Finite Group Databasebeta

← C67×#14906 #30694157

Subgroups — C67×#14906 #30694157

Order 17152.

1786 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C67C211yesmaximal, characteristic
2C2×#400×C67C211yesmaximal, characteristic, complemented
3C2×#398×C67C211yesmaximal, characteristic, complemented
4C2×#385×C67C211yesmaximal, characteristic
5C67×#1445C211yesmaximal, characteristic, complemented
6C67×#1445C211yesmaximal, characteristic, complemented
7C67×#1440C212yesmaximal
8C67×#1439C212yesmaximal
9C67×#1436C212yesmaximal, complemented
10C67×#1436C212yesmaximal, complemented
11C67×C22⋊#95C211yesmaximal, characteristic
12C24×C4×C67C2211yescharacteristic
13C2×#117×C67—24no—
14C2×#117×C67—24no—
15C2×#117×C67—24no—
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C4⋊C4×C67C2211yescharacteristic
18C22×C4⋊C4×C67C2212yes—
19C22×C4⋊C4×C67C2211yescharacteristic
20C22×C4⋊C4×C67C2211yescharacteristic
21C22×C22⋊C4×C67C2211yescharacteristic
22C22×C22⋊C4×C67C2211yescharacteristic
23C22×C22⋊C4×C67C2212yes—
24C22×C22⋊C4×C67C2212yes—
25C22×C42×C67C2212yes—
26#400×C67—24no—
27#398×C67—24no—
28#385×C67—24no—
29C2×#95×C67C2211yescharacteristic
30C2×#95×C67C2211yescharacteristic
31C2×#95×C67C2211yescharacteristic
32C2×#95×C67C2212yes—
33C2×#95×C67C2211yescharacteristic
34C2×#95×C67C2212yes—
35C2×#95×C67C2212yes—
36C2×#95×C67C2212yes—
37C2×#95×C67C2212yes—
38C2×#95×C67C2212yes—
39C2×#95×C67C2212yes—
40C2×#95×C67C2212yes—
41C25×C67C2311yescharacteristic
42C23×C4×C67—24no—
43C23×C4×C67—22no—
44C23×C4×C67—22no—
45C23×C4×C67—24no—
46C23×C4×C67D414yes—
47C23×C4×C67C2311yescharacteristic
48C23×C4×C67C2311yescharacteristic
49C23×C4×C67C2312yes—
50C23×C4×C67C2312yes—
51C23×C4×C67C2312yes—
52C23×C4×C67C2311yescharacteristic
53C23×C4×C67C2311yescharacteristic
54C23×C4×C67C2312yes—
55C23×C4×C67C2311yescharacteristic
56C23×C4×C67C2311yescharacteristic
57#117×C67—232no—
58C2×C4⋊C4×C67—22no—
59C2×C4⋊C4×C67—22no—
60C2×C4⋊C4×C67—22no—
61C2×C4⋊C4×C67—22no—
62C2×C4⋊C4×C67—22no—
63C2×C4⋊C4×C67—22no—
64C2×C4⋊C4×C67—24no—
65C2×C4⋊C4×C67—22no—
66C2×C4⋊C4×C67—22no—
67C2×C4⋊C4×C67—24no—
68C2×C4⋊C4×C67—24no—
69C2×C4⋊C4×C67—42no—
70C2×C4⋊C4×C67D414yes—
71C2×C4⋊C4×C67D414yes—
72C2×C22⋊C4×C67—24no—
73C2×C22⋊C4×C67—22no—
74C2×C22⋊C4×C67—24no—
75C2×C22⋊C4×C67—24no—
76C2×C22⋊C4×C67—22no—
77C2×C22⋊C4×C67—22no—
78C2×C22⋊C4×C67—22no—
79C2×C22⋊C4×C67—24no—
80C2×C22⋊C4×C67—24no—
81C2×C22⋊C4×C67—24no—
82C2×C22⋊C4×C67—42no—
83C2×C42×C67—24no—
84C2×C42×C67—24no—
85C2×C42×C67—24no—
86#95×C67—22no—
87#95×C67—22no—
88#95×C67—22no—
89#95×C67—24no—
90#95×C67—24no—
91#95×C67—24no—
92#95×C67—22no—
93#95×C67—24no—
94#95×C67—24no—
95#95×C67—24no—
96#95×C67—24no—
97#95×C67—42no—
98C24×C67—21no—
99C24×C67—21no—
100C24×C67—21no—
101C24×C67—21no—
102C24×C67—22no—
103C24×C67—22no—
104C24×C67—21no—
105C24×C67—22no—
106C24×C67—21no—
107C24×C67C2×D412yes—
108C24×C67#4114yes—
109C24×C67C2411yescharacteristic
110C22×C4×C67—216no—
111C22×C4×C67—216no—
112C22×C4×C67—28no—
113C22×C4×C67—21no—
114C22×C4×C67—21no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
117C22×C4×C67—22no—
118C22×C4×C67—28no—
119C22×C4×C67—28no—
120C22×C4×C67—22no—
121C22×C4×C67—22no—
