Finite Group Databasebeta

← C67×#33087 #30712338

Subgroups — C67×#33087 #30712338

Order 17152.

728 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2644C211yesmaximal, characteristic
2C67×#2276C211yesmaximal, characteristic, complemented
3C67×#1926C211yesmaximal, characteristic
4C67×#1925C211yesmaximal, characteristic
5C67×#1925C211yesmaximal, characteristic
6C67×#1921C211yesmaximal, characteristic, complemented
7C67×#1919C211yesmaximal, characteristic, complemented
8C67×#1918C211yesmaximal, characteristic, complemented
9C67×#1880C211yesmaximal, characteristic
10C67×#1880C211yesmaximal, characteristic
11C67×#1880C211yesmaximal, characteristic
12C67×#1865C211yesmaximal, characteristic, complemented
13C67×#1863C211yesmaximal, characteristic, complemented
14C67×#1863C211yesmaximal, characteristic, complemented
15C67×#1840C211yesmaximal, characteristic, complemented
16C22×SD16×C67C2211yescharacteristic
17C2×C4×Q8×C67C2211yescharacteristic
18C2×C4×Q8×C67C2211yescharacteristic
19C2×C4×Q8×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#397×C67—21no—
30#397×C67—21no—
31#397×C67—21no—
32#396×C67—21no—
33#390×C67C2211yescharacteristic
34#390×C67C2211yescharacteristic
35#390×C67C2211yescharacteristic
36#389×C67C2211yescharacteristic
37#389×C67C2211yescharacteristic
38#389×C67C2211yescharacteristic
39C4×C4⋊C4×C67—21no—
40C4×C4⋊C4×C67—21no—
41C4×C4⋊C4×C67—21no—
42C4×C4⋊C4×C67C2211yescharacteristic
43C4×C4⋊C4×C67C2211yescharacteristic
44C4×C4⋊C4×C67C2211yescharacteristic
45#376×C67—21no—
46#376×C67C2211yescharacteristic
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
61C2×SD16×C67—22no—
62C2×SD16×C67—22no—
63C2×SD16×C67—22no—
64C22×C8×C67C2311yescharacteristic
65C22×C8×C67C2311yescharacteristic
66C22×C8×C67C2311yescharacteristic
67C22×C8×C67C2311yescharacteristic
68C4×Q8×C67—22no—
69C4×Q8×C67—22no—
70C4×Q8×C67—22no—
71C4×Q8×C67—22no—
72C4×Q8×C67—22no—
73C4×Q8×C67—22no—
74C2×C4⋊C4×C67—21no—
75C2×C4⋊C4×C67—21no—
76C2×C4⋊C4×C67—21no—
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×C22⋊C4×C67—21no—
90C2×C22⋊C4×C67—21no—
91C2×C22⋊C4×C67—21no—
92C2×C42×C67—21no—
93C2×C42×C67—21no—
94C2×C42×C67—21no—
95C2×C42×C67—21no—
96C2×C42×C67—21no—
97C2×C42×C67—21no—
98C2×C42×C67C2311yescharacteristic
99C2×C42×C67C2311yescharacteristic
100C2×C42×C67C2311yescharacteristic
101#107×C67—22no—
102#107×C67—22no—
103#106×C67—22no—
104#103×C67—22no—
105#103×C67—22no—
106#103×C67—22no—
107#102×C67—22no—
108#102×C67—22no—
109#102×C67—22no—
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#4112yes—
125C2×Q8×C67#4112yes—
126C2×Q8×C67#4112yes—
127C2×D4×C67—21no—
128C2×D4×C67—21no—
129C2×D4×C67—21no—
130C2×D4×C67—21no—
131C2×D4×C67—21no—
132C2×D4×C67—21no—
133C22×C4×C67—21no—
134C22×C4×C67—21no—
135C22×C4×C67—21no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
138C22×C4×C67—21no—
139C22×C4×C67—21no—
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2×D411yescharacteristic
142C22×C4×C67C2×D411yescharacteristic
143C22×C4×C67C2×D411yescharacteristic
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67C2411yescharacteristic
147SD16×C67—44no—
148C2×C8×C67—21no—
149C2×C8×C67—21no—
150C2×C8×C67—21no—
151C2×C8×C67—21no—
152C2×C8×C67—21no—
153C2×C8×C67—21no—
154C2×C8×C67—21no—
155C2×C8×C67—21no—
156C2×C8×C67—21no—
157C2×C8×C67—21no—
158C2×C8×C67—21no—
159C2×C8×C67—21no—
160C4⋊C4×C67—24no—
161C4⋊C4×C67—24no—
162C4⋊C4×C67—24no—
163C4⋊C4×C67—21no—
164C4⋊C4×C67—21no—
165C4⋊C4×C67—21no—
166C4⋊C4×C67—21no—
167C4⋊C4×C67—21no—
168C4⋊C4×C67—21no—
169C4⋊C4×C67—21no—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—81no—
173C4⋊C4×C67—81no—
174C4⋊C4×C67—81no—
175C4⋊C4×C67#4112yes—
176C4⋊C4×C67#4112yes—
177C4⋊C4×C67#4112yes—
178C22⋊C4×C67—81no—
179C22⋊C4×C67—81no—
180C22⋊C4×C67—81no—
181C22⋊C4×C67—81no—
182C22⋊C4×C67—81no—
