Finite Group Databasebeta

← C67×#14882 #30694133

Subgroups — C67×#14882 #30694133

Order 17152.

1742 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×C4×Q8×C67C211yesmaximal, characteristic, complemented
2C2×#397×C67C211yesmaximal, characteristic
3C2×#393×C67C211yesmaximal, characteristic
4C2×#388×C67C211yesmaximal, characteristic
5C67×#1446C211yesmaximal, characteristic, complemented
6C67×#1445C211yesmaximal, characteristic, complemented
7C67×#1442C212yesmaximal, complemented
8C67×#1439C212yesmaximal
9C67×#1437C212yesmaximal
10C67×#1436C212yesmaximal, complemented
11C67×#1434C211yesmaximal, characteristic, complemented
12C23×Q8×C67C2211yescharacteristic
13C2×C4×Q8×C67—24no—
14C2×C4×Q8×C67—24no—
15C2×C4×Q8×C67—24no—
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C4⋊C4×C67C2212yes—
18C22×C4⋊C4×C67C2211yescharacteristic
19C22×C22⋊C4×C67C2211yescharacteristic
20C22×C22⋊C4×C67C2211yescharacteristic
21C22×C22⋊C4×C67C2211yescharacteristic
22C22×C22⋊C4×C67C2212yes—
23C22×C22⋊C4×C67C2212yes—
24C22×C42×C67C2212yes—
25C22×C42×C67C2211yescharacteristic
26#397×C67—24no—
27#393×C67—24no—
28#388×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
42C22×Q8×C67—22no—
43C22×Q8×C67—22no—
44C22×Q8×C67—24no—
45C22×Q8×C67—24no—
46C22×Q8×C67D414yes—
47C23×C4×C67C2311yescharacteristic
48C23×C4×C67C2311yescharacteristic
49C23×C4×C67C2312yes—
50C23×C4×C67C2312yes—
51C23×C4×C67C2311yescharacteristic
52C23×C4×C67C2312yes—
53C23×C4×C67C2311yescharacteristic
54C23×C4×C67C2311yescharacteristic
55C23×C4×C67C2311yescharacteristic
56C23×C4×C67C2312yes—
57C4×Q8×C67—232no—
58C2×C4⋊C4×C67—24no—
59C2×C4⋊C4×C67—24no—
60C2×C4⋊C4×C67—24no—
61C2×C4⋊C4×C67—22no—
62C2×C4⋊C4×C67—22no—
63C2×C4⋊C4×C67—22no—
64C2×C4⋊C4×C67—24no—
65C2×C4⋊C4×C67D414yes—
66C2×C22⋊C4×C67—22no—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—22no—
69C2×C22⋊C4×C67—24no—
70C2×C22⋊C4×C67—22no—
71C2×C22⋊C4×C67—22no—
72C2×C22⋊C4×C67—22no—
73C2×C22⋊C4×C67—24no—
74C2×C22⋊C4×C67—24no—
75C2×C22⋊C4×C67—22no—
76C2×C22⋊C4×C67—24no—
77C2×C22⋊C4×C67—24no—
78C2×C22⋊C4×C67—24no—
79C2×C22⋊C4×C67—42no—
80C2×C42×C67—24no—
81C2×C42×C67—24no—
82C2×C42×C67—24no—
83C2×C42×C67—22no—
84C2×C42×C67—22no—
85C2×C42×C67D414yes—
86#95×C67—22no—
87#95×C67—22no—
88#95×C67—22no—
89#95×C67—24no—
90#95×C67—24no—
91#95×C67—22no—
92#95×C67—24no—
93#95×C67—24no—
94#95×C67—24no—
95#95×C67—24no—
96#95×C67—42no—
97#95×C67—42no—
98C24×C67—21no—
99C24×C67—21no—
100C24×C67—21no—
101C24×C67—21no—
102C24×C67—22no—
103C24×C67—22no—
104C24×C67—22no—
105C24×C67—21no—
106C24×C67—21no—
107C24×C67—22no—
108C24×C67#4112yes—
109C24×C67C2411yescharacteristic
110C2×Q8×C67—28no—
111C2×Q8×C67—28no—
112C2×Q8×C67—216no—
113C2×Q8×C67—28no—
114C2×Q8×C67—216no—
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—22no—
123C22×C4×C67—22no—
124C22×C4×C67—22no—
125C22×C4×C67—21no—
126C22×C4×C67—22no—
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—21no—
