Finite Group Databasebeta

← C67×#14933 #30694184

Subgroups — C67×#14933 #30694184

Order 17152.

1614 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125×C67C211yesmaximal, characteristic, complemented
2C2×#388×C67C212yesmaximal
3C2×#388×C67C211yesmaximal, characteristic
4C67×#1442C212yesmaximal, complemented
5C67×#1441C211yesmaximal, characteristic, complemented
6C67×#1441C211yesmaximal, characteristic, complemented
7C67×#1439C211yesmaximal, characteristic
8C67×#1439C212yesmaximal
9C67×#1439C211yesmaximal, characteristic
10C67×#1436C212yesmaximal, complemented
11C4×#95×C67C211yesmaximal, characteristic, complemented
12C2×#125×C67—28no—
13C2×#125×C67—24no—
14C22×C4⋊C4×C67C2211yescharacteristic
15C22×C4⋊C4×C67C2211yescharacteristic
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C4⋊C4×C67C2212yes—
18C22×C4⋊C4×C67C2212yes—
19C22×C22⋊C4×C67C2211yescharacteristic
20C22×C22⋊C4×C67C2211yescharacteristic
21C22×C22⋊C4×C67C2212yes—
22C22×C22⋊C4×C67C2211yescharacteristic
23C22×C22⋊C4×C67C2212yes—
24C22×C42×C67C2212yes—
25C22×C42×C67C2211yescharacteristic
26#388×C67—28no—
27#388×C67—24no—
28C2×#95×C67C2211yescharacteristic
29C2×#95×C67C2212yes—
30C2×#95×C67C2212yes—
31C2×#95×C67C2211yescharacteristic
32C2×#95×C67C2211yescharacteristic
33C2×#95×C67C2211yescharacteristic
34C2×#95×C67C2212yes—
35C2×#95×C67C2212yes—
36C2×#95×C67C2212yes—
37C2×#95×C67C2212yes—
38C2×#95×C67C2212yes—
39C25×C67C2311yescharacteristic
40C23×C4×C67C2311yescharacteristic
41C23×C4×C67C2311yescharacteristic
42C23×C4×C67C2311yescharacteristic
43C23×C4×C67C2312yes—
44C23×C4×C67C2312yes—
45C23×C4×C67C2312yes—
46C23×C4×C67C2312yes—
47C23×C4×C67C2311yescharacteristic
48C23×C4×C67C2311yescharacteristic
49C23×C4×C67C2311yescharacteristic
50#125×C67—232no—
51C2×C4⋊C4×C67—24no—
52C2×C4⋊C4×C67—24no—
53C2×C4⋊C4×C67—24no—
54C2×C4⋊C4×C67—24no—
55C2×C4⋊C4×C67—24no—
56C2×C4⋊C4×C67—22no—
57C2×C4⋊C4×C67—22no—
58C2×C4⋊C4×C67—24no—
59C2×C4⋊C4×C67—24no—
60C2×C4⋊C4×C67—22no—
61C2×C4⋊C4×C67—22no—
62C2×C4⋊C4×C67—22no—
63C2×C4⋊C4×C67D418yes—
64C2×C22⋊C4×C67—22no—
65C2×C22⋊C4×C67—24no—
66C2×C22⋊C4×C67—24no—
67C2×C22⋊C4×C67—22no—
68C2×C22⋊C4×C67—24no—
69C2×C22⋊C4×C67—22no—
70C2×C22⋊C4×C67—24no—
71C2×C22⋊C4×C67—22no—
72C2×C22⋊C4×C67—24no—
73C2×C22⋊C4×C67—22no—
74C2×C22⋊C4×C67—24no—
75C2×C22⋊C4×C67—22no—
76C2×C22⋊C4×C67—22no—
77C2×C22⋊C4×C67—42no—
78C2×C42×C67—24no—
79C2×C42×C67—24no—
80C2×C42×C67—24no—
81C2×C42×C67—42no—
82C2×C42×C67D414yes—
83#95×C67—24no—
84#95×C67—24no—
85#95×C67—22no—
86#95×C67—22no—
87#95×C67—24no—
88#95×C67—24no—
89#95×C67—22no—
90#95×C67—22no—
91#95×C67—24no—
92#95×C67—24no—
93#95×C67—42no—
94C24×C67—21no—
95C24×C67—22no—
96C24×C67—22no—
97C24×C67—21no—
98C24×C67—21no—
99C24×C67—22no—
100C24×C67—21no—
101C24×C67—21no—
102C24×C67—22no—
103C24×C67—21no—
104C24×C67#4112yes—
105C24×C67C2411yescharacteristic
106C22×C4×C67—22no—
107C22×C4×C67—22no—
108C22×C4×C67—21no—
109C22×C4×C67—22no—
110C22×C4×C67—21no—
111C22×C4×C67—22no—
112C22×C4×C67—22no—
113C22×C4×C67—22no—
