Finite Group Databasebeta

← C67×#14506 #30693757

Subgroups — C67×#14506 #30693757

Order 17152.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C67C211yesmaximal, characteristic
2C2×#388×C67C212yesmaximal, complemented
3C67×#1445C212yesmaximal, complemented
4C67×#1439C212yesmaximal, complemented
5C67×#1438C211yesmaximal, characteristic
6C67×#1438C211yesmaximal, characteristic
7C67×#1437C212yesmaximal, complemented
8C67×C4⋊#95C212yesmaximal, complemented
9C67×C22⋊#95C212yesmaximal, complemented
10C23×D4×C67C2211yescharacteristic
11C24×C4×C67C2211yescharacteristic
12C2×#123×C67—28no—
13C2×#123×C67C418yescomplemented
14C22×C4⋊C4×C67C2212yes—
15C22×C4⋊C4×C67C2212yes—
16C22×C22⋊C4×C67C2212yes—
17C22×C22⋊C4×C67C2212yes—
18C22×C22⋊C4×C67C2211yescharacteristic
19C22×C22⋊C4×C67C2212yes—
20C22×C22⋊C4×C67C2212yes—
21C22×C22⋊C4×C67C2211yescharacteristic
22C22×C22⋊C4×C67C2211yescharacteristic
23C22×C42×C67C2212yes—
24#388×C67—28no—
25C2×#95×C67C2212yes—
26C2×#95×C67C2212yes—
27C2×#95×C67C2212yes—
28C2×#95×C67C2212yes—
29C2×#95×C67C2212yes—
30C2×#95×C67C2212yes—
31C2×#95×C67C2211yescharacteristic
32C2×#95×C67C2212yes—
33C2×#95×C67C2211yescharacteristic
34C25×C67C2311yescharacteristic
35C25×C67C2311yescharacteristic
36C22×D4×C67—24no—
37C22×D4×C67—42no—
38C22×D4×C67—41no—
39C22×D4×C67—41no—
40C22×D4×C67C2×C414yes—
41C23×C4×C67—24no—
42C23×C4×C67—42no—
43C23×C4×C67—41no—
44C23×C4×C67—41no—
45C23×C4×C67C2×C414yes—
46C23×C4×C67C2312yes—
47C23×C4×C67C2312yes—
48C23×C4×C67C2312yes—
49C23×C4×C67C2311yescharacteristic
50C23×C4×C67C2312yes—
51C23×C4×C67C2312yes—
52C23×C4×C67C2312yes—
53#123×C67—232no—
54C2×C4⋊C4×C67—24no—
55C2×C4⋊C4×C67—24no—
56C2×C4⋊C4×C67—24no—
57C2×C4⋊C4×C67—24no—
58C2×C4⋊C4×C67—42no—
59C2×C4⋊C4×C67C2×C418yes—
60C2×C22⋊C4×C67—24no—
61C2×C22⋊C4×C67—24no—
62C2×C22⋊C4×C67—24no—
63C2×C22⋊C4×C67—24no—
64C2×C22⋊C4×C67—24no—
65C2×C22⋊C4×C67—24no—
66C2×C22⋊C4×C67—24no—
67C2×C22⋊C4×C67—24no—
68C2×C22⋊C4×C67—41no—
69C2×C22⋊C4×C67—42no—
70C2×C22⋊C4×C67—42no—
71C2×C22⋊C4×C67—42no—
72C2×C22⋊C4×C67—42no—
73C2×C22⋊C4×C67—42no—
74C2×C22⋊C4×C67—41no—
75C2×C22⋊C4×C67C2×C418yes—
76C2×C22⋊C4×C67C2×C414yes—
77C2×C22⋊C4×C67D418yes—
78C2×C42×C67—24no—
79C2×C42×C67—24no—
80C2×C42×C67—42no—
81#95×C67—24no—
82#95×C67—24no—
83#95×C67—42no—
84#95×C67—42no—
85#95×C67—42no—
86#95×C67—42no—
87#95×C67—41no—
88#95×C67—42no—
89#95×C67—41no—
90C24×C67—22no—
91C24×C67—22no—
92C24×C67—22no—
93C24×C67—22no—
94C24×C67—21no—
95C24×C67—21no—
96C24×C67—22no—
97C24×C67—22no—
98C24×C67—22no—
99C24×C67—22no—
100C24×C67—22no—
101C24×C67—22no—
102C24×C67—21no—
103C24×C67—21no—
104C24×C67—22no—
105C24×C67C22×C412yes—
106C24×C67C22×C412yes—
107C24×C67C2×D414yes—
108C24×C67C2411yescharacteristic
109C2×D4×C67—28no—
110C2×D4×C67—44no—
111C2×D4×C67—44no—
112C2×D4×C67—48no—
113C2×D4×C67—48no—
114C22×C4×C67—22no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
117C22×C4×C67—22no—
118C22×C4×C67—28no—
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—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—22no—
133C22×C4×C67—22no—
134C22×C4×C67—22no—
135C22×C4×C67—22no—
136C22×C4×C67—21no—
137C22×C4×C67—21no—
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—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—22no—
152C22×C4×C67—22no—
153C22×C4×C67—22no—
154C22×C4×C67—22no—
155C22×C4×C67—44no—
156C22×C4×C67—44no—
157C22×C4×C67—48no—
158C22×C4×C67—48no—
159C22×C4×C67C22×C414yes—
160C22×C4×C67C22×C412yes—
161C22×C4×C67C22×C414yes—
162C22×C4×C67C2×D414yes—
163C22×C4×C67C2×D414yes—
