Finite Group Databasebeta

← C67×#33090 #30712341

Subgroups — C67×#33090 #30712341

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2645C211yesmaximal, characteristic, complemented
2C67×#2275C211yesmaximal, characteristic
3C67×#1928C211yesmaximal, characteristic, complemented
4C67×#1928C211yesmaximal, characteristic, complemented
5C67×#1927C211yesmaximal, characteristic, complemented
6C67×#1921C211yesmaximal, characteristic, complemented
7C67×#1920C211yesmaximal, characteristic, complemented
8C67×#1918C211yesmaximal, characteristic, complemented
9C67×#1877C211yesmaximal, characteristic
10C67×#1877C211yesmaximal, characteristic
11C67×#1877C211yesmaximal, characteristic
12C67×#1864C211yesmaximal, characteristic
13C67×#1862C211yesmaximal, characteristic
14C67×#1862C211yesmaximal, characteristic
15C67×#1840C211yesmaximal, characteristic, complemented
16C22×SD16×C67C2211yescharacteristic
17C2×C4×D4×C67C2211yescharacteristic
18C2×C4×D4×C67C2211yescharacteristic
19C2×C4×D4×C67C2211yescharacteristic
20C2×#107×C67C2211yescharacteristic
21C2×#107×C67C2211yescharacteristic
22C2×#106×C67C2211yescharacteristic
23C2×#103×C67C2211yescharacteristic
24C2×#103×C67C2211yescharacteristic
25C2×#103×C67C2211yescharacteristic
26C2×#102×C67C2211yescharacteristic
27C2×#102×C67C2211yescharacteristic
28C2×#102×C67C2211yescharacteristic
29#401×C67—21no—
30#400×C67—21no—
31#400×C67—21no—
32#400×C67—21no—
33#391×C67C2211yescharacteristic
34#391×C67C2211yescharacteristic
35#391×C67C2211yescharacteristic
36#389×C67C2211yescharacteristic
37#389×C67C2211yescharacteristic
38#389×C67C2211yescharacteristic
39C4×C4⋊C4×C67C2211yescharacteristic
40C4×C4⋊C4×C67C2211yescharacteristic
41C4×C4⋊C4×C67C2211yescharacteristic
42C4×C22⋊C4×C67—21no—
43C4×C22⋊C4×C67—21no—
44C4×C22⋊C4×C67—21no—
45#376×C67C2211yescharacteristic
46C2×#95×C67—21no—
47#340×C67C2211yescharacteristic
48#340×C67C2211yescharacteristic
49#340×C67C2211yescharacteristic
50#340×C67C2211yescharacteristic
51#340×C67C2211yescharacteristic
52#340×C67C2211yescharacteristic
53#336×C67C2211yescharacteristic
54#336×C67C2211yescharacteristic
55#336×C67C2211yescharacteristic
56#336×C67C2211yescharacteristic
57#336×C67C2211yescharacteristic
58#336×C67C2211yescharacteristic
59C22×Q8×C67C2311yescharacteristic
60C22×D4×C67C2311yescharacteristic
61C23×C4×C67—21no—
62C23×C4×C67—21no—
63C23×C4×C67—21no—
64C2×SD16×C67—22no—
65C2×SD16×C67—22no—
66C2×SD16×C67—22no—
67C22×C8×C67C2311yescharacteristic
68C22×C8×C67C2311yescharacteristic
69C22×C8×C67C2311yescharacteristic
70C22×C8×C67C2311yescharacteristic
71C4×D4×C67—22no—
72C4×D4×C67—22no—
73C4×D4×C67—22no—
74C4×D4×C67—22no—
75C4×D4×C67—22no—
76C4×D4×C67—22no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67—21no—
79C2×C4⋊C4×C67—21no—
80C2×C4⋊C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67C2311yescharacteristic
82C2×C4⋊C4×C67C2311yescharacteristic
83C2×C4⋊C4×C67C2311yescharacteristic
84C2×C4⋊C4×C67C2311yescharacteristic
85C2×C4⋊C4×C67C2311yescharacteristic
86C2×C22⋊C4×C67—21no—
87C2×C22⋊C4×C67—21no—
88C2×C22⋊C4×C67—21no—
89C2×C42×C67—21no—
90C2×C42×C67—21no—
91C2×C42×C67—21no—
92C2×C42×C67C2311yescharacteristic
93C2×C42×C67C2311yescharacteristic
94C2×C42×C67C2311yescharacteristic
95#107×C67—22no—
96#107×C67—22no—
97#106×C67—22no—
98#103×C67—22no—
