Finite Group Databasebeta

← C67×#14894 #30694145

Subgroups — C67×#14894 #30694145

Order 17152.

1426 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#397×C67C211yesmaximal, characteristic
2C2×#389×C67C211yesmaximal, characteristic, complemented
3C2×#388×C67C211yesmaximal, characteristic
4C2×#376×C67C211yesmaximal, characteristic, complemented
5C67×#1445C211yesmaximal, characteristic, complemented
6C67×#1442C211yesmaximal, characteristic, complemented
7C67×#1442C212yesmaximal, complemented
8C67×#1439C211yesmaximal, characteristic
9C67×#1439C212yesmaximal
10C67×#1437C212yesmaximal
11C67×#1436C212yesmaximal, complemented
12C22×C4⋊C4×C67C2211yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14C22×C4⋊C4×C67C2211yescharacteristic
15C22×C4⋊C4×C67C2212yes—
16C22×C22⋊C4×C67C2211yescharacteristic
17C22×C22⋊C4×C67C2211yescharacteristic
18C22×C22⋊C4×C67C2211yescharacteristic
19C22×C22⋊C4×C67C2212yes—
20C22×C22⋊C4×C67C2212yes—
21C22×C42×C67C2212yes—
22C22×C42×C67C2211yescharacteristic
23#397×C67—24no—
24#389×C67—24no—
25#388×C67—24no—
26#376×C67—24no—
27C2×#95×C67C2212yes—
28C2×#95×C67C2211yescharacteristic
29C2×#95×C67C2211yescharacteristic
30C2×#95×C67C2211yescharacteristic
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
40C23×C4×C67C2312yes—
41C23×C4×C67C2311yescharacteristic
42C23×C4×C67C2311yescharacteristic
43C23×C4×C67C2311yescharacteristic
44C23×C4×C67C2312yes—
45C23×C4×C67C2312yes—
46C23×C4×C67C2312yes—
47C23×C4×C67C2311yescharacteristic
48C23×C4×C67C2311yescharacteristic
49C23×C4×C67C2311yescharacteristic
50C2×C4⋊C4×C67—22no—
51C2×C4⋊C4×C67—24no—
52C2×C4⋊C4×C67—22no—
53C2×C4⋊C4×C67—24no—
54C2×C4⋊C4×C67—22no—
55C2×C4⋊C4×C67—22no—
56C2×C4⋊C4×C67—22no—
57C2×C4⋊C4×C67—24no—
58C2×C4⋊C4×C67—42no—
59C2×C4⋊C4×C67—42no—
60C2×C22⋊C4×C67—22no—
61C2×C22⋊C4×C67—22no—
62C2×C22⋊C4×C67—22no—
63C2×C22⋊C4×C67—24no—
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×C42×C67—24no—
75C2×C42×C67—22no—
76C2×C42×C67—24no—
77C2×C42×C67—24no—
78C2×C42×C67—22no—
79C2×C42×C67D414yes—
80#95×C67—24no—
81#95×C67—22no—
82#95×C67—22no—
83#95×C67—22no—
84#95×C67—24no—
85#95×C67—24no—
86#95×C67—24no—
87#95×C67—24no—
88#95×C67—24no—
89#95×C67—42no—
90#95×C67—42no—
91#95×C67D414yes—
92C24×C67—21no—
93C24×C67—21no—
94C24×C67—21no—
95C24×C67—22no—
96C24×C67—22no—
97C24×C67—22no—
98C24×C67—21no—
99C24×C67—21no—
100C24×C67—21no—
101C24×C67—22no—
102C24×C67#4112yes—
103C24×C67C2411yescharacteristic
104C22×C4×C67—21no—
105C22×C4×C67—21no—
106C22×C4×C67—22no—
107C22×C4×C67—22no—
108C22×C4×C67—22no—
109C22×C4×C67—21no—
110C22×C4×C67—22no—
111C22×C4×C67—21no—
112C22×C4×C67—22no—
113C22×C4×C67—21no—
114C22×C4×C67—22no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
117C22×C4×C67—21no—
118C22×C4×C67—21no—
119C22×C4×C67—21no—
120C22×C4×C67—22no—
121C22×C4×C67—22no—
122C22×C4×C67—22no—
123C22×C4×C67—22no—
124C22×C4×C67—22no—
125C22×C4×C67—22no—
126C22×C4×C67—22no—
127C22×C4×C67—22no—
128C22×C4×C67—22no—