164C22×C4×C67#4114yes—
165C22×C4×C67#4114yes—
166C4⋊C4×C67—216no—
167C4⋊C4×C67—48no—
168C22⋊C4×C67—216no—
169C22⋊C4×C67—44no—
170C22⋊C4×C67—48no—
171C22⋊C4×C67—48no—
172C22⋊C4×C67—48no—
173C22⋊C4×C67—44no—
174C22⋊C4×C67C22⋊C4116yescomplemented
175C42×C67—48no—
176C23×C67—24no—
177C23×C67—22no—
178C23×C67—22no—
179C23×C67—24no—
180C23×C67—24no—
181C23×C67—24no—
182C23×C67—24no—
183C23×C67—24no—
184C23×C67—24no—
185C23×C67—24no—
186C23×C67—22no—
187C23×C67—24no—
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—24no—
191C23×C67—42no—
192C23×C67—42no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—41no—
196C23×C67—42no—
197C23×C67—41no—
198C23×C67—41no—
199C23×C67—42no—
200C23×C67—41no—
201C23×C67—42no—
202C23×C67—42no—
203C23×C67—42no—
204C23×C67—42no—
205C23×C67—42no—
206C23×C67—42no—
207C23×C67—42no—
208C23×C67—42no—
209C23×C67—42no—
210C23×C67—42no—
211C23×C67—41no—
212C23×C67—42no—
213C23×C67—41no—
214C23×C67—41no—
215C23×C67—42no—
216C23×C67—41no—
217C23×C67C2×C22⋊C414yes—
218C23×C67C23×C411yescharacteristic
219C23×C67C22×D412yes—
220C23×C67C2×#4112yes—
221C23×C67C2×#4112yes—
222C23×C67H(5,𝔽2)12yes—
223C23×C67H(5,𝔽2)11yescharacteristic
224C23×C67H(5,𝔽2)11yescharacteristic
225C23×C67H(5,𝔽2)12yes—
226C23×C672−1+412yes—
227D4×C67—416no—
228C2×C4×C67—24no—
229C2×C4×C67—24no—
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—24no—
238C2×C4×C67—22no—
239C2×C4×C67—22no—
240C2×C4×C67—24no—
241C2×C4×C67—24no—
242C2×C4×C67—24no—
243C2×C4×C67—24no—
244C2×C4×C67—24no—
245C2×C4×C67—24no—
246C2×C4×C67—416no—
247C2×C4×C67—42no—
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—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—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×C67C2×C22⋊C418yes—
276C22×C67—24no—
277C22×C67—28no—
278C22×C67—44no—
279C22×C67—42no—
280C22×C67—42no—
281C22×C67—44no—
282C22×C67—44no—
283C22×C67—44no—
284C22×C67—44no—
285C22×C67—42no—
286C22×C67—44no—
287C22×C67—44no—
288C22×C67—42no—
289C22×C67—42no—
290C22×C67—82no—
291C22×C67—82no—
292C22×C67—82no—
293C22×C67—82no—
294C22×C67C22×C22⋊C411yescharacteristic
295C22×C67C4×#4112yes—
296C22×C67#51811yescharacteristic
297C22×C67#51811yescharacteristic
298C22×C67#52012yes—
299C22×C67#53512yes—
300C22×C67#53512yes—
301C22×C67#53812yes—
302C22×C67#53912yes—
303C22×C67#54012yes—
304C22×C67#54212yes—
305C22×C67#54612yes—
306C22×C67#54612yes—
307C22×C67#54712yes—
308C22×C67#54712yes—
309C22×C67#55111yescharacteristic
310C22×C67#55312yes—
311C22×C67#55911yescharacteristic
312C22×C67#56011yescharacteristic
313C22×C67#56212yes—
314C22×C67#56311yescharacteristic
315C4×C67—28no—
316C4×C67—44no—
317C4×C67—42no—
318C4×C67—44no—
319C4×C67—44no—
320C4×C67—82no—
321C4×C67—82no—
322#14506C6711yesmaximal, characteristic, direct product factor, Sylow
323C2×C67—44no—
324C2×C67—82no—
325C2×C67#230911yescharacteristic
326C2×C67#233512yes—
327C2×C67#233912yes—
328C2×C67#237411yescharacteristic
329C2×C67#237611yescharacteristic
330C2×C67#270312yes—
331C2×C67#270612yes—
332C2×C67#271012yes—
333C2×C67#271812yes—
334C22×#123C2×C6711yescharacteristic
335C2×#388C2×C6712yescomplemented
336#1445C2×C6712yescomplemented
337#1439C2×C6712yescomplemented
338#1438C2×C6711yescharacteristic
339#1438C2×C6711yescharacteristic
340#1437C2×C6712yescomplemented
341C4⋊#95C2×C6712yescomplemented
342C22⋊#95C2×C6712yescomplemented
343C67#1450611yescharacteristic, direct product factor, Sylow
344C23×D4C22×C6711yescharacteristic
345C24×C4C22×C6711yescharacteristic
346C2×#123—28no—
347C2×#123C4×C6718yescomplemented