122C22×C4×C67—22no—
123C22×C4×C67—22no—
124C22×C4×C67—21no—
125C22×C4×C67—21no—
126C22×C4×C67—21no—
127C22×C4×C67—21no—
128C22×C4×C67—22no—
129C22×C4×C67—21no—
130C22×C4×C67—21no—
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—21no—
142C22×C4×C67—22no—
143C22×C4×C67—22no—
144C22×C4×C67—22no—
145C22×C4×C67—22no—
146C22×C4×C67—21no—
147C22×C4×C67—22no—
148C22×C4×C67—21no—
149C22×C4×C67—21no—
150C22×C4×C67—21no—
151C22×C4×C67—21no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—22no—
156C22×C4×C67—22no—
157C22×C4×C67—22no—
158C22×C4×C67—22no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—22no—
162C22×C4×C67—21no—
163C22×C4×C67—21no—
164C22×C4×C67—22no—
165C22×C4×C67C2×D414yes—
166C22×C4×C67C2×D414yes—
167C22×C4×C67C2×D412yes—
168C22×C4×C67C2×D412yes—
169C22×C4×C67C2×D412yes—
170C22×C4×C67C2×D412yes—
171C22×C4×C67#4112yes—
172C22×C4×C67#4112yes—
173C22×C4×C67#4114yes—
174C22×C4×C67#4114yes—
175C4⋊C4×C67—28no—
176C4⋊C4×C67—28no—
177C4⋊C4×C67—48no—
178C4⋊C4×C67—44no—
179C4⋊C4×C67—44no—
180C22⋊C4×C67—28no—
181C22⋊C4×C67—216no—
182C22⋊C4×C67—48no—
183C22⋊C4×C67—44no—
184C42×C67—216no—
185C23×C67—22no—
186C23×C67—22no—
187C23×C67—22no—
188C23×C67—24no—
189C23×C67—24no—
190C23×C67—22no—
191C23×C67—22no—
192C23×C67—22no—
193C23×C67—24no—
194C23×C67—24no—
195C23×C67—24no—
196C23×C67—24no—
197C23×C67—42no—
198C23×C67—42no—
199C23×C67—42no—
200C23×C67—42no—
201C23×C67—41no—
202C23×C67—41no—
203C23×C67—42no—
204C23×C67—41no—
205C23×C67—42no—
206C23×C67—41no—
207C23×C67#12314yes—
208C23×C67C22×D411yescharacteristic
209C23×C67C22×D411yescharacteristic
210C23×C67C22×D411yescharacteristic
211C23×C67C2×#4112yes—
212C23×C67C2×#4112yes—
213C23×C67C2×#4111yescharacteristic
214C23×C67C2×#4111yescharacteristic
215C23×C67C2×#4112yes—
216C23×C67H(5,𝔽2)11yescharacteristic
217C23×C672−1+412yes—
218C23×C672−1+411yescharacteristic
219C2×C4×C67—232no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—22no—
224C2×C4×C67—24no—
225C2×C4×C67—24no—
226C2×C4×C67—24no—
227C2×C4×C67—24no—
228C2×C4×C67—24no—
229C2×C4×C67—24no—
230C2×C4×C67—22no—
231C2×C4×C67—22no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—22no—
235C2×C4×C67—24no—
236C2×C4×C67—24no—
237C2×C4×C67—24no—
238C2×C4×C67—24no—
239C2×C4×C67—22no—
240C2×C4×C67—24no—
241C2×C4×C67—24no—
242C2×C4×C67—41no—
243C2×C4×C67—41no—
244C2×C4×C67—42no—
245C2×C4×C67—42no—
246C2×C4×C67—42no—
247C2×C4×C67—41no—
248C2×C4×C67—42no—
249C2×C4×C67—42no—
250C2×C4×C67—42no—
251C2×C4×C67—41no—
252C2×C4×C67—42no—
253C2×C4×C67—42no—
254C2×C4×C67—41no—
255C2×C4×C67—42no—
256C2×C4×C67—42no—
257C2×C4×C67—42no—
258C2×C4×C67—42no—
259C2×C4×C67—41no—
260C2×C4×C67—42no—
261C2×C4×C67—41no—
262C2×C4×C67—41no—
263C2×C4×C67—41no—
264C2×C4×C67—41no—
265C2×C4×C67—42no—
266C2×C4×C67—42no—
267C2×C4×C67—42no—
268C2×C4×C67—42no—
269C2×C4×C67—41no—
270C2×C4×C67—41no—
271C2×C4×C67—42no—
272C2×C4×C67—42no—
273C2×C4×C67—42no—
274C2×C4×C67—42no—
275C2×C4×C67—42no—
276C2×C4×C67—42no—
277C2×C4×C67—42no—
278C2×C4×C67#12014yes—
279C2×C4×C67#12314yes—
280C22×C67—24no—
281C22×C67—24no—
282C22×C67—24no—
283C22×C67—42no—
284C22×C67—42no—
285C22×C67—42no—
286C22×C67—42no—
287C22×C67—44no—
288C22×C67—44no—
