Finite Group Databasebeta

← C67×#16473 #30695724

Subgroups — C67×#16473 #30695724

Order 17152.

1398 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399×C67C211yesmaximal, characteristic
2C2×#385×C67C211yesmaximal, characteristic
3C67×#1442C212yesmaximal, complemented
4C67×#1442C212yesmaximal, complemented
5C67×#1441C212yesmaximal, complemented
6C67×#1441C212yesmaximal, complemented
7C67×#1440C212yesmaximal
8C67×#1439C211yesmaximal, characteristic
9C67×C22⋊#95C212yesmaximal
10C24×C4×C67C2211yescharacteristic
11C22×C4⋊C4×C67C2212yes—
12C22×C4⋊C4×C67C2212yes—
13C22×C4⋊C4×C67C2211yescharacteristic
14C22×C4⋊C4×C67C2211yescharacteristic
15C22×C22⋊C4×C67C2212yes—
16C22×C22⋊C4×C67C2211yescharacteristic
17C22×C22⋊C4×C67C2212yes—
18C22×C22⋊C4×C67C2211yescharacteristic
19#399×C67—24no—
20#385×C67—24no—
21C2×#95×C67C2212yes—
22C2×#95×C67C2211yescharacteristic
23C2×#95×C67C2212yes—
24C2×#95×C67C2212yes—
25C2×#95×C67C2212yes—
26C2×#95×C67C2212yes—
27C2×#95×C67C2212yes—
28C2×#95×C67C2212yes—
29C2×#95×C67C2211yescharacteristic
30C2×#95×C67C2212yes—
31C2×#95×C67C2212yes—
32C2×#95×C67C2212yes—
33C25×C67C2311yescharacteristic
34C23×C4×C67—22no—
35C23×C4×C67—24no—
36C23×C4×C67—42no—
37C23×C4×C67—41no—
38C23×C4×C67—41no—
39C23×C4×C67C2312yes—
40C23×C4×C67C2312yes—
41C23×C4×C67C2312yes—
42C23×C4×C67C2311yescharacteristic
43C23×C4×C67C2311yescharacteristic
44C23×C4×C67C2312yes—
45C23×C4×C67C2312yes—
46C23×C4×C67C2312yes—
47C2×C4⋊C4×C67—24no—
48C2×C4⋊C4×C67—24no—
49C2×C4⋊C4×C67—22no—
50C2×C4⋊C4×C67—24no—
51C2×C4⋊C4×C67—24no—
52C2×C4⋊C4×C67—42no—
53C2×C4⋊C4×C67—42no—
54C2×C4⋊C4×C67—41no—
55C2×C4⋊C4×C67—41no—
56C2×C4⋊C4×C67—41no—
57C2×C4⋊C4×C67—42no—
58C2×C22⋊C4×C67—22no—
59C2×C22⋊C4×C67—22no—
60C2×C22⋊C4×C67—24no—
61C2×C22⋊C4×C67—24no—
62C2×C22⋊C4×C67—24no—
63C2×C22⋊C4×C67—41no—
64C2×C22⋊C4×C67—41no—
65C2×C22⋊C4×C67—42no—
66C2×C22⋊C4×C67—42no—
67C2×C22⋊C4×C67—42no—
68C2×C22⋊C4×C67D414yes—
69C2×C22⋊C4×C67Q814yes—
70#95×C67—24no—
71#95×C67—22no—
72#95×C67—24no—
73#95×C67—24no—
74#95×C67—41no—
75#95×C67—42no—
76#95×C67—42no—
77#95×C67—42no—
78#95×C67—42no—
79#95×C67—42no—
80#95×C67—42no—
81#95×C67—42no—
82C24×C67—22no—
83C24×C67—22no—
84C24×C67—22no—
85C24×C67—21no—
86C24×C67—21no—
87C24×C67—22no—
88C24×C67—21no—
89C24×C67—21no—
90C24×C67C2×D412yes—
91C24×C67C2×Q812yes—
92C24×C67#4112yes—
93C24×C67C2411yescharacteristic
94C22×C4×C67—22no—
95C22×C4×C67—22no—
96C22×C4×C67—22no—
97C22×C4×C67—22no—
98C22×C4×C67—22no—
99C22×C4×C67—22no—
100C22×C4×C67—22no—
101C22×C4×C67—22no—
102C22×C4×C67—22no—
103C22×C4×C67—22no—
104C22×C4×C67—21no—
105C22×C4×C67—21no—
106C22×C4×C67—21no—
107C22×C4×C67—22no—
108C22×C4×C67—21no—
109C22×C4×C67—22no—
110C22×C4×C67—22no—
111C22×C4×C67—22no—
112C22×C4×C67—22no—
113C22×C4×C67—22no—
114C22×C4×C67—22no—
115C22×C4×C67—22no—
116C22×C4×C67—22no—
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—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—22no—
137C22×C4×C67—22no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—22no—
141C22×C4×C67—44no—
142C22×C4×C67—48no—
143C22×C4×C67—48no—
144C22×C4×C67—44no—
145C22×C4×C67—44no—
146C22×C4×C67C2×D412yes—
147C22×C4×C67C2×D412yes—
148C22×C4×C67C2×Q814yes—
149C22×C4×C67#4114yes—
150C22×C4×C67#4114yes—
151C4⋊C4×C67—48no—
152C4⋊C4×C67—48no—
153C4⋊C4×C67—44no—
154C4⋊C4×C67—44no—
155C22⋊C4×C67—28no—
156C22⋊C4×C67—44no—
157C22⋊C4×C67—48no—
158C22⋊C4×C67—48no—
159C23×C67—24no—
160C23×C67—22no—
161C23×C67—22no—
162C23×C67—24no—
163C23×C67—22no—
164C23×C67—22no—