129C22×C4×C67—22no—
130C22×C4×C67—22no—
131C22×C4×C67—22no—
132C22×C4×C67—21no—
133C22×C4×C67—21no—
134C22×C4×C67—22no—
135C22×C4×C67—22no—
136C22×C4×C67—22no—
137C22×C4×C67—21no—
138C22×C4×C67—22no—
139C22×C4×C67—21no—
140C22×C4×C67—21no—
141C22×C4×C67—22no—
142C22×C4×C67—21no—
143C22×C4×C67—22no—
144C22×C4×C67—22no—
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—21no—
152C22×C4×C67—21no—
153C22×C4×C67—22no—
154C22×C4×C67C2×D414yes—
155C22×C4×C67C2×D412yes—
156C22×C4×C67C2×D412yes—
157C22×C4×C67C2×D412yes—
158C22×C4×C67C2×D412yes—
159C22×C4×C67#4114yes—
160C22×C4×C67#4114yes—
161C22×C4×C67#4112yes—
162C22×C4×C67#4112yes—
163C22×C4×C67#4114yes—
164C4⋊C4×C67—28no—
165C4⋊C4×C67—48no—
166C4⋊C4×C67—44no—
167C4⋊C4×C67—44no—
168C22⋊C4×C67—28no—
169C22⋊C4×C67—48no—
170C22⋊C4×C67—48no—
171C22⋊C4×C67—44no—
172C22⋊C4×C67—44no—
173C42×C67—216no—
174C42×C67—28no—
175C23×C67—22no—
176C23×C67—24no—
177C23×C67—22no—
178C23×C67—24no—
179C23×C67—22no—
180C23×C67—41no—
181C23×C67—41no—
182C23×C67—42no—
183C23×C67—41no—
184C23×C67—42no—
185C23×C67—42no—
186C23×C67—41no—
187C23×C67—41no—
188C23×C67—41no—
189C23×C67—42no—
190C23×C67—42no—
191C23×C67—42no—
192C23×C67—41no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—42no—
196C23×C67—42no—
197C23×C67—41no—
198C23×C67C22×D411yescharacteristic
199C23×C67C22×D411yescharacteristic
200C23×C67C2×#4111yescharacteristic
201C23×C67C2×#4112yes—
202C23×C67C2×#4111yescharacteristic
203C23×C67C2×#4112yes—
204C23×C67C2×#4111yescharacteristic
205C23×C67C2×#4112yes—
206C23×C67H(5,𝔽2)11yescharacteristic
207C23×C67H(5,𝔽2)11yescharacteristic
208C23×C672−1+412yes—
209C2×C4×C67—22no—
210C2×C4×C67—24no—
211C2×C4×C67—22no—
212C2×C4×C67—24no—
213C2×C4×C67—22no—
214C2×C4×C67—22no—
215C2×C4×C67—24no—
216C2×C4×C67—24no—
217C2×C4×C67—22no—
218C2×C4×C67—24no—
219C2×C4×C67—24no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—24no—
223C2×C4×C67—24no—
224C2×C4×C67—22no—
225C2×C4×C67—24no—
226C2×C4×C67—24no—
227C2×C4×C67—24no—
228C2×C4×C67—24no—
229C2×C4×C67—24no—
230C2×C4×C67—24no—
231C2×C4×C67—41no—
232C2×C4×C67—41no—
233C2×C4×C67—42no—
234C2×C4×C67—42no—
235C2×C4×C67—42no—
236C2×C4×C67—42no—
237C2×C4×C67—42no—
238C2×C4×C67—42no—
239C2×C4×C67—42no—
240C2×C4×C67—41no—
241C2×C4×C67—42no—
242C2×C4×C67—41no—
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—41no—
250C2×C4×C67—41no—
251C2×C4×C67—42no—
252C2×C4×C67—42no—
253C2×C4×C67—42no—
254C2×C4×C67—41no—
255C2×C4×C67—41no—
256C2×C4×C67—42no—
257C2×C4×C67—41no—
258C2×C4×C67—42no—
259C2×C4×C67—41no—
260C2×C4×C67—41no—
261C2×C4×C67—42no—
262C2×C4×C67—42no—
263C2×C4×C67—42no—
264C2×C4×C67—42no—
265C2×C4×C67—42no—
266C2×C4×C67—42no—
267C2×C4×C67#12114yes—
268C2×C4×C67#12414yes—
269C22×C67—42no—
270C22×C67—42no—
271C22×C67—42no—
272C22×C67—44no—
273C22×C67—44no—
274C22×C67—81no—
275C22×C67—81no—
276C22×C67—81no—
277C22×C67—81no—
278C22×C67—82no—
279C22×C67—82no—
280C22×C67C2×#12111yescharacteristic
281C22×C67#52512yes—
