Finite Group Databasebeta

← C67×#14919 #30694170

Subgroups — C67×#14919 #30694170

Order 17152.

1758 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#385×C67C211yesmaximal, characteristic
2C2×#381×C67C211yesmaximal, characteristic
3C2×#381×C67C211yesmaximal, characteristic
4C22×#95×C67C211yesmaximal, characteristic
5C67×#1445C212yesmaximal, complemented
6C67×#1443C212yesmaximal, complemented
7C67×#1443C212yesmaximal, complemented
8C67×#1442C212yesmaximal, complemented
9C67×C22⋊#95C211yesmaximal, characteristic
10C67×C22⋊#95C211yesmaximal, characteristic
11C67×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C67C2211yescharacteristic
13C24×C4×C67C2211yescharacteristic
14C24×C4×C67C2211yescharacteristic
15C22×C4⋊C4×C67C2212yes—
16C22×C22⋊C4×C67C2211yescharacteristic
17C22×C22⋊C4×C67C2211yescharacteristic
18C22×C22⋊C4×C67C2211yescharacteristic
19C22×C22⋊C4×C67C2211yescharacteristic
20#385×C67—24no—
21#381×C67—24no—
22#381×C67—24no—
23C2×#95×C67—24no—
24C2×#95×C67C2212yes—
25C2×#95×C67C2212yes—
26C2×#95×C67C2211yescharacteristic
27C2×#95×C67C2212yes—
28C2×#95×C67C2211yescharacteristic
29C2×#95×C67C2211yescharacteristic
30C2×#95×C67C2212yes—
31C2×#95×C67C2211yescharacteristic
32C2×#95×C67C2212yes—
33C2×#95×C67C2212yes—
34C2×#95×C67C2212yes—
35C2×#95×C67C2212yes—
36C2×#95×C67C2212yes—
37C2×#95×C67C2212yes—
38C2×#95×C67C2212yes—
39C25×C67C2311yescharacteristic
40C23×C4×C67—22no—
41C23×C4×C67—22no—
42C23×C4×C67—22no—
43C23×C4×C67—22no—
44C23×C4×C67—22no—
45C23×C4×C67—22no—
46C23×C4×C67—22no—
47C23×C4×C67—22no—
48C23×C4×C67—24no—
49C23×C4×C67—24no—
50C23×C4×C67—22no—
51C23×C4×C67—22no—
52C23×C4×C67—22no—
53C23×C4×C67—22no—
54C23×C4×C67—42no—
55C23×C4×C67—42no—
56C23×C4×C67D414yes—
57C23×C4×C67C2311yescharacteristic
58C23×C4×C67C2311yescharacteristic
59C23×C4×C67C2311yescharacteristic
60C23×C4×C67C2312yes—
61C23×C4×C67C2311yescharacteristic
62C23×C4×C67C2312yes—
63C23×C4×C67C2311yescharacteristic
64C23×C4×C67C2311yescharacteristic
65C23×C4×C67C2312yes—
66C23×C4×C67C2312yes—
67C2×C4⋊C4×C67—24no—
68C2×C4⋊C4×C67—24no—
69C2×C4⋊C4×C67—42no—
70C2×C22⋊C4×C67—22no—
71C2×C22⋊C4×C67—22no—
72C2×C22⋊C4×C67—22no—
73C2×C22⋊C4×C67—22no—
74C2×C22⋊C4×C67—22no—
75C2×C22⋊C4×C67—22no—
76C2×C22⋊C4×C67—22no—
77C2×C22⋊C4×C67—22no—
78C2×C22⋊C4×C67—22no—
79C2×C22⋊C4×C67—22no—
80C2×C22⋊C4×C67—22no—
81C2×C22⋊C4×C67D414yes—
82#95×C67—24no—
83#95×C67—24no—
84#95×C67—24no—
85#95×C67—24no—
86#95×C67—22no—
87#95×C67—22no—
88#95×C67—22no—
89#95×C67—22no—
90#95×C67—24no—
91#95×C67—24no—
92#95×C67—24no—
93#95×C67—24no—
94#95×C67—42no—
95#95×C67—42no—
96#95×C67—42no—
97C24×C67—22no—
98C24×C67—22no—
99C24×C67—22no—
100C24×C67—22no—
101C24×C67C2×D412yes—
102C24×C67C2×D412yes—
103C24×C67#4112yes—
104C24×C67#4112yes—
105C24×C67#4112yes—
106C24×C67#4112yes—
107C24×C67#4112yes—
108C24×C67C2411yescharacteristic
109C22×C4×C67—28no—
110C22×C4×C67—28no—
111C22×C4×C67—28no—
112C22×C4×C67—28no—
113C22×C4×C67—28no—
114C22×C4×C67—22no—
115C22×C4×C67—21no—
116C22×C4×C67—21no—
117C22×C4×C67—22no—
118C22×C4×C67—22no—
119C22×C4×C67—22no—
120C22×C4×C67—22no—
121C22×C4×C67—22no—
122C22×C4×C67—21no—
123C22×C4×C67—21no—
124C22×C4×C67—21no—