114C22×C4×C67—21no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
117C22×C4×C67—22no—
118C22×C4×C67—22no—
119C22×C4×C67—22no—
120C22×C4×C67—21no—
121C22×C4×C67—22no—
122C22×C4×C67—22no—
123C22×C4×C67—21no—
124C22×C4×C67—22no—
125C22×C4×C67—21no—
126C22×C4×C67—22no—
127C22×C4×C67—21no—
128C22×C4×C67—21no—
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—21no—
141C22×C4×C67—22no—
142C22×C4×C67—22no—
143C22×C4×C67—21no—
144C22×C4×C67—22no—
145C22×C4×C67—21no—
146C22×C4×C67—22no—
147C22×C4×C67—21no—
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×C67C2×D412yes—
156C22×C4×C67C2×D414yes—
157C22×C4×C67C2×D414yes—
158C22×C4×C67C2×D414yes—
159C22×C4×C67C2×D412yes—
160C22×C4×C67C2×D412yes—
161C22×C4×C67#4114yes—
162C22×C4×C67#4114yes—
163C22×C4×C67#4112yes—
164C22×C4×C67#4112yes—
165C22×C4×C67#4112yes—
166C4⋊C4×C67—28no—
167C4⋊C4×C67—28no—
168C4⋊C4×C67—216no—
169C4⋊C4×C67—216no—
170C4⋊C4×C67—44no—
171C22⋊C4×C67—28no—
172C22⋊C4×C67—216no—
173C22⋊C4×C67—48no—
174C22⋊C4×C67—44no—
175C22⋊C4×C67—44no—
176C42×C67—28no—
177C42×C67—48no—
178C23×C67—22no—
179C23×C67—24no—
180C23×C67—22no—
181C23×C67—24no—
182C23×C67—22no—
183C23×C67—41no—
184C23×C67—41no—
185C23×C67—42no—
186C23×C67—42no—
187C23×C67—41no—
188C23×C67—41no—
189C23×C67—42no—
190C23×C67—41no—
191C23×C67—41no—
192C23×C67—42no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—42no—
196C23×C67—42no—
197C23×C67—42no—
198C23×C67—41no—
199C23×C67—42no—
200C23×C67—41no—
201C23×C67C22×D411yescharacteristic
202C23×C67C22×D412yes—
203C23×C67C2×#4111yescharacteristic
204C23×C67C2×#4112yes—
205C23×C67C2×#4111yescharacteristic
206C23×C67C2×#4111yescharacteristic
207C23×C67C2×#4111yescharacteristic
208C23×C67C2×#4112yes—
209C23×C67H(5,𝔽2)11yescharacteristic
210C23×C672−1+412yes—
211C23×C672−1+411yescharacteristic
212C2×C4×C67—22no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—22no—
216C2×C4×C67—24no—
217C2×C4×C67—24no—
218C2×C4×C67—24no—
219C2×C4×C67—24no—
220C2×C4×C67—22no—
221C2×C4×C67—24no—
222C2×C4×C67—24no—
223C2×C4×C67—22no—
224C2×C4×C67—24no—
225C2×C4×C67—22no—
226C2×C4×C67—24no—
227C2×C4×C67—24no—
228C2×C4×C67—24no—
229C2×C4×C67—22no—
230C2×C4×C67—24no—
231C2×C4×C67—24no—
232C2×C4×C67—24no—
233C2×C4×C67—24no—
234C2×C4×C67—24no—
235C2×C4×C67—22no—
236C2×C4×C67—24no—
237C2×C4×C67—22no—
238C2×C4×C67—42no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—41no—
242C2×C4×C67—42no—
243C2×C4×C67—42no—
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—42no—
252C2×C4×C67—42no—
253C2×C4×C67—42no—
254C2×C4×C67—42no—
255C2×C4×C67—41no—
256C2×C4×C67—42no—
257C2×C4×C67—41no—
258C2×C4×C67—42no—
259C2×C4×C67—42no—
260C2×C4×C67—41no—
261C2×C4×C67—41no—
262C2×C4×C67—41no—
263C2×C4×C67—42no—
264C2×C4×C67—42no—
265C2×C4×C67—41no—
266C2×C4×C67—41no—
267C2×C4×C67—41no—
268C2×C4×C67—42no—
269C2×C4×C67—42no—
270C2×C4×C67#12014yes—
271C2×C4×C67#12414yes—
272C22×C67—42no—
273C22×C67—42no—
274C22×C67—42no—
275C22×C67—44no—
276C22×C67—44no—
277C22×C67—81no—
278C22×C67—81no—
