Finite Group Databasebeta

← C67×#32985 #30712236

Subgroups — C67×#32985 #30712236

Order 17152.

1430 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C67C211yesmaximal, characteristic
2C2×#485×C67C212yesmaximal, complemented
3C67×#2848C211yesmaximal, characteristic
4C67×#1873C212yesmaximal, complemented
5C67×#1872C211yesmaximal, characteristic, complemented
6C67×#1872C211yesmaximal, characteristic, complemented
7C67×#1854C212yesmaximal, complemented
8C67×#1853C211yesmaximal, characteristic, complemented
9C67×#1852C211yesmaximal, characteristic, complemented
10C67×#1825C212yesmaximal, complemented
11C67×#1779C211yesmaximal, characteristic
12C24×C4×C67C2211yescharacteristic
13C2×C22⋊Q8×C67C2211yescharacteristic
14C2×C22⋊Q8×C67C2211yescharacteristic
15C2×C22⋊Q8×C67C2211yescharacteristic
16C2×#117×C67—21no—
17C2×#117×C67—21no—
18C2×#117×C67—28no—
19C2×#117×C67C2212yes—
20C2×#117×C67C2212yes—
21C22×C4⋊C4×C67C2211yescharacteristic
22C22×C4⋊C4×C67C2211yescharacteristic
23C22×C22⋊C4×C67—21no—
24C22×C42×C67—21no—
25#485×C67—28no—
26C2×#107×C67C2212yescomplemented
27C2×#106×C67C2212yescomplemented
28C2×#103×C67C2212yescomplemented
29C2×#103×C67C2212yescomplemented
30C2×C22⋊C8×C67C2212yes—
31C2×C22⋊C8×C67C2211yescharacteristic
32C2×C22⋊C8×C67C2211yescharacteristic
33C2×C22⋊C8×C67C2212yes—
34#385×C67—21no—
35#385×C67—21no—
36#385×C67—21no—
37#380×C67C2211yescharacteristic
38#380×C67C2211yescharacteristic
39#380×C67C2211yescharacteristic
40#379×C67—21no—
41#340×C67C2212yescomplemented
42#340×C67C2212yescomplemented
43#340×C67C2212yescomplemented
44#340×C67C2212yescomplemented
45C25×C67D411yescharacteristic
46C22×Q8×C67C2311yescharacteristic
47C23×C4×C67—24no—
48C23×C4×C67—24no—
49C23×C4×C67—22no—
50C23×C4×C67—22no—
51C23×C4×C67—21no—
52C23×C4×C67—21no—
53C23×C4×C67D411yescharacteristic
54C23×C4×C67D411yescharacteristic
55C23×C4×C67D411yescharacteristic
56C23×C4×C67C2311yescharacteristic
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2311yescharacteristic
59C22×C8×C67C2312yes—
60C22×C8×C67C2312yes—
61C22⋊Q8×C67—24no—
62C22⋊Q8×C67—22no—
63C22⋊Q8×C67—22no—
64C22⋊Q8×C67—24no—
65#117×C67—28no—
66#117×C67—28no—
67#117×C67—24no—
68#117×C67—24no—
69#117×C67—24no—
70#117×C67D418yes—
71C2×C4⋊C4×C67—21no—
72C2×C4⋊C4×C67—21no—
73C2×C4⋊C4×C67—24no—
74C2×C4⋊C4×C67—24no—
75C2×C4⋊C4×C67—21no—
76C2×C4⋊C4×C67—21no—
77C2×C4⋊C4×C67—21no—
78C2×C4⋊C4×C67C2311yescharacteristic
79C2×C4⋊C4×C67C2311yescharacteristic
80C2×C4⋊C4×C67C2311yescharacteristic
81C2×C4⋊C4×C67C2312yes—
82C2×C4⋊C4×C67C2312yes—
83C2×C22⋊C4×C67—28no—
84C2×C22⋊C4×C67—22no—
85C2×C22⋊C4×C67—22no—
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—28no—
93C2×C42×C67—22no—
94C2×C42×C67—22no—
95#107×C67—24no—
96#106×C67—24no—
97#103×C67—24no—
98#103×C67—24no—
99C22⋊C8×C67—24no—
100C22⋊C8×C67—22no—
101C22⋊C8×C67—22no—
102C22⋊C8×C67—24no—
103#95×C67—21no—
104#95×C67—21no—
105#95×C67—21no—
106C24×C67—22no—
107C24×C67—22no—
108C24×C67—22no—
109C24×C67—22no—
110C24×C67—22no—
111C24×C67—41no—
112C24×C67—41no—
113C24×C67C2×D411yescharacteristic
114C24×C67C2×D411yescharacteristic
115C24×C67C2×D411yescharacteristic
116C2×Q8×C67—22no—
117C2×Q8×C67—21no—
118C2×Q8×C67—21no—
