C23×C6711yescharacteristic
359C4⋊Q8C23×C6711yescharacteristic
360C4⋊Q8C23×C6711yescharacteristic
361C4⋊Q8C23×C6711yescharacteristic
362#126—41no—
363#125—21no—
364#125—41no—
365#124—21no—
366C22⋊Q8—21no—
367C22⋊Q8—21no—
368C22⋊Q8—41no—
369C4×Q8—21no—
370C4×Q8—21no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8C23×C6711yescharacteristic
375C4×Q8C23×C6711yescharacteristic
376C4×Q8C23×C6711yescharacteristic
377C4×D4—21no—
378C4×D4C23×C6711yescharacteristic
379#117—21no—
380#107—21no—
381#107D4×C6711yescharacteristic
382#106—21no—
383#106—21no—
384#106—21no—
385#106—21no—
386#106—21no—
387#106—21no—
388#106—21no—
389#106—21no—
390#106—41no—
391#106D4×C6711yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8C23×C6711yescharacteristic
396C4⋊C8C23×C6711yescharacteristic
397C4⋊C8C23×C6711yescharacteristic
398C4⋊C8C23×C6711yescharacteristic
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#103—21no—
408#103—41no—
409#103D4×C6711yescharacteristic
410#103D4×C6711yescharacteristic
411#103D4×C6711yescharacteristic
412#102—21no—
413#102D4×C6711yescharacteristic
414C22⋊C8—21no—
415#97—21no—
416#97—21no—
417C4×C8—21no—
418C4×C8C23×C6711yescharacteristic
419C4×C8C23×C6711yescharacteristic
420C4×C8C23×C6711yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—41no—
426C2×Q8—41no—
427C2×Q8—42no—
428C2×Q8—81no—
429C2×Q8C2×D4×C6711yescharacteristic
430C2×Q8C2×D4×C6711yescharacteristic
431C2×Q8C2×D4×C6711yescharacteristic
432C2×D4C2×D4×C6711yescharacteristic
433C22×C4—21no—
434C22×C4—41no—
435Q16—22no—
436Q16—42no—
437Q16—81no—
438Q16—81no—
439Q16—81no—
440SD16—22no—
441SD16—81no—
442M16—81no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8C2×D4×C6711yescharacteristic
449C2×C8C2×D4×C6711yescharacteristic
450C2×C8C2×D4×C6711yescharacteristic
451C2×C8C2×D4×C6711yescharacteristic
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—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4C2×D4×C6711yescharacteristic
471C4⋊C4C2×D4×C6711yescharacteristic
472C4⋊C4C2×D4×C6711yescharacteristic
473C4⋊C4C2×D4×C6711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42C24×C6711yescharacteristic
485C23—21no—
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—82no—
498Q8—81no—
499Q8—161no—
500Q8#137×C6712yes—
501D4—21no—
502D4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4C22×D4×C6711yescharacteristic
518C2×C4C22×D4×C6711yescharacteristic
519C2×C4H(5,𝔽2)×C6711yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8#137×C6712yes—
529C22—41no—
530C22—41no—
531C22—41no—
532C22D42×C6711yescharacteristic
533C4—22no—
534C4—21no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4C2×#137×C6711yescharacteristic
546C4C2×#137×C6711yescharacteristic
547C4#577×C6711yescharacteristic
548C4#577×C6711yescharacteristic
549C2—81no—
550C2C67×#326611yescharacteristic
551C2C67×#327811yescharacteristic
552C2C67×#328911yescharacteristic
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×#3387C2semidirect
5. C67×#1716C2semidirect
6. C67×#1700C2semidirect
7. C67×#1692C2semidirect
9. C67×#1670C2semidirect
10. C67×#1653C2semidirect
12. C67×#1645C2semidirect
15. C67×#1584C2semidirect
272. #20185C67direct (N × H)
278. #3387C2×C67semidirect
281. #1716C2×C67semidirect
282. #1700C2×C67semidirect
283. #1692C2×C67semidirect
285. #1670C2×C67semidirect
286. #1653C2×C67semidirect
288. #1645C2×C67semidirect
291. #1584C2×C67semidirect
292. C67#20185direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#33982
C67×#33872
Q8×SD16×C672
C67×#33282
C67×#17162
C67×#17002
C67×#16922
C67×#16912
C67×#16702
C67×#16532
C67×#16512
C67×#16452
C67×#16382
C67×#15922
C67×#15842
#2018567