Finite Group Databasebeta

← C67×#32986 #30712237

Subgroups — C67×#32986 #30712237

Order 17152.

1358 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
1C2×#520×C67C211yesmaximal, characteristic, complemented
2C2×#482×C67C211yesmaximal, characteristic, complemented
3C2×#480×C67C211yesmaximal, characteristic, complemented
4C67×#2615C211yesmaximal, characteristic
5C67×#1875C212yesmaximal, complemented
6C67×#1874C212yesmaximal, complemented
7C67×#1854C212yesmaximal, complemented
8C67×#1852C212yesmaximal, complemented
9C67×#1825C211yesmaximal, characteristic, complemented
10C67×#1824C211yesmaximal, characteristic, complemented
11C67×#1780C211yesmaximal, characteristic, complemented
12C22×#41×C67C2211yescharacteristic
13C2×#121×C67C2211yescharacteristic
14C2×#121×C67C2212yes—
15#520×C67—24no—
16#520×C67—24no—
17C2×C4×Q8×C67—21no—
18C2×C4×D4×C67—22no—
19C2×C4×D4×C67—21no—
20C2×#117×C67C2212yes—
21C2×#117×C67C2211yescharacteristic
22C22×C4⋊C4×C67C2212yes—
23C22×C4⋊C4×C67C2211yescharacteristic
24#482×C67—24no—
25#480×C67—24no—
26C2×#107×C67C2212yes—
27C2×#107×C67C2211yescharacteristic
28C2×#106×C67C2211yescharacteristic
29C2×#102×C67C2212yes—
30C2×#102×C67C2211yescharacteristic
31C2×#102×C67C2211yescharacteristic
32C2×C22⋊C8×C67C2211yescharacteristic
33C2×C22⋊C8×C67C2211yescharacteristic
34C2×C22⋊C8×C67C2212yes—
35C2×C22⋊C8×C67C2212yes—
36#393×C67—21no—
37#392×C67—21no—
38#392×C67—22no—
39#380×C67C2211yescharacteristic
40#380×C67C2212yes—
41#340×C67C2212yes—
42#340×C67C2212yes—
43#340×C67C2212yes—
44#340×C67C2212yes—
45C2×#41×C67—22no—
46C2×#41×C67—22no—
47C2×#41×C67—24no—
48C2×#41×C67—24no—
49C22×Q8×C67D411yescharacteristic
50C22×D4×C67—21no—
51C22×D4×C67—22no—
52C22×D4×C67D411yescharacteristic
53C22×D4×C67D412yes—
54C22×D4×C67C2311yescharacteristic
55C23×C4×C67—22no—
56C23×C4×C67—21no—
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2312yes—
59C22×C8×C67C2311yescharacteristic
60C22×C8×C67C2311yescharacteristic
61C22×C8×C67C2312yes—
62#121×C67—22no—
63#121×C67—22no—
64#121×C67—24no—
65#121×C67—24no—
66C4×Q8×C67—28no—
67C4×D4×C67—216no—
68C4×D4×C67—28no—
69#117×C67—22no—
70#117×C67—24no—
71#117×C67—24no—
72#117×C67D414yescomplemented
73C2×C4⋊C4×C67—22no—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67—22no—
76C2×C4⋊C4×C67—24no—
77C2×C4⋊C4×C67—24no—
78C2×C4⋊C4×C67D414yescomplemented
79C2×C4⋊C4×C67C2312yes—
80C2×C4⋊C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67C2311yescharacteristic
82C2×C4⋊C4×C67C2312yes—
83C2×C4⋊C4×C67C2311yescharacteristic
84C2×C22⋊C4×C67—22no—
85C2×C22⋊C4×C67—21no—
86C2×C22⋊C4×C67—21no—
87C2×C22⋊C4×C67—22no—
88C2×C22⋊C4×C67—21no—
89C2×C22⋊C4×C67—22no—
90C2×C42×C67—22no—
91C2×C42×C67—21no—
92#107×C67—24no—
93#107×C67—22no—
94#106×C67—22no—
95#102×C67—24no—
96#102×C67—22no—
97#102×C67—22no—
98C22⋊C8×C67—22no—
99C22⋊C8×C67—22no—
100C22⋊C8×C67—24no—
101C22⋊C8×C67—24no—
102#95×C67—21no—
103#95×C67—22no—
104C24×C67—21no—
105C24×C67C2×D411yescharacteristic