99#103×C67—22no—
100#103×C67—22no—
101#102×C67—22no—
102#102×C67—22no—
103#102×C67—22no—
104#95×C67—21no—
105#95×C67—21no—
106#95×C67—21no—
107#95×C67—21no—
108#95×C67—21no—
109#95×C67—21no—
110#95×C67—21no—
111#95×C67—21no—
112#95×C67—21no—
113#95×C67—21no—
114#95×C67—21no—
115#95×C67—21no—
116#95×C67D411yescharacteristic
117#95×C67D411yescharacteristic
118#95×C67D411yescharacteristic
119#95×C67D411yescharacteristic
120C24×C67—21no—
121C2×Q8×C67—21no—
122C2×Q8×C67—21no—
123C2×Q8×C67—21no—
124C2×Q8×C67—21no—
125C2×Q8×C67—21no—
126C2×Q8×C67—21no—
127C2×D4×C67—21no—
128C2×D4×C67—21no—
129C2×D4×C67—21no—
130C2×D4×C67#4112yes—
131C2×D4×C67#4112yes—
132C2×D4×C67#4112yes—
133C22×C4×C67—21no—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—24no—
137C22×C4×C67—24no—
138C22×C4×C67—24no—
139C22×C4×C67—21no—
140C22×C4×C67—21no—
141C22×C4×C67—21no—
142C22×C4×C67—21no—
143C22×C4×C67—81no—
144C22×C4×C67—81no—
145C22×C4×C67—81no—
146C22×C4×C67—81no—
147C22×C4×C67—81no—
148C22×C4×C67—81no—
149C22×C4×C67C2×D411yescharacteristic
150C22×C4×C67C2×D411yescharacteristic
151C22×C4×C67C2×D411yescharacteristic
152C22×C4×C67C2×D411yescharacteristic
153C22×C4×C67C2×D411yescharacteristic
154C22×C4×C67C2×D411yescharacteristic
155C22×C4×C67C2411yescharacteristic
156SD16×C67—44no—
157C2×C8×C67—21no—
158C2×C8×C67—21no—
159C2×C8×C67—21no—
160C2×C8×C67—21no—
161C2×C8×C67—21no—
162C2×C8×C67—21no—
163C2×C8×C67—21no—
164C2×C8×C67—21no—
165C2×C8×C67—21no—
166C2×C8×C67—21no—
167C2×C8×C67—21no—
168C2×C8×C67—21no—
169C4⋊C4×C67—21no—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—21no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—21no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—81no—
179C4⋊C4×C67—81no—
180C4⋊C4×C67—81no—
181C4⋊C4×C67#4112yes—
182C4⋊C4×C67#4112yes—
183C4⋊C4×C67#4112yes—
184C22⋊C4×C67—24no—
185C22⋊C4×C67—24no—
186C22⋊C4×C67—24no—
187C42×C67—21no—
188C42×C67—21no—
189C42×C67—21no—
190C42×C67—21no—
191C42×C67—21no—
192C42×C67—21no—
193C42×C67—81no—
194C42×C67—81no—
195C42×C67—81no—
196C23×C67—22no—
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—41no—
200C23×C67—41no—
201C23×C67—41no—
202C23×C67—41no—
203C23×C67C22×D411yescharacteristic
204Q8×C67—41no—
205Q8×C67—41no—
206Q8×C67—41no—
207Q8×C67—41no—
208D4×C67—22no—
209D4×C67—22no—
210D4×C67—22no—
211D4×C67#12614yes—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—22no—
219C2×C4×C67—22no—
220C2×C4×C67—22no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
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—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—41no—
240C2×C4×C67—41no—
241C2×C4×C67—41no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—41no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—41no—
250C2×C4×C67—41no—
251C2×C4×C67—84no—
252C2×C4×C67—84no—
253C2×C4×C67—84no—
254C2×C4×C67C2×#4111yescharacteristic
255C2×C4×C67C2×#4111yescharacteristic
256C2×C4×C67C2×#4111yescharacteristic
257C2×C4×C67C2×#4111yescharacteristic
258C2×C4×C67C2×#4111yescharacteristic
259C2×C4×C67C2×#4111yescharacteristic
260C8×C67—41no—
261C8×C67—41no—
262C8×C67—41no—
263C8×C67—41no—
264C22×C67—24no—
265C22×C67—42no—
266C22×C67—42no—
267C22×C67—42no—
268C22×C67—42no—
269C22×C67—42no—
270C22×C67—42no—
271C22×C67#52511yescharacteristic
272C22×C67#52511yescharacteristic