183C22⋊C4×C67—81no—
184C42×C67—24no—
185C42×C67—24no—
186C42×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—41no—
197C23×C67—41no—
198C23×C67—41no—
199C23×C67—41no—
200C23×C67—41no—
201C23×C67—41no—
202C23×C67—41no—
203C23×C67C22×D411yescharacteristic
204Q8×C67—22no—
205Q8×C67—22no—
206Q8×C67—22no—
207Q8×C67#12614yes—
208D4×C67—41no—
209D4×C67—41no—
210D4×C67—41no—
211D4×C67—41no—
212C2×C4×C67—22no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—22no—
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—21no—
234C2×C4×C67—21no—
235C2×C4×C67—21no—
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×C67C2×#4111yescharacteristic
252C2×C4×C67C2×#4111yescharacteristic
253C2×C4×C67C2×#4111yescharacteristic
254C2×C4×C67C2×#4111yescharacteristic
255C2×C4×C67C2×#4111yescharacteristic
256C2×C4×C67C2×#4111yescharacteristic
257C8×C67—41no—
258C8×C67—41no—
259C8×C67—41no—
260C8×C67—41no—
261C22×C67—81no—
262C22×C67—81no—
263C22×C67—81no—
264C22×C67—81no—
265C22×C67—81no—
266C22×C67—81no—
267C22×C67—81no—
268C22×C67#52511yescharacteristic
269C22×C67#52511yescharacteristic
270C22×C67#52511yescharacteristic
271C22×C67#57611yescharacteristic
272C22×C67#57711yescharacteristic
273C22×C67#57711yescharacteristic
274C22×C67#57811yescharacteristic
275C4×C67—24no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—42no—
279C4×C67—42no—
280C4×C67—41no—
281C4×C67—41no—
282C4×C67—41no—
283C4×C67—42no—
284C4×C67—41no—
285C4×C67—81no—
286C4×C67—81no—
287C4×C67—81no—
288C4×C67C2×#12611yescharacteristic
289C4×C67#52911yescharacteristic
290C4×C67#52911yescharacteristic
291C4×C67#52911yescharacteristic
292#33087C6711yesmaximal, characteristic, direct product factor, Sylow
293C2×C67—161no—
294C2×C67#285211yescharacteristic
295C2×C67#311911yescharacteristic
296C2×C67#311911yescharacteristic
297C2×C67#312111yescharacteristic
298C2×C67#312111yescharacteristic
299C2×C67#312411yescharacteristic
300C2×C67#312611yescharacteristic
301#2644C2×C6711yescharacteristic
302#2276C2×C6711yescharacteristic, complemented
303#1926C2×C6711yescharacteristic
304#1925C2×C6711yescharacteristic
305#1925C2×C6711yescharacteristic
306#1921C2×C6711yescharacteristic, complemented
307#1919C2×C6711yescharacteristic, complemented
308#1918C2×C6711yescharacteristic, complemented
309#1880C2×C6711yescharacteristic
310#1880C2×C6711yescharacteristic
311#1880C2×C6711yescharacteristic
312#1865C2×C6711yescharacteristic, complemented
313#1863C2×C6711yescharacteristic, complemented
314#1863C2×C6711yescharacteristic, complemented
315#1840C2×C6711yescharacteristic, complemented
316C67#3308711yescharacteristic, direct product factor, Sylow
317C22×SD16C22×C6711yescharacteristic
318C2×C4×Q8C22×C6711yescharacteristic
319C2×C4×Q8C22×C6711yescharacteristic
320C2×C4×Q8C22×C6711yescharacteristic
321C2×#107C22×C6711yescharacteristic
322C2×#107C22×C6711yescharacteristic
323C2×#106C22×C6711yescharacteristic
324C2×#103C22×C6711yescharacteristic
325C2×#103C22×C6711yescharacteristic
326C2×#103C22×C6711yescharacteristic
327C2×#102C22×C6711yescharacteristic
328C2×#102C22×C6711yescharacteristic
329C2×#102C22×C6711yescharacteristic
330#397—21no—
331#397—21no—
332#397—21no—
333#396—21no—
334#390C22×C6711yescharacteristic
335#390C22×C6711yescharacteristic
336#390C22×C6711yescharacteristic
337#389C22×C6711yescharacteristic
338#389C22×C6711yescharacteristic
339#389C22×C6711yescharacteristic
340C4×C4⋊C4—21no—
341C4×C4⋊C4—21no—
342C4×C4⋊C4—21no—
343C4×C4⋊C4C22×C6711yescharacteristic
344C4×C4⋊C4C22×C6711yescharacteristic
345C4×C4⋊C4C22×C6711yescharacteristic
346#376—21no—
347#376C22×C6711yescharacteristic
348#340C22×C6711yescharacteristic
349#340C22×C6711yescharacteristic
350#340C22×C6711yescharacteristic
351#340C22×C6711yescharacteristic
352#340C22×C6711yescharacteristic
353#340C22×C6711yescharacteristic