119C2×Q8×C67—22no—
120C22×C4×C67—216no—
121C22×C4×C67—216no—
122C22×C4×C67—22no—
123C22×C4×C67—22no—
124C22×C4×C67—22no—
125C22×C4×C67—22no—
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—22no—
133C22×C4×C67—21no—
134C22×C4×C67—21no—
135C22×C4×C67—22no—
136C22×C4×C67—22no—
137C22×C4×C67—21no—
138C22×C4×C67—21no—
139C22×C4×C67—24no—
140C22×C4×C67—24no—
141C22×C4×C67—22no—
142C22×C4×C67—22no—
143C22×C4×C67—21no—
144C22×C4×C67—21no—
145C22×C4×C67—21no—
146C22×C4×C67—21no—
147C22×C4×C67—42no—
148C22×C4×C67—42no—
149C22×C4×C67—42no—
150C22×C4×C67—42no—
151C22×C4×C67C2×D411yescharacteristic
152C22×C4×C67C2×D414yes—
153C22×C4×C67C2×D411yescharacteristic
154C22×C4×C67C2×D411yescharacteristic
155C22×C4×C67#4114yes—
156C22×C4×C67#4112yes—
157C22×C4×C67#4112yes—
158C22×C4×C67C2411yescharacteristic
159C2×C8×C67—22no—
160C2×C8×C67—22no—
161C2×C8×C67—22no—
162C2×C8×C67—22no—
163C2×C8×C67—22no—
164C2×C8×C67—22no—
165C4⋊C4×C67—24no—
166C4⋊C4×C67—24no—
167C4⋊C4×C67—22no—
168C4⋊C4×C67—22no—
169C4⋊C4×C67—22no—
170C4⋊C4×C67—22no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—42no—
174C4⋊C4×C67—42no—
175C4⋊C4×C67—42no—
176C4⋊C4×C67C2×D414yes—
177C4⋊C4×C67C2×D414yes—
178C22⋊C4×C67—28no—
179C22⋊C4×C67—28no—
180C22⋊C4×C67—42no—
181C22⋊C4×C67—42no—
182C22⋊C4×C67—42no—
183C22⋊C4×C67—42no—
184C22⋊C4×C67—42no—
185C22⋊C4×C67—42no—
186C42×C67—28no—
187C42×C67—28no—
188C23×C67—21no—
189C23×C67—21no—
190C23×C67—21no—
191C23×C67—21no—
192C23×C67—28no—
193C23×C67—21no—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67—21no—
197C23×C67—21no—
198C23×C67—21no—
199C23×C67—21no—
200C23×C67—21no—
201C23×C67—21no—
202C23×C67—21no—
203C23×C67—21no—
204C23×C67—21no—
205C23×C67—44no—
206C23×C67—44no—
207C23×C67—44no—
208C23×C67—44no—
209C23×C67—44no—
210C23×C67—44no—
211C23×C67#12012yes—
212C23×C67#12312yes—
213C23×C67#12312yes—
214C23×C67#13512yes—
215C23×C67#13712yes—
216C23×C67C22×D411yescharacteristic
217Q8×C67—42no—
218Q8×C67—42no—
219C2×C4×C67—28no—
220C2×C4×C67—28no—
221C2×C4×C67—28no—
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—24no—
229C2×C4×C67—24no—
230C2×C4×C67—24no—
231C2×C4×C67—24no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—24no—
236C2×C4×C67—48no—
237C2×C4×C67—48no—
238C2×C4×C67—42no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—42no—
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—41no—
252C2×C4×C67—41no—
253C2×C4×C67—41no—
254C2×C4×C67#12012yes—
255C2×C4×C67#12312yes—
256C2×C4×C67#12312yes—
257C2×C4×C67C22×D412yes—
258C2×C4×C67C2×#4111yescharacteristic
259C2×C4×C67C2×#4111yescharacteristic
260C2×C4×C67C2×#4112yes—
261C8×C67—42no—
262C8×C67—42no—
263C22×C67—22no—
264C22×C67—22no—
265C22×C67—22no—
266C22×C67—22no—
267C22×C67—22no—
268C22×C67—22no—
269C22×C67—22no—
270C22×C67—22no—
271C22×C67—22no—
272C22×C67—22no—
273C22×C67—22no—
274C22×C67—22no—
275C22×C67—416no—
276C22×C67—41no—
277C22×C67—41no—
278C22×C67—41no—
279C22×C67—41no—
280C22×C67—41no—
281C22×C67—41no—
282C22×C67—41no—
283C22×C67—41no—
284C22×C67#48514yescomplemented
285C22×C67C2×#12011yescharacteristic
286C22×C67C2×#12311yescharacteristic
287C22×C67C2×#12311yescharacteristic
288C22×C67C2×#13511yescharacteristic
