Finite Group Databasebeta

← C67×#14866 #30694117

Subgroups — C67×#14866 #30694117

Order 17152.

1534 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393×C67C211yesmaximal, characteristic
2C2×#391×C67C211yesmaximal, characteristic, complemented
3C2×#388×C67C211yesmaximal, characteristic
4C2×C4×C4⋊C4×C67C211yesmaximal, characteristic, complemented
5C67×#1446C211yesmaximal, characteristic, complemented
6C67×#1442C211yesmaximal, characteristic, complemented
7C67×#1441C212yesmaximal, complemented
8C67×#1439C211yesmaximal, characteristic
9C67×#1439C212yesmaximal
10C67×#1437C212yesmaximal
11C67×#1434C212yesmaximal, complemented
12C23×Q8×C67C2211yescharacteristic
13C22×C4⋊C4×C67C2212yes—
14C22×C4⋊C4×C67C2211yescharacteristic
15C22×C4⋊C4×C67C2211yescharacteristic
16C22×C22⋊C4×C67C2211yescharacteristic
17C22×C22⋊C4×C67C2211yescharacteristic
18C22×C22⋊C4×C67C2212yes—
19C22×C22⋊C4×C67C2211yescharacteristic
20C22×C22⋊C4×C67C2212yes—
21C22×C42×C67C2212yes—
22C22×C42×C67C2211yescharacteristic
23#393×C67—24no—
24#391×C67—24no—
25#388×C67—24no—
26C4×C4⋊C4×C67—24no—
27C2×#95×C67C2211yescharacteristic
28C2×#95×C67C2211yescharacteristic
29C2×#95×C67C2211yescharacteristic
30C2×#95×C67C2212yes—
31C2×#95×C67C2212yes—
32C2×#95×C67C2212yes—
33C2×#95×C67C2212yes—
34C2×#95×C67C2212yes—
35C2×#95×C67C2212yes—
36C2×#95×C67C2212yes—
37C2×#95×C67C2212yes—
38C2×#95×C67C2211yescharacteristic
39C25×C67C2311yescharacteristic
40C22×Q8×C67—22no—
41C22×Q8×C67—22no—
42C22×Q8×C67—24no—
43C22×Q8×C67—22no—
44C22×Q8×C67—42no—
45C23×C4×C67C2311yescharacteristic
46C23×C4×C67C2312yes—
47C23×C4×C67C2311yescharacteristic
48C23×C4×C67C2312yes—
49C23×C4×C67C2311yescharacteristic
50C23×C4×C67C2312yes—
51C23×C4×C67C2312yes—
52C23×C4×C67C2311yescharacteristic
53C23×C4×C67C2311yescharacteristic
54C23×C4×C67C2311yescharacteristic
55C2×C4⋊C4×C67—24no—
56C2×C4⋊C4×C67—24no—
57C2×C4⋊C4×C67—24no—
58C2×C4⋊C4×C67—22no—
59C2×C4⋊C4×C67—22no—
60C2×C4⋊C4×C67—24no—
61C2×C4⋊C4×C67—42no—
62C2×C22⋊C4×C67—22no—
63C2×C22⋊C4×C67—22no—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—22no—
66C2×C22⋊C4×C67—24no—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—24no—
69C2×C22⋊C4×C67—24no—
70C2×C22⋊C4×C67—22no—
71C2×C22⋊C4×C67—24no—
72C2×C22⋊C4×C67—42no—
73C2×C22⋊C4×C67—42no—
74C2×C22⋊C4×C67—42no—
75C2×C22⋊C4×C67Q814yescomplemented
76C2×C42×C67—22no—
77C2×C42×C67—22no—
78C2×C42×C67—24no—
79C2×C42×C67—24no—
80C2×C42×C67—42no—
81C2×C42×C67D414yes—
82#95×C67—22no—
83#95×C67—22no—
84#95×C67—22no—
85#95×C67—24no—
86#95×C67—24no—
87#95×C67—24no—
88#95×C67—24no—
89#95×C67—24no—
90#95×C67—42no—
91#95×C67—42no—
92#95×C67—42no—
93#95×C67D414yes—
94C24×C67—21no—
95C24×C67—21no—
96C24×C67—21no—
97C24×C67—22no—
98C24×C67—22no—
99C24×C67—21no—
100C24×C67—22no—
101C24×C67—22no—
102C24×C67—21no—
103C24×C67—21no—
104C24×C67C2×Q812yes—
105C24×C67C2411yescharacteristic
106C2×Q8×C67—28no—
107C2×Q8×C67—48no—
108C2×Q8×C67—48no—
109C2×Q8×C67—44no—
110C2×Q8×C67—44no—
111C22×C4×C67—21no—
112C22×C4×C67—21no—
113C22×C4×C67—21no—
114C22×C4×C67—21no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
117C22×C4×C67—21no—
118C22×C4×C67—22no—
119C22×C4×C67—21no—
120C22×C4×C67—22no—