138C22×C4×C67—21no—
139C22×C4×C67—22no—
140C22×C4×C67—21no—
141C22×C4×C67—21no—
142C22×C4×C67—21no—
143C22×C4×C67—22no—
144C22×C4×C67—21no—
145C22×C4×C67—22no—
146C22×C4×C67—22no—
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—22no—
157C22×C4×C67—22no—
158C22×C4×C67—21no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—21no—
162C22×C4×C67—22no—
163C22×C4×C67—22no—
164C22×C4×C67C2×D414yes—
165C22×C4×C67C2×D412yes—
166C22×C4×C67C2×D412yes—
167C22×C4×C67C2×D412yes—
168C22×C4×C67C2×D412yes—
169C22×C4×C67C2×D414yes—
170C22×C4×C67C2×D412yes—
171C22×C4×C67#4112yes—
172C22×C4×C67#4114yes—
173C22×C4×C67#4114yes—
174C22×C4×C67#4114yes—
175C4⋊C4×C67—216no—
176C4⋊C4×C67—28no—
177C4⋊C4×C67—44no—
178C22⋊C4×C67—28no—
179C22⋊C4×C67—48no—
180C22⋊C4×C67—44no—
181C22⋊C4×C67—48no—
182C22⋊C4×C67—44no—
183C42×C67—216no—
184C42×C67—28no—
185C23×C67—22no—
186C23×C67—24no—
187C23×C67—22no—
188C23×C67—22no—
189C23×C67—24no—
190C23×C67—41no—
191C23×C67—42no—
192C23×C67—41no—
193C23×C67—41no—
194C23×C67—42no—
195C23×C67—41no—
196C23×C67—42no—
197C23×C67—41no—
198C23×C67—41no—
199C23×C67—42no—
200C23×C67—42no—
201C23×C67—41no—
202C23×C67—42no—
203C23×C67—42no—
204C23×C67—41no—
205C23×C67—42no—
206C23×C67—42no—
207C23×C67—42no—
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
219Q8×C67—232no—
220C2×C4×C67—22no—
221C2×C4×C67—24no—
222C2×C4×C67—24no—
223C2×C4×C67—24no—
224C2×C4×C67—22no—
225C2×C4×C67—22no—
226C2×C4×C67—24no—
227C2×C4×C67—24no—
228C2×C4×C67—24no—
229C2×C4×C67—24no—
230C2×C4×C67—24no—
231C2×C4×C67—22no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—22no—
236C2×C4×C67—22no—
237C2×C4×C67—22no—
238C2×C4×C67—24no—
239C2×C4×C67—22no—
240C2×C4×C67—24no—
241C2×C4×C67—24no—
242C2×C4×C67—41no—
243C2×C4×C67—42no—
244C2×C4×C67—41no—
245C2×C4×C67—42no—
246C2×C4×C67—42no—
247C2×C4×C67—41no—
248C2×C4×C67—41no—
249C2×C4×C67—41no—
250C2×C4×C67—42no—
251C2×C4×C67—41no—
252C2×C4×C67—42no—
253C2×C4×C67—41no—
254C2×C4×C67—42no—
255C2×C4×C67—41no—
256C2×C4×C67—42no—
257C2×C4×C67—41no—
258C2×C4×C67—41no—
259C2×C4×C67—42no—
260C2×C4×C67—42no—
261C2×C4×C67—41no—
262C2×C4×C67—41no—
263C2×C4×C67—42no—
264C2×C4×C67—42no—
265C2×C4×C67—42no—
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—42no—
274C2×C4×C67—42no—
275C2×C4×C67—42no—
276C2×C4×C67—42no—
277C2×C4×C67#12014yes—
278C2×C4×C67#12318yes—
279C2×C4×C67#12414yes—
280C22×C67—42no—
281C22×C67—42no—
282C22×C67—42no—
283C22×C67—44no—
284C22×C67—44no—
285C22×C67—81no—
286C22×C67—81no—
287C22×C67—81no—
288C22×C67—81no—
289C22×C67—82no—
290C22×C67—82no—
291C22×C67C2×#12011yescharacteristic
292C22×C67C2×#12312yes—
293C22×C67#52511yescharacteristic
294C22×C67#52511yescharacteristic
295C22×C67#52512yes—
296C22×C67#52511yescharacteristic
297C22×C67C2×#12411yescharacteristic
298C22×C67#52912yes—
299C22×C67#52912yes—
300C22×C67#53611yescharacteristic
301C22×C67#53711yescharacteristic
302C22×C67#54112yes—