279C22×C67—81no—
280C22×C67—81no—
281C22×C67—82no—
282C22×C67—82no—
283C22×C67C2×#12011yescharacteristic
284C22×C67#52511yescharacteristic
285C22×C67#52512yes—
286C22×C67C2×#12411yescharacteristic
287C22×C67#52912yes—
288C22×C67#52912yes—
289C22×C67#53211yescharacteristic
290C22×C67#53211yescharacteristic
291C22×C67#53212yes—
292C22×C67#53611yescharacteristic
293C22×C67#53711yescharacteristic
294C22×C67#53912yes—
295C22×C67#54112yes—
296C22×C67#54612yes—
297C22×C67#54712yes—
298C22×C67#54812yes—
299C22×C67#54812yes—
300C22×C67#55211yescharacteristic
301C22×C67#55211yescharacteristic
302C22×C67#55212yes—
303C22×C67#55212yes—
304C22×C67#55311yescharacteristic
305C22×C67#55311yescharacteristic
306C4×C67—24no—
307C4×C67—24no—
308C4×C67—42no—
309C4×C67—42no—
310C4×C67—44no—
311C4×C67—44no—
312C4×C67—44no—
313C4×C67—44no—
314C4×C67—81no—
315C4×C67—81no—
316#14933C6711yesmaximal, characteristic, direct product factor, Sylow
317C2×C67—82no—
318C2×C67#238811yescharacteristic
319C2×C67#243011yescharacteristic
320C2×C67#243611yescharacteristic
321C2×C67#246012yes—
322C2×C67#247512yes—
323C2×C67#251912yes—
324C2×C67#255812yes—
325C2×C67#256211yescharacteristic
326C2×C67#261111yescharacteristic
327C2×C67#266411yescharacteristic
328C2×C67#286111yescharacteristic
329C22×#125C2×C6711yescharacteristic, complemented
330C2×#388C2×C6711yescharacteristic
331C2×#388C2×C6712yes—
332#1442C2×C6712yescomplemented
333#1441C2×C6711yescharacteristic, complemented
334#1441C2×C6711yescharacteristic, complemented
335#1439C2×C6711yescharacteristic
336#1439C2×C6712yes—
337#1439C2×C6711yescharacteristic
338#1436C2×C6712yescomplemented
339C4×#95C2×C6711yescharacteristic, complemented
340C67#1493311yescharacteristic, direct product factor, Sylow
341C2×#125—28no—
342C2×#125—24no—
343C22×C4⋊C4C22×C6711yescharacteristic
344C22×C4⋊C4C22×C6711yescharacteristic
345C22×C4⋊C4C22×C6712yes—
346C22×C4⋊C4C22×C6711yescharacteristic
347C22×C4⋊C4C22×C6712yes—
348C22×C22⋊C4C22×C6712yes—
349C22×C22⋊C4C22×C6711yescharacteristic
350C22×C22⋊C4C22×C6711yescharacteristic
351C22×C22⋊C4C22×C6711yescharacteristic
352C22×C22⋊C4C22×C6712yes—
353C22×C42C22×C6712yes—
354C22×C42C22×C6711yescharacteristic
355#388—28no—
356#388—24no—
357C2×#95C22×C6711yescharacteristic
358C2×#95C22×C6711yescharacteristic
359C2×#95C22×C6712yes—
360C2×#95C22×C6711yescharacteristic
361C2×#95C22×C6712yes—
362C2×#95C22×C6712yes—
363C2×#95C22×C6712yes—
364C2×#95C22×C6712yes—
365C2×#95C22×C6712yes—
366C2×#95C22×C6711yescharacteristic
367C2×#95C22×C6712yes—
368C25C23×C6711yescharacteristic
369C23×C4C23×C6711yescharacteristic
370C23×C4C23×C6711yescharacteristic
371C23×C4C23×C6712yes—
372C23×C4C23×C6712yes—
373C23×C4C23×C6711yescharacteristic
374C23×C4C23×C6711yescharacteristic
375C23×C4C23×C6712yes—
376C23×C4C23×C6712yes—
377C23×C4C23×C6711yescharacteristic
378C23×C4C23×C6711yescharacteristic
379#125—232no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4D4×C6718yes—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—42no—