121C22×C4×C67—21no—
122C22×C4×C67—22no—
123C22×C4×C67—22no—
124C22×C4×C67—22no—
125C22×C4×C67—22no—
126C22×C4×C67—21no—
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—22no—
134C22×C4×C67—22no—
135C22×C4×C67—22no—
136C22×C4×C67—22no—
137C22×C4×C67—22no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—22no—
141C22×C4×C67—22no—
142C22×C4×C67—21no—
143C22×C4×C67—21no—
144C22×C4×C67—21no—
145C22×C4×C67—22no—
146C22×C4×C67—21no—
147C22×C4×C67—22no—
148C22×C4×C67—22no—
149C22×C4×C67—22no—
150C22×C4×C67—22no—
151C22×C4×C67—22no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—21no—
157C22×C4×C67—22no—
158C22×C4×C67—21no—
159C22×C4×C67C2×D414yes—
160C22×C4×C67C2×D412yes—
161C22×C4×C67C2×D412yes—
162C22×C4×C67C2×D412yes—
163C22×C4×C67C2×D412yes—
164C22×C4×C67C2×Q812yes—
165C22×C4×C67C2×Q812yes—
166C22×C4×C67#4114yes—
167C22×C4×C67#4114yes—
168C22×C4×C67#4112yes—
169C22×C4×C67#4112yes—
170C22×C4×C67#4114yes—
171C4⋊C4×C67—216no—
172C4⋊C4×C67—44no—
173C4⋊C4×C67—44no—
174C22⋊C4×C67—28no—
175C22⋊C4×C67—48no—
176C22⋊C4×C67—48no—
177C22⋊C4×C67—44no—
178C22⋊C4×C67—44no—
179C42×C67—28no—
180C42×C67—48no—
181C23×C67—22no—
182C23×C67—24no—
183C23×C67—22no—
184C23×C67—24no—
185C23×C67—22no—
186C23×C67—41no—
187C23×C67—41no—
188C23×C67—42no—
189C23×C67—41no—
190C23×C67—42no—
191C23×C67—42no—
192C23×C67—41no—
193C23×C67—41no—
194C23×C67—41no—
195C23×C67—42no—
196C23×C67—42no—
197C23×C67—42no—
198C23×C67—41no—
199C23×C67—42no—
200C23×C67—41no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67—42no—
204C23×C67C22×D411yescharacteristic
205C23×C67C22×D411yescharacteristic
206C23×C67C22×Q811yescharacteristic
207C23×C67C2×#4111yescharacteristic
208C23×C67C2×#4111yescharacteristic
209C23×C67C2×#4112yes—
210C23×C67C2×#4112yes—
211C23×C67C2×#4112yes—
212C23×C67H(5,𝔽2)12yes—
213C23×C67H(5,𝔽2)11yescharacteristic
214C23×C672−1+411yescharacteristic
215Q8×C67—416no—
216C2×C4×C67—22no—
217C2×C4×C67—22no—
218C2×C4×C67—24no—
219C2×C4×C67—22no—
220C2×C4×C67—24no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—22no—
224C2×C4×C67—22no—
225C2×C4×C67—24no—
226C2×C4×C67—22no—
227C2×C4×C67—24no—
228C2×C4×C67—22no—
229C2×C4×C67—22no—
230C2×C4×C67—22no—
231C2×C4×C67—24no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—24no—
236C2×C4×C67—24no—
237C2×C4×C67—24no—
238C2×C4×C67—24no—
239C2×C4×C67—24no—
240C2×C4×C67—24no—
241C2×C4×C67—42no—
242C2×C4×C67—41no—
243C2×C4×C67—42no—
244C2×C4×C67—41no—
245C2×C4×C67—42no—
246C2×C4×C67—42no—
247C2×C4×C67—42no—
248C2×C4×C67—42no—
249C2×C4×C67—42no—
250C2×C4×C67—42no—
251C2×C4×C67—42no—
252C2×C4×C67—42no—
253C2×C4×C67—42no—
254C2×C4×C67—42no—
255C2×C4×C67—42no—
256C2×C4×C67—41no—
257C2×C4×C67—41no—
258C2×C4×C67—42no—
259C2×C4×C67—41no—
260C2×C4×C67—41no—
261C2×C4×C67—42no—
262C2×C4×C67—41no—
263C2×C4×C67—42no—
264C2×C4×C67—42no—
265C2×C4×C67—41no—
266C2×C4×C67—42no—
267C2×C4×C67—42no—
268C2×C4×C67—42no—
269C2×C4×C67—42no—
270C2×C4×C67—42no—
271C2×C4×C67—42no—
272C2×C4×C67—42no—
273C2×C4×C67#12114yes—
274C2×C4×C67C22⋊Q814yes—
275C2×C4×C67#12414yes—
276C22×C67—42no—
277C22×C67—44no—
278C22×C67—42no—
279C22×C67—42no—
280C22×C67—44no—
281C22×C67—81no—
282C22×C67—81no—
283C22×C67—81no—
284C22×C67—81no—