303C22×C67#54212yes—
304C22×C67#54611yescharacteristic
305C22×C67#54711yescharacteristic
306C22×C67#54811yescharacteristic
307C22×C67#54812yes—
308C22×C67#54812yes—
309C22×C67#55011yescharacteristic
310C22×C67#55212yes—
311C22×C67#55612yes—
312C22×C67#55612yes—
313C22×C67#55912yes—
314C4×C67—24no—
315C4×C67—28no—
316C4×C67—24no—
317C4×C67—42no—
318C4×C67—42no—
319C4×C67—44no—
320C4×C67—81no—
321C4×C67—81no—
322C4×C67—82no—
323C4×C67—82no—
324#14882C6711yesmaximal, characteristic, direct product factor, Sylow
325C2×C67—82no—
326C2×C67#239811yescharacteristic
327C2×C67#240711yescharacteristic
328C2×C67#242112yes—
329C2×C67#243611yescharacteristic
330C2×C67#244812yes—
331C2×C67#245812yes—
332C2×C67#246912yes—
333C2×C67#259211yescharacteristic
334C2×C67#259911yescharacteristic
335C2×C67#263711yescharacteristic
336C2×C67#284911yescharacteristic
337C22×C4×Q8C2×C6711yescharacteristic, complemented
338C2×#397C2×C6711yescharacteristic
339C2×#393C2×C6711yescharacteristic
340C2×#388C2×C6711yescharacteristic
341#1446C2×C6711yescharacteristic, complemented
342#1445C2×C6711yescharacteristic, complemented
343#1442C2×C6712yescomplemented
344#1439C2×C6712yes—
345#1437C2×C6712yes—
346#1436C2×C6712yescomplemented
347#1434C2×C6711yescharacteristic, complemented
348C67#1488211yescharacteristic, direct product factor, Sylow
349C23×Q8C22×C6711yescharacteristic
350C2×C4×Q8—24no—
351C2×C4×Q8—24no—
352C2×C4×Q8—24no—
353C22×C4⋊C4C22×C6712yes—
354C22×C4⋊C4C22×C6711yescharacteristic
355C22×C4⋊C4C22×C6711yescharacteristic
356C22×C22⋊C4C22×C6711yescharacteristic
357C22×C22⋊C4C22×C6712yes—
358C22×C22⋊C4C22×C6711yescharacteristic
359C22×C22⋊C4C22×C6711yescharacteristic
360C22×C22⋊C4C22×C6712yes—
361C22×C42C22×C6712yes—
362C22×C42C22×C6711yescharacteristic
363#397—24no—
364#393—24no—
365#388—24no—
366C2×#95C22×C6711yescharacteristic
367C2×#95C22×C6712yes—
368C2×#95C22×C6711yescharacteristic
369C2×#95C22×C6712yes—
370C2×#95C22×C6712yes—
371C2×#95C22×C6712yes—
372C2×#95C22×C6712yes—
373C2×#95C22×C6712yes—
374C2×#95C22×C6711yescharacteristic
375C2×#95C22×C6711yescharacteristic
376C2×#95C22×C6712yes—
377C2×#95C22×C6712yes—
378C25C23×C6711yescharacteristic
379C22×Q8—22no—
380C22×Q8—22no—
381C22×Q8—24no—
382C22×Q8—24no—
383C22×Q8D4×C6714yes—
384C23×C4C23×C6711yescharacteristic
385C23×C4C23×C6712yes—
386C23×C4C23×C6712yes—
387C23×C4C23×C6711yescharacteristic
388C23×C4C23×C6712yes—
389C23×C4C23×C6711yescharacteristic
390C23×C4C23×C6711yescharacteristic
391C23×C4C23×C6711yescharacteristic
392C23×C4C23×C6712yes—
393C23×C4C23×C6711yescharacteristic
394C4×Q8—232no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—24no—
402C2×C4⋊C4D4×C6714yes—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—24no—
415C2×C22⋊C4—24no—
416C2×C22⋊C4—42no—
417C2×C42—24no—
418C2×C42—24no—
419C2×C42—24no—
420C2×C42—22no—
421C2×C42—22no—
422C2×C42D4×C6714yes—
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—42no—