165C23×C67—22no—
166C23×C67—22no—
167C23×C67—24no—
168C23×C67—24no—
169C23×C67—24no—
170C23×C67—24no—
171C23×C67—42no—
172C23×C67—42no—
173C23×C67—42no—
174C23×C67—42no—
175C23×C67—41no—
176C23×C67—42no—
177C23×C67—41no—
178C23×C67—41no—
179C23×C67—41no—
180C23×C67—42no—
181C23×C67C22⋊Q814yes—
182C23×C67C22×D411yescharacteristic
183C23×C67C22×Q811yescharacteristic
184C23×C67C2×#4111yescharacteristic
185C23×C67C2×#4112yes—
186C23×C67C2×#4112yes—
187C23×C67H(5,𝔽2)11yescharacteristic
188C23×C67H(5,𝔽2)11yescharacteristic
189C23×C67H(5,𝔽2)12yes—
190C23×C67H(5,𝔽2)12yes—
191C23×C672−1+411yescharacteristic
192C23×C672−1+411yescharacteristic
193C2×C4×C67—24no—
194C2×C4×C67—22no—
195C2×C4×C67—22no—
196C2×C4×C67—24no—
197C2×C4×C67—24no—
198C2×C4×C67—24no—
199C2×C4×C67—24no—
200C2×C4×C67—24no—
201C2×C4×C67—24no—
202C2×C4×C67—24no—
203C2×C4×C67—24no—
204C2×C4×C67—24no—
205C2×C4×C67—24no—
206C2×C4×C67—416no—
207C2×C4×C67—42no—
208C2×C4×C67—42no—
209C2×C4×C67—42no—
210C2×C4×C67—42no—
211C2×C4×C67—42no—
212C2×C4×C67—42no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67—42no—
216C2×C4×C67—42no—
217C2×C4×C67—42no—
218C2×C4×C67—42no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—42no—
224C2×C4×C67—42no—
225C2×C4×C67—42no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67—42no—
229C2×C4×C67—42no—
230C2×C4×C67—42no—
231C2×C4×C67—42no—
232C2×C4×C67—42no—
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—42no—
241C2×C4×C67—42no—
242C2×C4×C67—42no—
243C2×C4×C67—42no—
244C2×C4×C67—42no—
245C2×C4×C67—42no—
246C22×C67—24no—
247C22×C67—28no—
248C22×C67—42no—
249C22×C67—44no—
250C22×C67—44no—
251C22×C67—42no—
252C22×C67—42no—
253C22×C67—42no—
254C22×C67—44no—
255C22×C67—42no—
256C22×C67—42no—
257C22×C67C2×C22⋊Q811yescharacteristic
258C22×C67#52912yes—
259C22×C67#53411yescharacteristic
260C22×C67#53611yescharacteristic
261C22×C67#53812yes—
262C22×C67#53812yes—
263C22×C67#53911yescharacteristic
264C22×C67#53911yescharacteristic
265C22×C67#53912yes—
266C22×C67#54012yes—
267C22×C67#54212yes—
268C22×C67#54411yescharacteristic
269C22×C67#54411yescharacteristic
270C22×C67#54612yes—
271C22×C67#54612yes—
272C22×C67#55111yescharacteristic
273C22×C67#55212yes—
274C22×C67#55412yes—
275C22×C67#55412yes—
276C22×C67#55911yescharacteristic
277C22×C67#56011yescharacteristic
278C22×C67#56011yescharacteristic
279C22×C67#56212yes—
280C4×C67—44no—
281C4×C67—42no—
282C4×C67—44no—
283C4×C67—44no—
284C4×C67—42no—
285C4×C67—82no—
286C4×C67—82no—
287C4×C67—82no—
288#16473C6711yesmaximal, characteristic, direct product factor, Sylow
289C2×C67—44no—
290C2×C67#245012yes—
291C2×C67#251711yescharacteristic
292C2×C67#262711yescharacteristic
293C2×C67#264912yes—
294C2×C67#268712yes—
295C2×C67#269711yescharacteristic
296C2×C67#270911yescharacteristic
297C2×C67#274211yescharacteristic
298C2×C67#276812yes—
299C2×C67#280711yescharacteristic
300C2×C67#282611yescharacteristic
301C2×#399C2×C6711yescharacteristic
302C2×#385C2×C6711yescharacteristic
303#1442C2×C6712yescomplemented
304#1442C2×C6712yescomplemented
305#1441C2×C6712yescomplemented
306#1441C2×C6712yescomplemented
307#1440C2×C6712yes—
308#1439C2×C6711yescharacteristic
309C22⋊#95C2×C6712yes—
310C67#1647311yescharacteristic, direct product factor, Sylow
311C24×C4C22×C6711yescharacteristic
312C22×C4⋊C4C22×C6712yes—
313C22×C4⋊C4C22×C6712yes—
314C22×C4⋊C4C22×C6711yescharacteristic
315C22×C4⋊C4C22×C6711yescharacteristic
316C22×C22⋊C4C22×C6712yes—
317C22×C22⋊C4C22×C6711yescharacteristic
318C22×C22⋊C4C22×C6712yes—