285C22×C67—82no—
286C22×C67—82no—
287C22×C67C2×#12111yescharacteristic
288C22×C67C2×C22⋊Q811yescharacteristic
289C22×C67#52511yescharacteristic
290C22×C67#52512yes—
291C22×C67C2×#12411yescharacteristic
292C22×C67#52912yes—
293C22×C67#52912yes—
294C22×C67#53211yescharacteristic
295C22×C67#53312yes—
296C22×C67#53511yescharacteristic
297C22×C67#53711yescharacteristic
298C22×C67#53812yes—
299C22×C67#54212yes—
300C22×C67#54612yes—
301C22×C67#54611yescharacteristic
302C22×C67D4×Q811yescharacteristic
303C22×C67#55112yes—
304C22×C67#55211yescharacteristic
305C22×C67#55412yes—
306C22×C67#55411yescharacteristic
307C22×C67#55512yes—
308C22×C67#55612yes—
309C22×C67#55912yes—
310C4×C67—24no—
311C4×C67—24no—
312C4×C67—24no—
313C4×C67—42no—
314C4×C67—44no—
315C4×C67—44no—
316C4×C67—44no—
317C4×C67—81no—
318C4×C67—81no—
319C4×C67—82no—
320#14866C6711yesmaximal, characteristic, direct product factor, Sylow
321C2×C67—82no—
322C2×C67#239611yescharacteristic
323C2×C67#241111yescharacteristic
324C2×C67#241711yescharacteristic
325C2×C67#245912yes—
326C2×C67#246512yes—
327C2×C67#253212yes—
328C2×C67#256311yescharacteristic
329C2×C67#258912yes—
330C2×C67#259211yescharacteristic
331C2×C67#266211yescharacteristic
332C2×C67#285011yescharacteristic
333C2×#393C2×C6711yescharacteristic
334C2×#391C2×C6711yescharacteristic, complemented
335C2×#388C2×C6711yescharacteristic
336C2×C4×C4⋊C4C2×C6711yescharacteristic, complemented
337#1446C2×C6711yescharacteristic, complemented
338#1442C2×C6711yescharacteristic, complemented
339#1441C2×C6712yescomplemented
340#1439C2×C6711yescharacteristic
341#1439C2×C6712yes—
342#1437C2×C6712yes—
343#1434C2×C6712yescomplemented
344C67#1486611yescharacteristic, direct product factor, Sylow
345C23×Q8C22×C6711yescharacteristic
346C22×C4⋊C4C22×C6712yes—
347C22×C4⋊C4C22×C6711yescharacteristic
348C22×C4⋊C4C22×C6711yescharacteristic
349C22×C22⋊C4C22×C6711yescharacteristic
350C22×C22⋊C4C22×C6711yescharacteristic
351C22×C22⋊C4C22×C6712yes—
352C22×C22⋊C4C22×C6711yescharacteristic
353C22×C22⋊C4C22×C6712yes—
354C22×C42C22×C6712yes—
355C22×C42C22×C6711yescharacteristic
356#393—24no—
357#391—24no—
358#388—24no—
359C4×C4⋊C4—24no—
360C2×#95C22×C6712yes—
361C2×#95C22×C6711yescharacteristic
362C2×#95C22×C6712yes—
363C2×#95C22×C6712yes—
364C2×#95C22×C6712yes—
365C2×#95C22×C6712yes—
366C2×#95C22×C6711yescharacteristic
367C2×#95C22×C6711yescharacteristic
368C2×#95C22×C6712yes—
369C2×#95C22×C6712yes—
370C2×#95C22×C6712yes—
371C2×#95C22×C6711yescharacteristic
372C25C23×C6711yescharacteristic
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—24no—
376C22×Q8—22no—
377C22×Q8—42no—
378C23×C4C23×C6711yescharacteristic
379C23×C4C23×C6712yes—
380C23×C4C23×C6711yescharacteristic
381C23×C4C23×C6712yes—
382C23×C4C23×C6711yescharacteristic
383C23×C4C23×C6712yes—
384C23×C4C23×C6712yes—
385C23×C4C23×C6711yescharacteristic
386C23×C4C23×C6711yescharacteristic
387C23×C4C23×C6711yescharacteristic
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—42no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4Q8×C6714yescomplemented
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—24no—
412C2×C42—24no—
413C2×C42—42no—