289C22×C67—44no—
290C22×C67—44no—
291C22×C67C2×#12011yescharacteristic
292C22×C67C2×#12311yescharacteristic
293C22×C67C2×#12311yescharacteristic
294C22×C67#52512yes—
295C22×C67#52512yes—
296C22×C67#52511yescharacteristic
297C22×C67#52912yes—
298C22×C67#52912yes—
299C22×C67#53211yescharacteristic
300C22×C67#53611yescharacteristic
301C22×C67#53711yescharacteristic
302C22×C67#54112yes—
303C22×C67#54212yes—
304C22×C67#54611yescharacteristic
305C22×C67#54711yescharacteristic
306C22×C67#54712yes—
307C22×C67#54811yescharacteristic
308C22×C67#54812yes—
309C22×C67#55011yescharacteristic
310C22×C67#55112yes—
311C22×C67#55212yes—
312C22×C67#55212yes—
313C22×C67#55212yes—
314C4×C67—24no—
315C4×C67—24no—
316C4×C67—42no—
317C4×C67—44no—
318C4×C67—42no—
319C4×C67—44no—
320C4×C67—44no—
321C4×C67—81no—
322C4×C67—82no—
323C4×C67—81no—
324#14906C6711yesmaximal, characteristic, direct product factor, Sylow
325C2×C67—44no—
326C2×C67#239511yescharacteristic
327C2×C67#240711yescharacteristic
328C2×C67#243211yescharacteristic
329C2×C67#247512yes—
330C2×C67#247512yes—
331C2×C67#247712yes—
332C2×C67#247912yes—
333C2×C67#248411yescharacteristic
334C2×C67#249811yescharacteristic
335C2×C67#262511yescharacteristic
336C2×C67#284911yescharacteristic
337C22×#117C2×C6711yescharacteristic
338C2×#400C2×C6711yescharacteristic, complemented
339C2×#398C2×C6711yescharacteristic, complemented
340C2×#385C2×C6711yescharacteristic
341#1445C2×C6711yescharacteristic, complemented
342#1445C2×C6711yescharacteristic, complemented
343#1440C2×C6712yes—
344#1439C2×C6712yes—
345#1436C2×C6712yescomplemented
346#1436C2×C6712yescomplemented
347C22⋊#95C2×C6711yescharacteristic
348C67#1490611yescharacteristic, direct product factor, Sylow
349C24×C4C22×C6711yescharacteristic
350C2×#117—24no—
351C2×#117—24no—
352C2×#117—24no—
353C22×C4⋊C4C22×C6711yescharacteristic
354C22×C4⋊C4C22×C6711yescharacteristic
355C22×C4⋊C4C22×C6712yes—
356C22×C4⋊C4C22×C6711yescharacteristic
357C22×C4⋊C4C22×C6711yescharacteristic
358C22×C22⋊C4C22×C6712yes—
359C22×C22⋊C4C22×C6711yescharacteristic
360C22×C22⋊C4C22×C6711yescharacteristic
361C22×C22⋊C4C22×C6712yes—
362C22×C42C22×C6712yes—
363#400—24no—
364#398—24no—
365#385—24no—
366C2×#95C22×C6711yescharacteristic
367C2×#95C22×C6712yes—
368C2×#95C22×C6711yescharacteristic
369C2×#95C22×C6712yes—
370C2×#95C22×C6712yes—
371C2×#95C22×C6712yes—
372C2×#95C22×C6711yescharacteristic
373C2×#95C22×C6712yes—
374C2×#95C22×C6712yes—
375C2×#95C22×C6712yes—
376C2×#95C22×C6711yescharacteristic
377C2×#95C22×C6712yes—
378C25C23×C6711yescharacteristic
379C23×C4—24no—
380C23×C4—22no—
381C23×C4—22no—
382C23×C4—24no—
383C23×C4D4×C6714yes—
384C23×C4C23×C6711yescharacteristic
385C23×C4C23×C6712yes—
386C23×C4C23×C6711yescharacteristic
387C23×C4C23×C6712yes—
388C23×C4C23×C6712yes—
389C23×C4C23×C6711yescharacteristic
390C23×C4C23×C6712yes—
391C23×C4C23×C6711yescharacteristic
392C23×C4C23×C6711yescharacteristic
393C23×C4C23×C6711yescharacteristic
394#117—232no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—24no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—42no—
407C2×C4⋊C4D4×C6714yes—