106C24×C67C2×D412yes—
107#41×C67—28no—
108#41×C67—28no—
109#41×C67—216no—
110C2×Q8×C67—22no—
111C2×Q8×C67—24no—
112C2×D4×C67—24no—
113C2×D4×C67—24no—
114C2×D4×C67—24no—
115C2×D4×C67—22no—
116C2×D4×C67—24no—
117C2×D4×C67—21no—
118C2×D4×C67—21no—
119C2×D4×C67—22no—
120C2×D4×C67—22no—
121C2×D4×C67—42no—
122C2×D4×C67—44no—
123C2×D4×C67—82no—
124C2×D4×C67—82no—
125C2×D4×C67—82no—
126C22×C4×C67—22no—
127C22×C4×C67—22no—
128C22×C4×C67—22no—
129C22×C4×C67—22no—
130C22×C4×C67—22no—
131C22×C4×C67—22no—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—21no—
135C22×C4×C67—28no—
136C22×C4×C67—24no—
137C22×C4×C67—21no—
138C22×C4×C67—22no—
139C22×C4×C67—82no—
140C22×C4×C67—82no—
141C22×C4×C67—82no—
142C22×C4×C67C2×D411yescharacteristic
143C22×C4×C67C2×D412yes—
144C22×C4×C67C2×D412yes—
145C22×C4×C67C2×D412yes—
146C22×C4×C67#4114yes—
147C22×C4×C67#4114yes—
148C22×C4×C67C2411yescharacteristic
149C2×C8×C67—21no—
150C2×C8×C67—22no—
151C2×C8×C67—21no—
152C2×C8×C67—22no—
153C2×C8×C67—22no—
154C2×C8×C67—22no—
155C2×C8×C67—22no—
156C4⋊C4×C67—28no—
157C4⋊C4×C67—24no—
158C4⋊C4×C67—22no—
159C4⋊C4×C67—21no—
160C4⋊C4×C67—21no—
161C4⋊C4×C67—21no—
162C4⋊C4×C67—21no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—22no—
165C4⋊C4×C67C2×D412yes—
166C4⋊C4×C67C2×D412yes—
167C4⋊C4×C67C2×D414yes—
168C22⋊C4×C67—28no—
169C22⋊C4×C67—24no—
170C22⋊C4×C67—42no—
171C22⋊C4×C67—44no—
172C22⋊C4×C67—82no—
173C22⋊C4×C67—81no—
174C42×C67—28no—
175C42×C67—24no—
176C23×C67—22no—
177C23×C67—22no—
178C23×C67—22no—
179C23×C67—22no—
180C23×C67—22no—
181C23×C67—22no—
182C23×C67—21no—
183C23×C67—22no—
184C23×C67—22no—
185C23×C67—21no—
186C23×C67—41no—
187C23×C67—42no—
188C23×C67—42no—
189C23×C67—41no—
190C23×C67—41no—
191C23×C67#12012yes—
192C23×C67#12314yes—
193C23×C67C22×D411yescharacteristic
194Q8×C67—28no—
195D4×C67—28no—
196D4×C67—216no—
197D4×C67—42no—
198D4×C67—41no—
199D4×C67—41no—
200D4×C67—88no—
201D4×C67—84no—
202C2×C4×C67—22no—
203C2×C4×C67—22no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—24no—
208C2×C4×C67—24no—
209C2×C4×C67—24no—
210C2×C4×C67—24no—
211C2×C4×C67—24no—
212C2×C4×C67—22no—
213C2×C4×C67—24no—
214C2×C4×C67—22no—
215C2×C4×C67—42no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—42no—
222C2×C4×C67—42no—
223C2×C4×C67—42no—
224C2×C4×C67—42no—
225C2×C4×C67—84no—
226C2×C4×C67—88no—
227C2×C4×C67#12012yes—
228C2×C4×C67#12314yes—
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C22×D411yescharacteristic
231C2×C4×C67C2×#4112yes—
232C2×C4×C67C2×#4112yes—
233C8×C67—41no—
234C8×C67—42no—
235C8×C67—41no—
236C22×C67—24no—
237C22×C67—24no—
238C22×C67—22no—
239C22×C67—24no—
240C22×C67—22no—
241C22×C67—22no—
242C22×C67—42no—
243C22×C67—42no—
244C22×C67—42no—
245C22×C67—42no—
246C22×C67—42no—
247C22×C67—42no—
248C22×C67—82no—
249C22×C67—82no—
250C22×C67—81no—
251C22×C67—81no—
252C22×C67—81no—
253C22×C67C2×#12011yescharacteristic
254C22×C67C2×#12312yes—
255C22×C67#57612yes—