434#95—42no—
435C24—21no—
436C24—21no—
437C24—21no—
438C24—21no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—21no—
443C24—21no—
444C24—22no—
445C24#41×C6712yes—
446C24C24×C6711yescharacteristic
447C2×Q8—28no—
448C2×Q8—28no—
449C2×Q8—216no—
450C2×Q8—28no—
451C2×Q8—216no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
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—22no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4C2×D4×C6714yes—
502C22×C4C2×D4×C6712yes—
503C22×C4C2×D4×C6712yes—
504C22×C4C2×D4×C6712yes—
505C22×C4C2×D4×C6712yes—
506C22×C4C2×D4×C6714yes—
507C22×C4C2×D4×C6712yes—
508C22×C4#41×C6712yes—
509C22×C4#41×C6714yes—
510C22×C4#41×C6714yes—
511C22×C4#41×C6714yes—
512C4⋊C4—216no—
513C4⋊C4—28no—
514C4⋊C4—44no—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C42—216no—
521C42—28no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—41no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
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
556Q8—232no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
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—42no—
614C2×C4#120×C6714yes—
615C2×C4#123×C6718yes—
616C2×C4#124×C6714yes—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22C2×#120×C6711yescharacteristic
629C22C2×#123×C6712yes—
630C22#525×C6711yescharacteristic
631C22#525×C6711yescharacteristic
632C22#525×C6712yes—
633C22#525×C6711yescharacteristic
634C22C2×#124×C6711yescharacteristic
635C22#529×C6712yes—
636C22#529×C6712yes—
637C22#536×C6711yescharacteristic
638C22#537×C6711yescharacteristic
639C22#541×C6712yes—
640C22#542×C6712yes—
641C22#546×C6711yescharacteristic
642C22#547×C6711yescharacteristic
643C22#548×C6711yescharacteristic
644C22#548×C6712yes—
645C22#548×C6712yes—
646C22#550×C6711yescharacteristic
647C22#552×C6712yes—
648C22#556×C6712yes—
649C22#556×C6712yes—
650C22#559×C6712yes—
651C4—24no—
652C4—28no—
653C4—24no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4—82no—
661C2—82no—
662C2C67×#239811yescharacteristic
663C2C67×#240711yescharacteristic
664C2C67×#242112yes—
665C2C67×#243611yescharacteristic
666C2C67×#244812yes—
667C2C67×#245812yes—
668C2C67×#246912yes—
669C2C67×#259211yescharacteristic
670C2C67×#259911yescharacteristic
671C2C67×#263711yescharacteristic
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
1. C22×C4×Q8×C67C2semidirect
5. C67×#1446C2semidirect
6. C67×#1445C2semidirect
7. C67×#1442C2semidirect
10. C67×#1436C2semidirect
11. C67×#1434C2semidirect
324. #14882C67direct (N × H)
337. C22×C4×Q8C2×C67semidirect
341. #1446C2×C67semidirect
342. #1445C2×C67semidirect
343. #1442C2×C67semidirect
346. #1436C2×C67semidirect
347. #1434C2×C67semidirect
348. C67#14882direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C672
C2×#397×C672
C2×#393×C672
C2×#388×C672
C67×#14462
C67×#14452
C67×#14422
C67×#14392
C67×#14372
C67×#14362
C67×#14342
#1488267