125C22×C4×C67—21no—
126C22×C4×C67—21no—
127C22×C4×C67—21no—
128C22×C4×C67—21no—
129C22×C4×C67—21no—
130C22×C4×C67—21no—
131C22×C4×C67—21no—
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—22no—
143C22×C4×C67—22no—
144C22×C4×C67—21no—
145C22×C4×C67—21no—
146C22×C4×C67—21no—
147C22×C4×C67—21no—
148C22×C4×C67—21no—
149C22×C4×C67—21no—
150C22×C4×C67—22no—
151C22×C4×C67—21no—
152C22×C4×C67—21no—
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67—21no—
156C22×C4×C67—21no—
157C22×C4×C67—22no—
158C22×C4×C67—22no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—22no—
162C22×C4×C67—22no—
163C22×C4×C67—22no—
164C22×C4×C67—22no—
165C22×C4×C67—22no—
166C22×C4×C67—22no—
167C22×C4×C67—22no—
168C22×C4×C67—22no—
169C22×C4×C67—22no—
170C22×C4×C67—22no—
171C22×C4×C67—48no—
172C22×C4×C67—48no—
173C22×C4×C67—48no—
174C22×C4×C67—48no—
175C22×C4×C67—44no—
176C22×C4×C67—44no—
177C22×C4×C67—44no—
178C22×C4×C67—44no—
179C22×C4×C67—44no—
180C22×C4×C67—44no—
181C22×C4×C67—44no—
182C22×C4×C67—44no—
183C22×C4×C67C2×D414yes—
184C22×C4×C67C2×D412yes—
185C22×C4×C67C2×D412yes—
186C22×C4×C67#4114yes—
187C22×C4×C67#4114yes—
188C4⋊C4×C67—48no—
189C22⋊C4×C67—28no—
190C22⋊C4×C67—44no—
191C22⋊C4×C67—44no—
192C22⋊C4×C67—44no—
193C23×C67—22no—
194C23×C67—22no—
195C23×C67—24no—
196C23×C67—24no—
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—24no—
200C23×C67—24no—
201C23×C67—24no—
202C23×C67—24no—
203C23×C67—24no—
204C23×C67—24no—
205C23×C67—24no—
206C23×C67—24no—
207C23×C67—24no—
208C23×C67—24no—
209C23×C67#12114yes—
210C23×C67#12314yes—
211C23×C67#12314yes—
212C23×C67#12314yes—
213C23×C67#12414yes—
214C23×C67#12614yes—
215C23×C67#12614yes—
216C23×C67C22×D411yescharacteristic
217C23×C67C22×D411yescharacteristic
218C23×C67C2×#4111yescharacteristic
219C23×C67C2×#4111yescharacteristic
220C23×C67C2×#4111yescharacteristic
221C23×C67C2×#4111yescharacteristic
222C23×C67C2×#4111yescharacteristic
223C23×C67C2×#4112yes—
224C23×C67C2×#4112yes—
225C23×C67H(5,𝔽2)12yes—
226C23×C672−1+412yes—
227C2×C4×C67—22no—
228C2×C4×C67—22no—
229C2×C4×C67—24no—
230C2×C4×C67—22no—
231C2×C4×C67—22no—
232C2×C4×C67—22no—
233C2×C4×C67—22no—
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—24no—
242C2×C4×C67—416no—
243C2×C4×C67—416no—
244C2×C4×C67—42no—
245C2×C4×C67—41no—
246C2×C4×C67—41no—
247C2×C4×C67—42no—
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—42no—
255C2×C4×C67—42no—
256C2×C4×C67—42no—
257C2×C4×C67—42no—
258C2×C4×C67—42no—
259C2×C4×C67—42no—
260C2×C4×C67—42no—
261C2×C4×C67—42no—
262C2×C4×C67—42no—
263C2×C4×C67—42no—
264C2×C4×C67—416no—
265C2×C4×C67—41no—
266C2×C4×C67—41no—
267C2×C4×C67—41no—
268C2×C4×C67—41no—
269C2×C4×C67—41no—
270C2×C4×C67—41no—
271C2×C4×C67—41no—
272C2×C4×C67—41no—
273C2×C4×C67—42no—
274C2×C4×C67—42no—
275C2×C4×C67—42no—
276C2×C4×C67—42no—
277C2×C4×C67—41no—
278C2×C4×C67—41no—
279C2×C4×C67—41no—
280C2×C4×C67—41no—
281C2×C4×C67—42no—
282C2×C4×C67—42no—
283C2×C4×C67—42no—
284C2×C4×C67—42no—
285C2×C4×C67—42no—
286C2×C4×C67—42no—
287C2×C4×C67—42no—
288C2×C4×C67—42no—
289C2×C4×C67—42no—
290C2×C4×C67—42no—
291C2×C4×C67—42no—
292C22×C67—24no—
293C22×C67—24no—
294C22×C67—24no—
295C22×C67—24no—