256C22×C67#57711yescharacteristic
257C22×C67#57811yescharacteristic
258C4×C67—22no—
259C4×C67—24no—
260C4×C67—44no—
261C4×C67—42no—
262C4×C67—42no—
263C4×C67—82no—
264C4×C67—81no—
265C4×C67#52511yescharacteristic
266C4×C67#52511yescharacteristic
267C4×C67#52512yes—
268#32986C6711yesmaximal, characteristic, direct product factor, Sylow
269C2×C67—44no—
270C2×C67—42no—
271C2×C67—161no—
272C2×C67#284911yescharacteristic
273C2×C67#300811yescharacteristic
274C2×C67#301311yescharacteristic
275C2×C67#309612yes—
276C2×C67#309812yes—
277C2×#520C2×C6711yescharacteristic, complemented
278C2×#482C2×C6711yescharacteristic, complemented
279C2×#480C2×C6711yescharacteristic, complemented
280#2615C2×C6711yescharacteristic
281#1875C2×C6712yescomplemented
282#1874C2×C6712yescomplemented
283#1854C2×C6712yescomplemented
284#1852C2×C6712yescomplemented
285#1825C2×C6711yescharacteristic, complemented
286#1824C2×C6711yescharacteristic, complemented
287#1780C2×C6711yescharacteristic, complemented
288C67#3298611yescharacteristic, direct product factor, Sylow
289C22×#41C22×C6711yescharacteristic
290C2×#121C22×C6711yescharacteristic
291C2×#121C22×C6712yes—
292#520—24no—
293#520—24no—
294C2×C4×Q8—21no—
295C2×C4×D4—22no—
296C2×C4×D4—21no—
297C2×#117C22×C6712yes—
298C2×#117C22×C6711yescharacteristic
299C22×C4⋊C4C22×C6712yes—
300C22×C4⋊C4C22×C6711yescharacteristic
301#482—24no—
302#480—24no—
303C2×#107C22×C6712yes—
304C2×#107C22×C6711yescharacteristic
305C2×#106C22×C6711yescharacteristic
306C2×#102C22×C6712yes—
307C2×#102C22×C6711yescharacteristic
308C2×#102C22×C6711yescharacteristic
309C2×C22⋊C8C22×C6711yescharacteristic
310C2×C22⋊C8C22×C6711yescharacteristic
311C2×C22⋊C8C22×C6712yes—
312C2×C22⋊C8C22×C6712yes—
313#393—21no—
314#392—21no—
315#392—22no—
316#380C22×C6711yescharacteristic
317#380C22×C6712yes—
318#340C22×C6712yes—
319#340C22×C6712yes—
320#340C22×C6712yes—
321#340C22×C6712yes—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—24no—
325C2×#41—24no—
326C22×Q8D4×C6711yescharacteristic
327C22×D4—21no—
328C22×D4—22no—
329C22×D4D4×C6711yescharacteristic
330C22×D4D4×C6712yes—
331C22×D4C23×C6711yescharacteristic
332C23×C4—22no—
333C23×C4—21no—
334C23×C4C23×C6711yescharacteristic
335C23×C4C23×C6712yes—
336C22×C8C23×C6711yescharacteristic
337C22×C8C23×C6711yescharacteristic
338C22×C8C23×C6712yes—
339#121—22no—
340#121—22no—
341#121—24no—
342#121—24no—
343C4×Q8—28no—
344C4×D4—216no—
345C4×D4—28no—
346#117—22no—
347#117—24no—
348#117—24no—
349#117D4×C6714yescomplemented
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4D4×C6714yescomplemented
356C2×C4⋊C4C23×C6712yes—
357C2×C4⋊C4C23×C6711yescharacteristic
358C2×C4⋊C4C23×C6711yescharacteristic
359C2×C4⋊C4C23×C6712yes—
360C2×C4⋊C4C23×C6711yescharacteristic
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C42—22no—
368C2×C42—21no—
369#107—24no—
370#107—22no—
371#106—22no—
372#102—24no—
373#102—22no—
374#102—22no—
375C22⋊C8—22no—