282C22×C67#52511yescharacteristic
283C22×C67#52511yescharacteristic
284C22×C67C2×#12411yescharacteristic
285C22×C67#52912yes—
286C22×C67#52912yes—
287C22×C67#52912yes—
288C22×C67#53211yescharacteristic
289C22×C67#53511yescharacteristic
290C22×C67#53611yescharacteristic
291C22×C67#54212yes—
292C22×C67#54212yes—
293C22×C67#54611yescharacteristic
294C22×C67#54712yes—
295C22×C67#55011yescharacteristic
296C22×C67#55112yes—
297C22×C67#55212yes—
298C22×C67#55212yes—
299C22×C67#55212yes—
300C22×C67#55312yes—
301C22×C67#55311yescharacteristic
302C22×C67#55511yescharacteristic
303C4×C67—24no—
304C4×C67—24no—
305C4×C67—44no—
306C4×C67—42no—
307C4×C67—42no—
308C4×C67—44no—
309C4×C67—44no—
310C4×C67—81no—
311C4×C67—81no—
312C4×C67—82no—
313#14894C6711yesmaximal, characteristic, direct product factor, Sylow
314C2×C67—82no—
315C2×C67#239811yescharacteristic
316C2×C67#241311yescharacteristic
317C2×C67#241411yescharacteristic
318C2×C67#247012yes—
319C2×C67#247912yes—
320C2×C67#252912yes—
321C2×C67#253012yes—
322C2×C67#260311yescharacteristic
323C2×C67#261111yescharacteristic
324C2×C67#264011yescharacteristic
325C2×C67#285011yescharacteristic
326C2×#397C2×C6711yescharacteristic
327C2×#389C2×C6711yescharacteristic, complemented
328C2×#388C2×C6711yescharacteristic
329C2×#376C2×C6711yescharacteristic, complemented
330#1445C2×C6711yescharacteristic, complemented
331#1442C2×C6711yescharacteristic, complemented
332#1442C2×C6712yescomplemented
333#1439C2×C6711yescharacteristic
334#1439C2×C6712yes—
335#1437C2×C6712yes—
336#1436C2×C6712yescomplemented
337C67#1489411yescharacteristic, direct product factor, Sylow
338C22×C4⋊C4C22×C6711yescharacteristic
339C22×C4⋊C4C22×C6711yescharacteristic
340C22×C4⋊C4C22×C6711yescharacteristic
341C22×C4⋊C4C22×C6712yes—
342C22×C22⋊C4C22×C6711yescharacteristic
343C22×C22⋊C4C22×C6711yescharacteristic
344C22×C22⋊C4C22×C6711yescharacteristic
345C22×C22⋊C4C22×C6712yes—
346C22×C22⋊C4C22×C6712yes—
347C22×C42C22×C6712yes—
348C22×C42C22×C6711yescharacteristic
349#397—24no—
350#389—24no—
351#388—24no—
352#376—24no—
353C2×#95C22×C6712yes—
354C2×#95C22×C6712yes—
355C2×#95C22×C6712yes—
356C2×#95C22×C6712yes—
357C2×#95C22×C6712yes—
358C2×#95C22×C6712yes—
359C2×#95C22×C6711yescharacteristic
360C2×#95C22×C6711yescharacteristic
361C2×#95C22×C6711yescharacteristic
362C2×#95C22×C6712yes—
363C2×#95C22×C6712yes—
364C2×#95C22×C6711yescharacteristic
365C25C23×C6711yescharacteristic
366C23×C4C23×C6712yes—
367C23×C4C23×C6711yescharacteristic
368C23×C4C23×C6711yescharacteristic
369C23×C4C23×C6711yescharacteristic
370C23×C4C23×C6712yes—
371C23×C4C23×C6712yes—
372C23×C4C23×C6711yescharacteristic
373C23×C4C23×C6712yes—
374C23×C4C23×C6711yescharacteristic
375C23×C4C23×C6711yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—42no—
385C2×C4⋊C4—42no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C42—24no—
401C2×C42—22no—
402C2×C42—24no—
403C2×C42—24no—
404C2×C42—22no—
405C2×C42D4×C6714yes—