273C22×C67#52511yescharacteristic
274C22×C67#57611yescharacteristic
275C22×C67#57711yescharacteristic
276C22×C67#57711yescharacteristic
277C22×C67#57811yescharacteristic
278C4×C67—41no—
279C4×C67—41no—
280C4×C67—42no—
281C4×C67—42no—
282C4×C67—41no—
283C4×C67—41no—
284C4×C67—41no—
285C4×C67—42no—
286C4×C67—41no—
287C4×C67—81no—
288C4×C67—81no—
289C4×C67—81no—
290C4×C67—81no—
291C4×C67C2×#12611yescharacteristic
292C4×C67#52911yescharacteristic
293C4×C67#52911yescharacteristic
294C4×C67#52911yescharacteristic
295#33090C6711yesmaximal, characteristic, direct product factor, Sylow
296C2×C67—44no—
297C2×C67#285211yescharacteristic
298C2×C67#311811yescharacteristic
299C2×C67#311811yescharacteristic
300C2×C67#312011yescharacteristic
301C2×C67#312011yescharacteristic
302C2×C67#312311yescharacteristic
303C2×C67#312411yescharacteristic
304#2645C2×C6711yescharacteristic, complemented
305#2275C2×C6711yescharacteristic
306#1928C2×C6711yescharacteristic, complemented
307#1928C2×C6711yescharacteristic, complemented
308#1927C2×C6711yescharacteristic, complemented
309#1921C2×C6711yescharacteristic, complemented
310#1920C2×C6711yescharacteristic, complemented
311#1918C2×C6711yescharacteristic, complemented
312#1877C2×C6711yescharacteristic
313#1877C2×C6711yescharacteristic
314#1877C2×C6711yescharacteristic
315#1864C2×C6711yescharacteristic
316#1862C2×C6711yescharacteristic
317#1862C2×C6711yescharacteristic
318#1840C2×C6711yescharacteristic, complemented
319C67#3309011yescharacteristic, direct product factor, Sylow
320C22×SD16C22×C6711yescharacteristic
321C2×C4×D4C22×C6711yescharacteristic
322C2×C4×D4C22×C6711yescharacteristic
323C2×C4×D4C22×C6711yescharacteristic
324C2×#107C22×C6711yescharacteristic
325C2×#107C22×C6711yescharacteristic
326C2×#106C22×C6711yescharacteristic
327C2×#103C22×C6711yescharacteristic
328C2×#103C22×C6711yescharacteristic
329C2×#103C22×C6711yescharacteristic
330C2×#102C22×C6711yescharacteristic
331C2×#102C22×C6711yescharacteristic
332C2×#102C22×C6711yescharacteristic
333#401—21no—
334#400—21no—
335#400—21no—
336#400—21no—
337#391C22×C6711yescharacteristic
338#391C22×C6711yescharacteristic
339#391C22×C6711yescharacteristic
340#389C22×C6711yescharacteristic
341#389C22×C6711yescharacteristic
342#389C22×C6711yescharacteristic
343C4×C4⋊C4C22×C6711yescharacteristic
344C4×C4⋊C4C22×C6711yescharacteristic
345C4×C4⋊C4C22×C6711yescharacteristic
346C4×C22⋊C4—21no—
347C4×C22⋊C4—21no—
348C4×C22⋊C4—21no—
349#376C22×C6711yescharacteristic
350C2×#95—21no—
351#340C22×C6711yescharacteristic
352#340C22×C6711yescharacteristic
353#340C22×C6711yescharacteristic
354#340C22×C6711yescharacteristic
355#340C22×C6711yescharacteristic
356#340C22×C6711yescharacteristic
357#336C22×C6711yescharacteristic
358#336C22×C6711yescharacteristic
359#336C22×C6711yescharacteristic
360#336C22×C6711yescharacteristic
361#336C22×C6711yescharacteristic
362#336C22×C6711yescharacteristic
363C22×Q8C23×C6711yescharacteristic
364C22×D4C23×C6711yescharacteristic
365C23×C4—21no—
366C23×C4—21no—
367C23×C4—21no—
368C2×SD16—22no—
369C2×SD16—22no—
370C2×SD16—22no—
371C22×C8C23×C6711yescharacteristic
372C22×C8C23×C6711yescharacteristic
373C22×C8C23×C6711yescharacteristic
374C22×C8C23×C6711yescharacteristic
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4C23×C6711yescharacteristic