348C22×C4⋊C4C22×C6712yes—
349C22×C4⋊C4C22×C6712yes—
350C22×C22⋊C4C22×C6712yes—
351C22×C22⋊C4C22×C6712yes—
352C22×C22⋊C4C22×C6711yescharacteristic
353C22×C22⋊C4C22×C6712yes—
354C22×C22⋊C4C22×C6712yes—
355C22×C22⋊C4C22×C6711yescharacteristic
356C22×C22⋊C4C22×C6711yescharacteristic
357C22×C42C22×C6712yes—
358#388—28no—
359C2×#95C22×C6712yes—
360C2×#95C22×C6712yes—
361C2×#95C22×C6712yes—
362C2×#95C22×C6712yes—
363C2×#95C22×C6712yes—
364C2×#95C22×C6712yes—
365C2×#95C22×C6711yescharacteristic
366C2×#95C22×C6712yes—
367C2×#95C22×C6711yescharacteristic
368C25C23×C6711yescharacteristic
369C25C23×C6711yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C6714yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C6714yes—
380C23×C4C23×C6712yes—
381C23×C4C23×C6712yes—
382C23×C4C23×C6712yes—
383C23×C4C23×C6711yescharacteristic
384C23×C4C23×C6712yes—
385C23×C4C23×C6712yes—
386C23×C4C23×C6712yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C6718yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C6718yes—
410C2×C22⋊C4C2×C4×C6714yes—
411C2×C22⋊C4D4×C6718yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C6712yes—
440C24C22×C4×C6712yes—
441C24C2×D4×C6714yes—
442C24C24×C6711yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C6714yes—
494C22×C4C22×C4×C6712yes—
495C22×C4C22×C4×C6714yes—
496C22×C4C2×D4×C6714yes—
497C22×C4C2×D4×C6714yes—
498C22×C4#41×C6714yes—
499C22×C4#41×C6714yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C67116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C6714yes—
552C23C23×C4×C6711yescharacteristic
553C23C22×D4×C6712yes—
554C23C2×#41×C6712yes—
555C23C2×#41×C6712yes—
556C23H(5,𝔽2)×C6712yes—
557C23H(5,𝔽2)×C6711yescharacteristic
558C23H(5,𝔽2)×C6711yescharacteristic
559C23H(5,𝔽2)×C6712yes—
560C232−1+4×C6712yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
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×C4C2×C22⋊C4×C6718yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C6711yescharacteristic
629C22C4×#41×C6712yes—
630C22#518×C6711yescharacteristic
631C22#518×C6711yescharacteristic
632C22#520×C6712yes—
633C22#535×C6712yes—
634C22#535×C6712yes—
635C22#538×C6712yes—
636C22#539×C6712yes—
637C22#540×C6712yes—
638C22#542×C6712yes—
639C22#546×C6712yes—
640C22#546×C6712yes—
641C22#547×C6712yes—
642C22#547×C6712yes—
643C22#551×C6711yescharacteristic
644C22#553×C6712yes—
645C22#559×C6711yescharacteristic
646C22#560×C6711yescharacteristic
647C22#562×C6712yes—
648C22#563×C6711yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C67×#230911yescharacteristic
659C2C67×#233512yes—
660C2C67×#233912yes—
661C2C67×#237411yescharacteristic
662C2C67×#237611yescharacteristic
663C2C67×#270312yes—
664C2C67×#270612yes—
665C2C67×#271012yes—
666C2C67×#271812yes—
subgroupG/H
center, socleC24×C67C24
commutatorC23C23×C4×C67
Frattini, cosocleC24C24×C67
prime radicalC23×D4×C67C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#388×C67C2semidirect
3. C67×#1445C2semidirect
4. C67×#1439C2semidirect
7. C67×#1437C2semidirect
8. C67×C4⋊#95C2semidirect
9. C67×C22⋊#95C2semidirect
13. C2×#123×C67C4semidirect
174. C22⋊C4×C67C22⋊C4semidirect
322. #14506C67direct (N × H)
335. C2×#388C2×C67semidirect
336. #1445C2×C67semidirect
337. #1439C2×C67semidirect
340. #1437C2×C67semidirect
341. C4⋊#95C2×C67semidirect
342. C22⋊#95C2×C67semidirect
343. C67#14506direct (N × H)
347. C2×#123C4×C67semidirect
508. C22⋊C4C22⋊C4×C67semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C672
C2×#388×C672
C67×#14452
C67×#14392
C67×#14382
C67×#14372
C67×C4⋊#952
C67×C22⋊#952
#1450667