296C22×C67—28no—
297C22×C67—24no—
298C22×C67—24no—
299C22×C67—28no—
300C22×C67—28no—
301C22×C67—28no—
302C22×C67#39718yes—
303C22×C67C2×#12111yescharacteristic
304C22×C67C2×#12311yescharacteristic
305C22×C67C2×#12311yescharacteristic
306C22×C67C2×#12311yescharacteristic
307C22×C67#52511yescharacteristic
308C22×C67#52511yescharacteristic
309C22×C67C2×#12411yescharacteristic
310C22×C67C2×#12611yescharacteristic
311C22×C67C2×#12611yescharacteristic
312C22×C67#52912yes—
313C22×C67#52912yes—
314C22×C67#53511yescharacteristic
315C22×C67#53611yescharacteristic
316C22×C67#53912yes—
317C22×C67#53912yes—
318C22×C67#54612yes—
319C22×C67#54712yes—
320C22×C67#55112yes—
321C22×C67#55112yes—
322C22×C67#55112yes—
323C22×C67#55212yes—
324C22×C67#55212yes—
325C22×C67#55212yes—
326C4×C67—42no—
327C4×C67—42no—
328C4×C67—44no—
329C4×C67—44no—
330C4×C67—44no—
331C4×C67—81no—
332C4×C67—81no—
333C4×C67—81no—
334C4×C67—81no—
335C4×C67—82no—
336#14919C6711yesmaximal, characteristic, direct product factor, Sylow
337C2×C67—28no—
338C2×C67C2×#39711yescharacteristic
339C2×C67#241111yescharacteristic
340C2×C67#241311yescharacteristic
341C2×C67#241912yes—
342C2×C67#248211yescharacteristic
343C2×C67#248411yescharacteristic
344C2×C67#248812yes—
345C2×C67#249312yes—
346C2×C67#249312yes—
347C2×C67#262311yescharacteristic
348C2×C67#285311yescharacteristic
349C2×#385C2×C6711yescharacteristic
350C2×#381C2×C6711yescharacteristic
351C2×#381C2×C6711yescharacteristic
352C22×#95C2×C6711yescharacteristic
353#1445C2×C6712yescomplemented
354#1443C2×C6712yescomplemented
355#1443C2×C6712yescomplemented
356#1442C2×C6712yescomplemented
357C22⋊#95C2×C6711yescharacteristic
358C22⋊#95C2×C6711yescharacteristic
359C22⋊#95C2×C6711yescharacteristic
360C67#1491911yescharacteristic, direct product factor, Sylow
361C24×C4C22×C6711yescharacteristic
362C24×C4C22×C6711yescharacteristic
363C24×C4C22×C6711yescharacteristic
364C22×C4⋊C4C22×C6712yes—
365C22×C22⋊C4C22×C6711yescharacteristic
366C22×C22⋊C4C22×C6711yescharacteristic
367C22×C22⋊C4C22×C6711yescharacteristic
368C22×C22⋊C4C22×C6711yescharacteristic
369#385—24no—
370#381—24no—
371#381—24no—
372C2×#95—24no—
373C2×#95C22×C6712yes—
374C2×#95C22×C6712yes—
375C2×#95C22×C6712yes—
376C2×#95C22×C6711yescharacteristic
377C2×#95C22×C6712yes—
378C2×#95C22×C6712yes—
379C2×#95C22×C6711yescharacteristic
380C2×#95C22×C6711yescharacteristic
381C2×#95C22×C6712yes—
382C2×#95C22×C6712yes—
383C2×#95C22×C6712yes—
384C2×#95C22×C6711yescharacteristic
385C2×#95C22×C6712yes—
386C2×#95C22×C6712yes—
387C2×#95C22×C6712yes—
388C25C23×C6711yescharacteristic
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—22no—
394C23×C4—22no—
395C23×C4—22no—
396C23×C4—22no—
397C23×C4—24no—
398C23×C4—24no—
399C23×C4—22no—
400C23×C4—22no—
401C23×C4—22no—
402C23×C4—22no—
403C23×C4—42no—
404C23×C4—42no—
405C23×C4D4×C6714yes—
406C23×C4C23×C6711yescharacteristic
407C23×C4C23×C6711yescharacteristic
408C23×C4C23×C6711yescharacteristic
409C23×C4C23×C6712yes—
410C23×C4C23×C6711yescharacteristic
411C23×C4C23×C6712yes—
412C23×C4C23×C6711yescharacteristic
413C23×C4C23×C6711yescharacteristic
414C23×C4C23×C6712yes—