385C2×C4⋊C4C23×C6711yescharacteristic
386C2×C4⋊C4C23×C6711yescharacteristic
387C2×C4⋊C4C23×C6711yescharacteristic
388C2×C4⋊C4C23×C6711yescharacteristic
389C2×C4⋊C4C23×C6711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42C23×C6711yescharacteristic
397C2×C42C23×C6711yescharacteristic
398C2×C42C23×C6711yescharacteristic
399#107—22no—
400#107—22no—
401#106—22no—
402#103—22no—
403#103—22no—
404#103—22no—
405#102—22no—
406#102—22no—
407#102—22no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95D4×C6711yescharacteristic
421#95D4×C6711yescharacteristic
422#95D4×C6711yescharacteristic
423#95D4×C6711yescharacteristic
424C24—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4#41×C6712yes—
435C2×D4#41×C6712yes—
436C2×D4#41×C6712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4C2×D4×C6711yescharacteristic
454C22×C4C2×D4×C6711yescharacteristic
455C22×C4C2×D4×C6711yescharacteristic
456C22×C4C2×D4×C6711yescharacteristic
457C22×C4C2×D4×C6711yescharacteristic
458C22×C4C2×D4×C6711yescharacteristic
459C22×C4C24×C6711yescharacteristic
460SD16—44no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4#41×C6712yes—
486C4⋊C4#41×C6712yes—
487C4⋊C4#41×C6712yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—81no—
498C42—81no—
499C42—81no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×D4×C6711yescharacteristic
508Q8—41no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4#126×C6714yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4C2×#41×C6711yescharacteristic
559C2×C4C2×#41×C6711yescharacteristic
560C2×C4C2×#41×C6711yescharacteristic
561C2×C4C2×#41×C6711yescharacteristic
562C2×C4C2×#41×C6711yescharacteristic
563C2×C4C2×#41×C6711yescharacteristic
564C8—41no—
565C8—41no—
566C8—41no—
567C8—41no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22#525×C6711yescharacteristic
576C22#525×C6711yescharacteristic
577C22#525×C6711yescharacteristic
578C22#576×C6711yescharacteristic
579C22#577×C6711yescharacteristic
580C22#577×C6711yescharacteristic
581C22#578×C6711yescharacteristic
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4C2×#126×C6711yescharacteristic
596C4#529×C6711yescharacteristic
597C4#529×C6711yescharacteristic
598C4#529×C6711yescharacteristic
599C2—44no—
600C2C67×#285211yescharacteristic
601C2C67×#311811yescharacteristic
602C2C67×#311811yescharacteristic
603C2C67×#312011yescharacteristic
604C2C67×#312011yescharacteristic
605C2C67×#312311yescharacteristic
606C2C67×#312411yescharacteristic
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC22×D4×C67C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ #376×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2645C2semidirect
3. C67×#1928C2semidirect
4. C67×#1928C2semidirect
5. C67×#1927C2semidirect
6. C67×#1921C2semidirect
7. C67×#1920C2semidirect
8. C67×#1918C2semidirect
15. C67×#1840C2semidirect
295. #33090C67direct (N × H)
304. #2645C2×C67semidirect
306. #1928C2×C67semidirect
307. #1928C2×C67semidirect
308. #1927C2×C67semidirect
309. #1921C2×C67semidirect
310. #1920C2×C67semidirect
311. #1918C2×C67semidirect
318. #1840C2×C67semidirect
319. C67#33090direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#26452
C67×#22752
C67×#19282
C67×#19272
C67×#19212
C67×#19202
C67×#19182
C67×#18772
C67×#18642
C67×#18622
C67×#18402
#3309067