289C22×C67C2×#13711yescharacteristic
290C22×C67#57511yescharacteristic
291C22×C67#57511yescharacteristic
292C4×C67—22no—
293C4×C67—22no—
294C4×C67—22no—
295C4×C67—44no—
296C4×C67—44no—
297C4×C67—81no—
298C4×C67—81no—
299C4×C67—81no—
300C4×C67—81no—
301C4×C67#52512yes—
302C4×C67#52512yes—
303#32985C6711yesmaximal, characteristic, direct product factor, Sylow
304C2×C67—24no—
305C2×C67—42no—
306C2×C67—42no—
307C2×C67#284911yescharacteristic
308C2×C67#300211yescharacteristic
309C2×C67#300711yescharacteristic
310C2×C67C2×#48511yescharacteristic
311C2×C67#309111yescharacteristic
312C2×C67#309311yescharacteristic
313C2×C67#309511yescharacteristic
314C22×#117C2×C6711yescharacteristic
315C2×#485C2×C6712yescomplemented
316#2848C2×C6711yescharacteristic
317#1873C2×C6712yescomplemented
318#1872C2×C6711yescharacteristic, complemented
319#1872C2×C6711yescharacteristic, complemented
320#1854C2×C6712yescomplemented
321#1853C2×C6711yescharacteristic, complemented
322#1852C2×C6711yescharacteristic, complemented
323#1825C2×C6712yescomplemented
324#1779C2×C6711yescharacteristic
325C67#3298511yescharacteristic, direct product factor, Sylow
326C24×C4C22×C6711yescharacteristic
327C2×C22⋊Q8C22×C6711yescharacteristic
328C2×C22⋊Q8C22×C6711yescharacteristic
329C2×C22⋊Q8C22×C6711yescharacteristic
330C2×#117—21no—
331C2×#117—21no—
332C2×#117—28no—
333C2×#117C22×C6712yes—
334C2×#117C22×C6712yes—
335C22×C4⋊C4C22×C6711yescharacteristic
336C22×C4⋊C4C22×C6711yescharacteristic
337C22×C22⋊C4—21no—
338C22×C42—21no—
339#485—28no—
340C2×#107C22×C6712yescomplemented
341C2×#106C22×C6712yescomplemented
342C2×#103C22×C6712yescomplemented
343C2×#103C22×C6712yescomplemented
344C2×C22⋊C8C22×C6712yes—
345C2×C22⋊C8C22×C6711yescharacteristic
346C2×C22⋊C8C22×C6711yescharacteristic
347C2×C22⋊C8C22×C6712yes—
348#385—21no—
349#385—21no—
350#385—21no—
351#380C22×C6711yescharacteristic
352#380C22×C6711yescharacteristic
353#380C22×C6711yescharacteristic
354#379—21no—
355#340C22×C6712yescomplemented
356#340C22×C6712yescomplemented
357#340C22×C6712yescomplemented
358#340C22×C6712yescomplemented
359C25D4×C6711yescharacteristic
360C22×Q8C23×C6711yescharacteristic
361C23×C4—24no—
362C23×C4—24no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—21no—
366C23×C4—21no—
367C23×C4D4×C6711yescharacteristic
368C23×C4D4×C6711yescharacteristic
369C23×C4D4×C6711yescharacteristic
370C23×C4C23×C6711yescharacteristic
371C23×C4C23×C6711yescharacteristic
372C23×C4C23×C6711yescharacteristic
373C22×C8C23×C6712yes—
374C22×C8C23×C6712yes—
375C22⋊Q8—24no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—24no—
379#117—28no—
380#117—28no—
381#117—24no—
382#117—24no—
383#117—24no—
384#117D4×C6718yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4C23×C6711yescharacteristic
393C2×C4⋊C4C23×C6711yescharacteristic
394C2×C4⋊C4C23×C6711yescharacteristic
395C2×C4⋊C4C23×C6712yes—
396C2×C4⋊C4C23×C6712yes—
397C2×C22⋊C4—28no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C42—28no—
407C2×C42—22no—
408C2×C42—22no—
409#107—24no—
410#106—24no—
411#103—24no—
412#103—24no—
413C22⋊C8—24no—
414C22⋊C8—22no—
415C22⋊C8—22no—
416C22⋊C8—24no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—41no—
426C24—41no—