415C23×C4C23×C6712yes—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—24no—
418C2×C4⋊C4—42no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—22no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4D4×C6714yes—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—24no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95—24no—
443#95—42no—
444#95—42no—
445#95—42no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24C2×D4×C6712yes—
451C24C2×D4×C6712yes—
452C24#41×C6712yes—
453C24#41×C6712yes—
454C24#41×C6712yes—
455C24#41×C6712yes—
456C24#41×C6712yes—
457C24C24×C6711yescharacteristic
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—48no—
521C22×C4—48no—
522C22×C4—48no—
523C22×C4—48no—
524C22×C4—44no—
525C22×C4—44no—
526C22×C4—44no—
527C22×C4—44no—
528C22×C4—44no—
529C22×C4—44no—
530C22×C4—44no—
531C22×C4—44no—
532C22×C4C2×D4×C6712yes—
533C22×C4C2×D4×C6712yes—
534C22×C4C2×D4×C6714yes—
535C22×C4#41×C6714yes—
536C22×C4#41×C6714yes—
537C4⋊C4—48no—
538C22⋊C4—28no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23#121×C6714yes—
559C23#123×C6714yes—
560C23#123×C6714yes—
561C23#123×C6714yes—
562C23#124×C6714yes—
563C23#126×C6714yes—
564C23#126×C6714yes—
565C23C22×D4×C6711yescharacteristic
566C23C22×D4×C6711yescharacteristic
567C23C2×#41×C6711yescharacteristic
568C23C2×#41×C6711yescharacteristic
569C23C2×#41×C6711yescharacteristic
570C23C2×#41×C6711yescharacteristic
571C23C2×#41×C6711yescharacteristic
572C23C2×#41×C6712yes—
573C23C2×#41×C6712yes—
574C23H(5,𝔽2)×C6712yes—
575C232−1+4×C6712yes—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—416no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22#397×C6718yes—
652C22C2×#121×C6711yescharacteristic
653C22C2×#123×C6711yescharacteristic
654C22C2×#123×C6711yescharacteristic
655C22C2×#123×C6711yescharacteristic
656C22#525×C6711yescharacteristic
657C22#525×C6711yescharacteristic
658C22C2×#124×C6711yescharacteristic
659C22C2×#126×C6711yescharacteristic
660C22C2×#126×C6711yescharacteristic
661C22#529×C6712yes—
662C22#529×C6712yes—
663C22#535×C6711yescharacteristic
664C22#536×C6711yescharacteristic
665C22#539×C6712yes—
666C22#539×C6712yes—
667C22#546×C6712yes—
668C22#547×C6712yes—
669C22#551×C6712yes—
670C22#551×C6712yes—
671C22#551×C6712yes—
672C22#552×C6712yes—
673C22#552×C6712yes—
674C22#552×C6712yes—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—82no—
685C2—28no—
686C2C2×#397×C6711yescharacteristic
687C2C67×#241111yescharacteristic
688C2C67×#241311yescharacteristic
689C2C67×#241912yes—
690C2C67×#248211yescharacteristic
691C2C67×#248411yescharacteristic
692C2C67×#248812yes—
693C2C67×#249312yes—
694C2C67×#249312yes—
695C2C67×#262311yescharacteristic
696C2C67×#285311yescharacteristic
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
5. C67×#1445C2semidirect
6. C67×#1443C2semidirect
7. C67×#1443C2semidirect
8. C67×#1442C2semidirect
336. #14919C67direct (N × H)
353. #1445C2×C67semidirect
354. #1443C2×C67semidirect
355. #1443C2×C67semidirect
356. #1442C2×C67semidirect
360. C67#14919direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#385×C672
C2×#381×C672
C22×#95×C672
C67×#14452
C67×#14432
C67×#14422
C67×C22⋊#952
#1491967