376C22⋊C8—22no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379#95—21no—
380#95—22no—
381C24—21no—
382C24C2×D4×C6711yescharacteristic
383C24C2×D4×C6712yes—
384#41—28no—
385#41—28no—
386#41—216no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—44no—
400C2×D4—82no—
401C2×D4—82no—
402C2×D4—82no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4C2×D4×C6711yescharacteristic
420C22×C4C2×D4×C6712yes—
421C22×C4C2×D4×C6712yes—
422C22×C4C2×D4×C6712yes—
423C22×C4#41×C6714yes—
424C22×C4#41×C6714yes—
425C22×C4C24×C6711yescharacteristic
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4C2×D4×C6712yes—
443C4⋊C4C2×D4×C6712yes—
444C4⋊C4C2×D4×C6714yes—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—82no—
450C22⋊C4—81no—
451C42—28no—
452C42—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—22no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23#120×C6712yes—
469C23#123×C6714yes—
470C23C22×D4×C6711yescharacteristic
471Q8—28no—
472D4—28no—
473D4—216no—
474D4—42no—
475D4—41no—
476D4—41no—
477D4—88no—
478D4—84no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4#120×C6712yes—
505C2×C4#123×C6714yes—
506C2×C4C22×D4×C6711yescharacteristic
507C2×C4C22×D4×C6711yescharacteristic
508C2×C4C2×#41×C6712yes—
509C2×C4C2×#41×C6712yes—
510C8—41no—
511C8—42no—
512C8—41no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—81no—
526C22—82no—
527C22—82no—
528C22—81no—
529C22—81no—
530C22C2×#120×C6711yescharacteristic
531C22C2×#123×C6712yes—
532C22#576×C6712yes—
533C22#577×C6711yescharacteristic
534C22#578×C6711yescharacteristic
535C4—22no—
536C4—24no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—82no—
541C4—81no—
542C4#525×C6711yescharacteristic
543C4#525×C6711yescharacteristic
544C4#525×C6712yes—
545C2—42no—
546C2—44no—
547C2—161no—
548C2C67×#284911yescharacteristic
549C2C67×#300811yescharacteristic
550C2C67×#301311yescharacteristic
551C2C67×#309612yes—
552C2C67×#309812yes—
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC67×#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C22×#41×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#520×C67C2semidirect
2. C2×#482×C67C2semidirect
3. C2×#480×C67C2semidirect
5. C67×#1875C2semidirect
6. C67×#1874C2semidirect
7. C67×#1854C2semidirect
8. C67×#1852C2semidirect
9. C67×#1825C2semidirect
10. C67×#1824C2semidirect
11. C67×#1780C2semidirect
72. #117×C67D4semidirect
78. C2×C4⋊C4×C67D4semidirect
268. #32986C67direct (N × H)
277. C2×#520C2×C67semidirect
278. C2×#482C2×C67semidirect
279. C2×#480C2×C67semidirect
281. #1875C2×C67semidirect
282. #1874C2×C67semidirect
283. #1854C2×C67semidirect
284. #1852C2×C67semidirect
285. #1825C2×C67semidirect
286. #1824C2×C67semidirect
287. #1780C2×C67semidirect
288. C67#32986direct (N × H)
349. #117D4×C67semidirect
355. C2×C4⋊C4D4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#520×C672
C2×#482×C672
C2×#480×C672
C67×#26152
C67×#18752
C67×#18742
C67×#18542
C67×#18522
C67×#18252
C67×#18242
C67×#17802
#3298667