427C24C2×D4×C6711yescharacteristic
428C24C2×D4×C6711yescharacteristic
429C24C2×D4×C6711yescharacteristic
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4C2×D4×C6711yescharacteristic
466C22×C4C2×D4×C6714yes—
467C22×C4C2×D4×C6711yescharacteristic
468C22×C4C2×D4×C6711yescharacteristic
469C22×C4#41×C6714yes—
470C22×C4#41×C6712yes—
471C22×C4#41×C6712yes—
472C22×C4C24×C6711yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4C2×D4×C6714yes—
491C4⋊C4C2×D4×C6714yes—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C42—28no—
501C42—28no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—28no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23#120×C6712yes—
526C23#123×C6712yes—
527C23#123×C6712yes—
528C23#135×C6712yes—
529C23#137×C6712yes—
530C23C22×D4×C6711yescharacteristic
531Q8—42no—
532Q8—42no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4#120×C6712yes—
569C2×C4#123×C6712yes—
570C2×C4#123×C6712yes—
571C2×C4C22×D4×C6712yes—
572C2×C4C2×#41×C6711yescharacteristic
573C2×C4C2×#41×C6711yescharacteristic
574C2×C4C2×#41×C6712yes—
575C8—42no—
576C8—42no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—416no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22#485×C6714yescomplemented
599C22C2×#120×C6711yescharacteristic
600C22C2×#123×C6711yescharacteristic
601C22C2×#123×C6711yescharacteristic
602C22C2×#135×C6711yescharacteristic
603C22C2×#137×C6711yescharacteristic
604C22#575×C6711yescharacteristic
605C22#575×C6711yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—44no—
610C4—44no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4#525×C6712yes—
616C4#525×C6712yes—
617C2—24no—
618C2—42no—
619C2—42no—
620C2C67×#284911yescharacteristic
621C2C67×#300211yescharacteristic
622C2C67×#300711yescharacteristic
623C2C2×#485×C6711yescharacteristic
624C2C67×#309111yescharacteristic
625C2C67×#309311yescharacteristic
626C2C67×#309511yescharacteristic
subgroupG/H
center, socleC23×C67C22×D4
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC25×C67D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#485×C67C2semidirect
4. C67×#1873C2semidirect
5. C67×#1872C2semidirect
6. C67×#1872C2semidirect
7. C67×#1854C2semidirect
8. C67×#1853C2semidirect
9. C67×#1852C2semidirect
10. C67×#1825C2semidirect
26. C2×#107×C67C22semidirect
27. C2×#106×C67C22semidirect
28. C2×#103×C67C22semidirect
29. C2×#103×C67C22semidirect
41. #340×C67C22semidirect
42. #340×C67C22semidirect
43. #340×C67C22semidirect
44. #340×C67C22semidirect
284. C22×C67#485semidirect
303. #32985C67direct (N × H)
315. C2×#485C2×C67semidirect
317. #1873C2×C67semidirect
318. #1872C2×C67semidirect
319. #1872C2×C67semidirect
320. #1854C2×C67semidirect
321. #1853C2×C67semidirect
322. #1852C2×C67semidirect
323. #1825C2×C67semidirect
325. C67#32985direct (N × H)
340. C2×#107C22×C67semidirect
341. C2×#106C22×C67semidirect
342. C2×#103C22×C67semidirect
343. C2×#103C22×C67semidirect
355. #340C22×C67semidirect
356. #340C22×C67semidirect
357. #340C22×C67semidirect
358. #340C22×C67semidirect
598. C22#485×C67semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C672
C2×#485×C672
C67×#28482
C67×#18732
C67×#18722
C67×#18542
C67×#18532
C67×#18522
C67×#18252
C67×#17792
#3298567