407C2×C42—24no—
408C2×C42—24no—
409C2×C42—24no—
410C2×C42—42no—
411C2×C42D4×C6714yes—
412#95—24no—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95—42no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24#41×C6712yes—
434C24C24×C6711yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4C2×D4×C6712yes—
485C22×C4C2×D4×C6714yes—
486C22×C4C2×D4×C6714yes—
487C22×C4C2×D4×C6712yes—
488C22×C4C2×D4×C6714yes—
489C22×C4C2×D4×C6712yes—
490C22×C4#41×C6714yes—
491C22×C4#41×C6712yes—
492C22×C4#41×C6714yes—
493C22×C4#41×C6712yes—
494C22×C4#41×C6712yes—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—44no—
500C22⋊C4—28no—
501C22⋊C4—216no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C42—28no—
506C42—48no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—41no—
530C23C22×D4×C6711yescharacteristic
531C23C22×D4×C6712yes—
532C23C2×#41×C6711yescharacteristic
533C23C2×#41×C6712yes—
534C23C2×#41×C6711yescharacteristic
535C23C2×#41×C6711yescharacteristic
536C23C2×#41×C6711yescharacteristic
537C23C2×#41×C6712yes—
538C23H(5,𝔽2)×C6711yescharacteristic
539C232−1+4×C6712yes—
540C232−1+4×C6711yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
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—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4#120×C6714yes—
600C2×C4#124×C6714yes—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#120×C6711yescharacteristic
613C22#525×C6711yescharacteristic
614C22#525×C6712yes—
615C22C2×#124×C6711yescharacteristic
616C22#529×C6712yes—
617C22#529×C6712yes—
618C22#532×C6712yes—
619C22#532×C6711yescharacteristic
620C22#532×C6711yescharacteristic
621C22#536×C6711yescharacteristic
622C22#537×C6711yescharacteristic
623C22#539×C6712yes—
624C22#541×C6712yes—
625C22#546×C6712yes—
626C22#547×C6712yes—
627C22#548×C6712yes—
628C22#548×C6712yes—
629C22#552×C6711yescharacteristic
630C22#552×C6712yes—
631C22#552×C6711yescharacteristic
632C22#552×C6712yes—
633C22#553×C6711yescharacteristic
634C22#553×C6711yescharacteristic
635C4—24no—
636C4—24no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—44no—
643C4—81no—
644C4—81no—
645C2—82no—
646C2C67×#238811yescharacteristic
647C2C67×#243011yescharacteristic
648C2C67×#243611yescharacteristic
649C2C67×#246012yes—
650C2C67×#247512yes—
651C2C67×#251912yes—
652C2C67×#255812yes—
653C2C67×#256211yescharacteristic
654C2C67×#261111yescharacteristic
655C2C67×#266411yescharacteristic
656C2C67×#286111yescharacteristic
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×#125×C67C2semidirect
4. C67×#1442C2semidirect
5. C67×#1441C2semidirect
6. C67×#1441C2semidirect
10. C67×#1436C2semidirect
11. C4×#95×C67C2semidirect
316. #14933C67direct (N × H)
329. C22×#125C2×C67semidirect
332. #1442C2×C67semidirect
333. #1441C2×C67semidirect
334. #1441C2×C67semidirect
338. #1436C2×C67semidirect
339. C4×#95C2×C67semidirect
340. C67#14933direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#125×C672
C2×#388×C672
C67×#14422
C67×#14412
C67×#14392
C67×#14362
C4×#95×C672
#1493367