319C22×C22⋊C4C22×C6711yescharacteristic
320#399—24no—
321#385—24no—
322C2×#95C22×C6711yescharacteristic
323C2×#95C22×C6712yes—
324C2×#95C22×C6712yes—
325C2×#95C22×C6712yes—
326C2×#95C22×C6712yes—
327C2×#95C22×C6712yes—
328C2×#95C22×C6712yes—
329C2×#95C22×C6712yes—
330C2×#95C22×C6711yescharacteristic
331C2×#95C22×C6712yes—
332C2×#95C22×C6712yes—
333C2×#95C22×C6712yes—
334C25C23×C6711yescharacteristic
335C23×C4—22no—
336C23×C4—24no—
337C23×C4—41no—
338C23×C4—41no—
339C23×C4—42no—
340C23×C4C23×C6712yes—
341C23×C4C23×C6712yes—
342C23×C4C23×C6712yes—
343C23×C4C23×C6711yescharacteristic
344C23×C4C23×C6711yescharacteristic
345C23×C4C23×C6712yes—
346C23×C4C23×C6712yes—
347C23×C4C23×C6712yes—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—42no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4D4×C6714yes—
370C2×C22⋊C4Q8×C6714yes—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95—42no—
376#95—42no—
377#95—42no—
378#95—42no—
379#95—41no—
380#95—42no—
381#95—42no—
382#95—42no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24C2×D4×C6712yes—
392C24C2×Q8×C6712yes—
393C24#41×C6712yes—
394C24C24×C6711yescharacteristic
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C2×D4×C6712yes—
448C22×C4C2×D4×C6712yes—
449C22×C4C2×Q8×C6714yes—
450C22×C4#41×C6714yes—
451C22×C4#41×C6714yes—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C22⋊C4—28no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23C22⋊Q8×C6714yes—
483C23C22×D4×C6711yescharacteristic
484C23C22×Q8×C6711yescharacteristic
485C23C2×#41×C6711yescharacteristic
486C23C2×#41×C6712yes—
487C23C2×#41×C6712yes—
488C23H(5,𝔽2)×C6711yescharacteristic
489C23H(5,𝔽2)×C6711yescharacteristic
490C23H(5,𝔽2)×C6712yes—
491C23H(5,𝔽2)×C6712yes—
492C232−1+4×C6711yescharacteristic
493C232−1+4×C6711yescharacteristic
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—416no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C22—24no—
548C22—28no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22C2×C22⋊Q8×C6711yescharacteristic
559C22#529×C6712yes—
560C22#534×C6711yescharacteristic
561C22#536×C6711yescharacteristic
562C22#538×C6712yes—
563C22#538×C6712yes—
564C22#539×C6711yescharacteristic
565C22#539×C6712yes—
566C22#539×C6711yescharacteristic
567C22#540×C6712yes—
568C22#542×C6712yes—
569C22#544×C6711yescharacteristic
570C22#544×C6711yescharacteristic
571C22#546×C6712yes—
572C22#546×C6712yes—
573C22#551×C6711yescharacteristic
574C22#552×C6712yes—
575C22#554×C6712yes—
576C22#554×C6712yes—
577C22#559×C6711yescharacteristic
578C22#560×C6711yescharacteristic
579C22#560×C6711yescharacteristic
580C22#562×C6712yes—
581C4—44no—
582C4—44no—
583C4—42no—
584C4—44no—
585C4—42no—
586C4—82no—
587C4—82no—
588C4—82no—
589C2—44no—
590C2C67×#245012yes—
591C2C67×#251711yescharacteristic
592C2C67×#262711yescharacteristic
593C2C67×#264912yes—
594C2C67×#268712yes—
595C2C67×#269711yescharacteristic
596C2C67×#270911yescharacteristic
597C2C67×#274211yescharacteristic
598C2C67×#276812yes—
599C2C67×#280711yescharacteristic
600C2C67×#282611yescharacteristic
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
3. C67×#1442C2semidirect
4. C67×#1442C2semidirect
5. C67×#1441C2semidirect
6. C67×#1441C2semidirect
288. #16473C67direct (N × H)
303. #1442C2×C67semidirect
304. #1442C2×C67semidirect
305. #1441C2×C67semidirect
306. #1441C2×C67semidirect
310. C67#16473direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#399×C672
C2×#385×C672
C67×#14422
C67×#14412
C67×#14402
C67×#14392
C67×C22⋊#952
#1647367