414C2×C42D4×C6714yes—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—42no—
424#95—42no—
425#95—42no—
426#95D4×C6714yes—
427C24—21no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24C2×Q8×C6712yes—
438C24C24×C6711yescharacteristic
439C2×Q8—28no—
440C2×Q8—48no—
441C2×Q8—48no—
442C2×Q8—44no—
443C2×Q8—44no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4C2×D4×C6714yes—
493C22×C4C2×D4×C6712yes—
494C22×C4C2×D4×C6712yes—
495C22×C4C2×D4×C6712yes—
496C22×C4C2×D4×C6712yes—
497C22×C4C2×Q8×C6712yes—
498C22×C4C2×Q8×C6712yes—
499C22×C4#41×C6714yes—
500C22×C4#41×C6714yes—
501C22×C4#41×C6712yes—
502C22×C4#41×C6712yes—
503C22×C4#41×C6714yes—
504C4⋊C4—216no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—28no—
513C42—48no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C22×D4×C6711yescharacteristic
538C23C22×D4×C6711yescharacteristic
539C23C22×Q8×C6711yescharacteristic
540C23C2×#41×C6711yescharacteristic
541C23C2×#41×C6711yescharacteristic
542C23C2×#41×C6712yes—
543C23C2×#41×C6712yes—
544C23C2×#41×C6712yes—
545C23H(5,𝔽2)×C6712yes—
546C23H(5,𝔽2)×C6711yescharacteristic
547C232−1+4×C6711yescharacteristic
548Q8—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4#121×C6714yes—
607C2×C4C22⋊Q8×C6714yes—
608C2×C4#124×C6714yes—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—82no—
620C22C2×#121×C6711yescharacteristic
621C22C2×C22⋊Q8×C6711yescharacteristic
622C22#525×C6711yescharacteristic
623C22#525×C6712yes—
624C22C2×#124×C6711yescharacteristic
625C22#529×C6712yes—
626C22#529×C6712yes—
627C22#532×C6711yescharacteristic
628C22#533×C6712yes—
629C22#535×C6711yescharacteristic
630C22#537×C6711yescharacteristic
631C22#538×C6712yes—
632C22#542×C6712yes—
633C22#546×C6712yes—
634C22#546×C6711yescharacteristic
635C22D4×Q8×C6711yescharacteristic
636C22#551×C6712yes—
637C22#552×C6711yescharacteristic
638C22#554×C6712yes—
639C22#554×C6711yescharacteristic
640C22#555×C6712yes—
641C22#556×C6712yes—
642C22#559×C6712yes—
643C4—24no—
644C4—24no—
645C4—24no—
646C4—42no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—81no—
651C4—81no—
652C4—82no—
653C2—82no—
654C2C67×#239611yescharacteristic
655C2C67×#241111yescharacteristic
656C2C67×#241711yescharacteristic
657C2C67×#245912yes—
658C2C67×#246512yes—
659C2C67×#253212yes—
660C2C67×#256311yescharacteristic
661C2C67×#258912yes—
662C2C67×#259211yescharacteristic
663C2C67×#266211yescharacteristic
664C2C67×#285011yescharacteristic
subgroupG/H
center, socleC24×C67C24
commutator, Frattini, cosocleC24C24×C67
prime radicalC25×C67C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#391×C67C2semidirect
4. C2×C4×C4⋊C4×C67C2semidirect
5. C67×#1446C2semidirect
6. C67×#1442C2semidirect
7. C67×#1441C2semidirect
11. C67×#1434C2semidirect
75. C2×C22⋊C4×C67Q8semidirect
320. #14866C67direct (N × H)
334. C2×#391C2×C67semidirect
336. C2×C4×C4⋊C4C2×C67semidirect
337. #1446C2×C67semidirect
338. #1442C2×C67semidirect
339. #1441C2×C67semidirect
343. #1434C2×C67semidirect
344. C67#14866direct (N × H)
408. C2×C22⋊C4Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#393×C672
C2×#391×C672
C2×#388×C672
C2×C4×C4⋊C4×C672
C67×#14462
C67×#14422
C67×#14412
C67×#14392
C67×#14372
C67×#14342
#1486667