354#336C22×C6711yescharacteristic
355#336C22×C6711yescharacteristic
356#336C22×C6711yescharacteristic
357#336C22×C6711yescharacteristic
358#336C22×C6711yescharacteristic
359#336C22×C6711yescharacteristic
360C22×Q8C23×C6711yescharacteristic
361C22×D4C23×C6711yescharacteristic
362C2×SD16—22no—
363C2×SD16—22no—
364C2×SD16—22no—
365C22×C8C23×C6711yescharacteristic
366C22×C8C23×C6711yescharacteristic
367C22×C8C23×C6711yescharacteristic
368C22×C8C23×C6711yescharacteristic
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4C23×C6711yescharacteristic
382C2×C4⋊C4C23×C6711yescharacteristic
383C2×C4⋊C4C23×C6711yescharacteristic
384C2×C4⋊C4C23×C6711yescharacteristic
385C2×C4⋊C4C23×C6711yescharacteristic
386C2×C4⋊C4C23×C6711yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42C23×C6711yescharacteristic
400C2×C42C23×C6711yescharacteristic
401C2×C42C23×C6711yescharacteristic
402#107—22no—
403#107—22no—
404#106—22no—
405#103—22no—
406#103—22no—
407#103—22no—
408#102—22no—
409#102—22no—
410#102—22no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95D4×C6711yescharacteristic
418#95D4×C6711yescharacteristic
419#95D4×C6711yescharacteristic
420#95D4×C6711yescharacteristic
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#41×C6712yes—
426C2×Q8#41×C6712yes—
427C2×Q8#41×C6712yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4C2×D4×C6711yescharacteristic
442C22×C4C2×D4×C6711yescharacteristic
443C22×C4C2×D4×C6711yescharacteristic
444C22×C4C2×D4×C6711yescharacteristic
445C22×C4C2×D4×C6711yescharacteristic
446C22×C4C2×D4×C6711yescharacteristic
447C22×C4C24×C6711yescharacteristic
448SD16—44no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
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—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4#41×C6712yes—
477C4⋊C4#41×C6712yes—
478C4⋊C4#41×C6712yes—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—81no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C22×D4×C6711yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8#126×C6714yes—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
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×C4C2×#41×C6711yescharacteristic
553C2×C4C2×#41×C6711yescharacteristic
554C2×C4C2×#41×C6711yescharacteristic
555C2×C4C2×#41×C6711yescharacteristic
556C2×C4C2×#41×C6711yescharacteristic
557C2×C4C2×#41×C6711yescharacteristic
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#525×C6711yescharacteristic
570C22#525×C6711yescharacteristic
571C22#525×C6711yescharacteristic
572C22#576×C6711yescharacteristic
573C22#577×C6711yescharacteristic
574C22#577×C6711yescharacteristic
575C22#578×C6711yescharacteristic
576C4—24no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4C2×#126×C6711yescharacteristic
590C4#529×C6711yescharacteristic
591C4#529×C6711yescharacteristic
592C4#529×C6711yescharacteristic
593C2—161no—
594C2C67×#285211yescharacteristic
595C2C67×#311911yescharacteristic
596C2C67×#311911yescharacteristic
597C2C67×#312111yescharacteristic
598C2C67×#312111yescharacteristic
599C2C67×#312411yescharacteristic
600C2C67×#312611yescharacteristic
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
2. C67×#2276C2semidirect
6. C67×#1921C2semidirect
7. C67×#1919C2semidirect
8. C67×#1918C2semidirect
12. C67×#1865C2semidirect
13. C67×#1863C2semidirect
14. C67×#1863C2semidirect
15. C67×#1840C2semidirect
292. #33087C67direct (N × H)
302. #2276C2×C67semidirect
306. #1921C2×C67semidirect
307. #1919C2×C67semidirect
308. #1918C2×C67semidirect
312. #1865C2×C67semidirect
313. #1863C2×C67semidirect
314. #1863C2×C67semidirect
315. #1840C2×C67semidirect
316. C67#33087direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#26442
C67×#22762
C67×#19262
C67×#19252
C67×#19212
C67×#19192
C67×#19182
C67×#18802
C67×#18652
C67×#18632
C67×#18402
#3308767