408C2×C4⋊C4D4×C6714yes—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—24no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—24no—
419C2×C22⋊C4—42no—
420C2×C42—24no—
421C2×C42—24no—
422C2×C42—24no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95—22no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—42no—
435C24—21no—
436C24—21no—
437C24—21no—
438C24—21no—
439C24—22no—
440C24—22no—
441C24—21no—
442C24—22no—
443C24—21no—
444C24C2×D4×C6712yes—
445C24#41×C6714yes—
446C24C24×C6711yescharacteristic
447C22×C4—216no—
448C22×C4—216no—
449C22×C4—28no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—22no—
502C22×C4C2×D4×C6714yes—
503C22×C4C2×D4×C6714yes—
504C22×C4C2×D4×C6712yes—
505C22×C4C2×D4×C6712yes—
506C22×C4C2×D4×C6712yes—
507C22×C4C2×D4×C6712yes—
508C22×C4#41×C6712yes—
509C22×C4#41×C6714yes—
510C22×C4#41×C6712yes—
511C22×C4#41×C6714yes—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C22⋊C4—216no—
518C22⋊C4—28no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C42—216no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23#123×C6714yes—
545C23C22×D4×C6711yescharacteristic
546C23C22×D4×C6711yescharacteristic
547C23C22×D4×C6711yescharacteristic
548C23C2×#41×C6712yes—
549C23C2×#41×C6712yes—
550C23C2×#41×C6711yescharacteristic
551C23C2×#41×C6711yescharacteristic
552C23C2×#41×C6712yes—
553C23H(5,𝔽2)×C6711yescharacteristic
554C232−1+4×C6712yes—
555C232−1+4×C6711yescharacteristic
556C2×C4—232no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4#120×C6714yes—
616C2×C4#123×C6714yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22C2×#120×C6711yescharacteristic
629C22C2×#123×C6711yescharacteristic
630C22C2×#123×C6711yescharacteristic
631C22#525×C6712yes—
632C22#525×C6712yes—
633C22#525×C6711yescharacteristic
634C22#529×C6712yes—
635C22#529×C6712yes—
636C22#532×C6711yescharacteristic
637C22#536×C6711yescharacteristic
638C22#537×C6711yescharacteristic
639C22#541×C6712yes—
640C22#542×C6712yes—
641C22#546×C6711yescharacteristic
642C22#547×C6711yescharacteristic
643C22#547×C6712yes—
644C22#548×C6711yescharacteristic
645C22#548×C6712yes—
646C22#550×C6711yescharacteristic
647C22#551×C6712yes—
648C22#552×C6712yes—
649C22#552×C6712yes—
650C22#552×C6712yes—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—81no—
659C4—81no—
660C4—82no—
661C2—44no—
662C2C67×#239511yescharacteristic
663C2C67×#240711yescharacteristic
664C2C67×#243211yescharacteristic
665C2C67×#247512yes—
666C2C67×#247512yes—
667C2C67×#247712yes—
668C2C67×#247912yes—
669C2C67×#248411yescharacteristic
670C2C67×#249811yescharacteristic
671C2C67×#262511yescharacteristic
672C2C67×#284911yescharacteristic
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×#400×C67C2semidirect
3. C2×#398×C67C2semidirect
5. C67×#1445C2semidirect
6. C67×#1445C2semidirect
9. C67×#1436C2semidirect
10. C67×#1436C2semidirect
324. #14906C67direct (N × H)
338. C2×#400C2×C67semidirect
339. C2×#398C2×C67semidirect
341. #1445C2×C67semidirect
342. #1445C2×C67semidirect
345. #1436C2×C67semidirect
346. #1436C2×C67semidirect
348. C67#14906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#117×C672
C2×#400×C672
C2×#398×C672
C2×#385×C672
C67×#14452
C67×#14402
C67×#14392
C67×#14362
C67×C22⋊#952
#1490667