406#95—24no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—42no—
416#95—42no—
417#95D4×C6714yes—
418C24—21no—
419C24—21no—
420C24—21no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24#41×C6712yes—
429C24C24×C6711yescharacteristic
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
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—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4C2×D4×C6714yes—
481C22×C4C2×D4×C6712yes—
482C22×C4C2×D4×C6712yes—
483C22×C4C2×D4×C6712yes—
484C22×C4C2×D4×C6712yes—
485C22×C4#41×C6712yes—
486C22×C4#41×C6714yes—
487C22×C4#41×C6714yes—
488C22×C4#41×C6712yes—
489C22×C4#41×C6714yes—
490C4⋊C4—28no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C22⋊C4—28no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C42—216no—
500C42—28no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23C22×D4×C6711yescharacteristic
525C23C22×D4×C6711yescharacteristic
526C23C2×#41×C6711yescharacteristic
527C23C2×#41×C6712yes—
528C23C2×#41×C6711yescharacteristic
529C23C2×#41×C6712yes—
530C23C2×#41×C6711yescharacteristic
531C23C2×#41×C6712yes—
532C23H(5,𝔽2)×C6711yescharacteristic
533C23H(5,𝔽2)×C6711yescharacteristic
534C232−1+4×C6712yes—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4#121×C6714yes—
594C2×C4#124×C6714yes—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22C2×#121×C6711yescharacteristic
607C22#525×C6712yes—
608C22#525×C6711yescharacteristic
609C22#525×C6711yescharacteristic
610C22C2×#124×C6711yescharacteristic
611C22#529×C6712yes—
612C22#529×C6712yes—
613C22#529×C6712yes—
614C22#532×C6711yescharacteristic
615C22#535×C6711yescharacteristic
616C22#536×C6711yescharacteristic
617C22#542×C6712yes—
618C22#542×C6712yes—
619C22#546×C6711yescharacteristic
620C22#547×C6712yes—
621C22#550×C6711yescharacteristic
622C22#551×C6712yes—
623C22#552×C6712yes—
624C22#552×C6712yes—
625C22#552×C6712yes—
626C22#553×C6712yes—
627C22#553×C6711yescharacteristic
628C22#555×C6711yescharacteristic
629C4—24no—
630C4—24no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—44no—
636C4—81no—
637C4—81no—
638C4—82no—
639C2—82no—
640C2C67×#239811yescharacteristic
641C2C67×#241311yescharacteristic
642C2C67×#241411yescharacteristic
643C2C67×#247012yes—
644C2C67×#247912yes—
645C2C67×#252912yes—
646C2C67×#253012yes—
647C2C67×#260311yescharacteristic
648C2C67×#261111yescharacteristic
649C2C67×#264011yescharacteristic
650C2C67×#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×#389×C67C2semidirect
4. C2×#376×C67C2semidirect
5. C67×#1445C2semidirect
6. C67×#1442C2semidirect
7. C67×#1442C2semidirect
11. C67×#1436C2semidirect
313. #14894C67direct (N × H)
327. C2×#389C2×C67semidirect
329. C2×#376C2×C67semidirect
330. #1445C2×C67semidirect
331. #1442C2×C67semidirect
332. #1442C2×C67semidirect
336. #1436C2×C67semidirect
337. C67#14894direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#397×C672
C2×#389×C672
C2×#388×C672
C2×#376×C672
C67×#14452
C67×#14422
C67×#14392
C67×#14372
C67×#14362
#1489467