Finite Group Databasebeta

← C4×#16864137 #31018521

Subgroups — C4×#16864137 #31018521

Order 17152.

2568 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864137C211yesmaximal, characteristic
2#22563401C211yesmaximal, characteristic, complemented
3#22563398C211yesmaximal, characteristic, complemented
4#22563385C211yesmaximal, characteristic, complemented
5#22563364C213yesmaximal, complemented
6#22563362C213yesmaximal, complemented
7#22563356C213yesmaximal, complemented
8#22563349C213yesmaximal, complemented
9C4×#14524040C211yesmaximal, characteristic, complemented
10C4×#14524037C211yesmaximal, characteristic, complemented
11C4×#14524028C213yesmaximal, complemented
12C4×#14524018C213yesmaximal, complemented
13C4×#14524016C213yesmaximal, complemented
14C4×#14524013C213yesmaximal, complemented
15C4×#126×C67C211yesmaximal, characteristic, complemented
16C23×C4×D67C2211yescharacteristic
17#16864137C4116yesdirect product factor
18C2×#14524040C2211yescharacteristic
19C2×#14524037C2211yescharacteristic
20C2×#14524028C2213yes—
21C2×#14524018C2213yes—
22C2×#14524016C2213yes—
23C2×#14524013C2213yes—
24C2×C4×D268C2213yes—
25C2×C4×D268C2213yes—
26C2×C4×D268C2213yescomplemented
27C2×C4×C22⋊D67C2213yes—
28C2×C42×D67C2213yescomplemented
29#16863298C2213yes—
30#16863298C2213yescomplemented
31#16863297C2213yes—
32#16863296C2213yes—
33#16863295C2213yescomplemented
34#16863293C2213yescomplemented
35#16863289C2213yes—
36#16863288C2213yes—
37#16863288C2213yescomplemented
38#16863287C2213yes—
39#16863286C2213yes—
40#16863285C2213yes—
41#16863285C2213yescomplemented
42#16863283C2213yescomplemented
43#16863282C2213yescomplemented
44#16863280C2213yescomplemented
45#16863274C2213yes—
46#16863274C2213yes—
47#16863274C2213yes—
48#16863274C2213yes—
49#16863274C2213yescomplemented
50#16863274C2213yescomplemented
51#16863273C2213yes—
52#16863272C2213yes—
53#16863271C2213yes—
54#16863270C2213yes—
55#16863270C2213yes—
56#16863267C2213yescomplemented
57#16863266C2211yescharacteristic, complemented
58#16863264C2211yescharacteristic, complemented
59#16863258C2213yescomplemented
60#16863255C2213yescomplemented
61#16863253C2213yescomplemented
62#16863251C2213yes—
63C4×Dic67⋊C4C2213yescomplemented
64C4×(C2×D67)⋊C4C2213yes—
65C4×(C2×D67)⋊C4C2213yes—
66C4×(C2×D67)⋊C4C2213yescomplemented
67C2×#126×C67C2211yescharacteristic
68#388×C67C2213yes—
69#387×C67C2213yescomplemented
70#383×C67C2211yescharacteristic, complemented
71C4×C4⋊C4×C67C2213yescomplemented
72C4×C22⋊C4×C67C2213yes—
73C43×C67C2211yescharacteristic, complemented
74C24×D67C2311yescharacteristic
75C22×D268C2313yes—
76C22×D268C2313yes—
77C22×D268C2313yes—
78C22×C22⋊D67C2313yes—
79C23×Dic67C2311yescharacteristic
80C22×C4×D67—46no—
81C22×C4×D67C2311yescharacteristic
82C22×C4×D67C2311yescharacteristic
83C22×C4×D67C2313yes—
84C22×C4×D67C2313yes—
85C22×C4×D67C2311yescharacteristic
86C22×C4×D67C2311yescharacteristic
87C22×C4×D67C2313yes—
88#14524040C2×C418yescomplemented
89#14524037C2×C418yescomplemented
90#14524028C2×C4124yescomplemented
91#14524018C2×C4124yescomplemented
92#14524016C2×C4124yescomplemented
93#14524013C2×C4124yescomplemented
94C4×D268—46no—
95C4×D268—46no—
96C4×D268—46no—
97C4×C22⋊D67—46no—
98C2×C4⋊Dic67C2313yes—
99C2×C4⋊Dic67C2313yes—
100C2×C4⋊Dic67C2313yes—
101C2×Dic67⋊C4C2313yes—
102C2×Dic67⋊C4C2313yes—
103C2×Dic67⋊C4C2313yes—
104C2×C22⋊Dic67C2313yes—
105C2×(C2×D67)⋊C4C2313yes—
106C2×(C2×D67)⋊C4C2313yes—
107C2×(C2×D67)⋊C4C2313yes—
108C2×(C2×D67)⋊C4C2313yes—
109C2×(C2×D67)⋊C4C2313yes—
110C2×(C2×D67)⋊C4C2313yes—
111C2×(C2×D67)⋊C4C2313yes—
112C2×(C2×D67)⋊C4C2313yes—
113C2×(C2×D67)⋊C4C2313yes—
114C2×(C2×D67)⋊C4C2313yes—
115C2×(C2×D67)⋊C4C2313yes—
116C2×(C2×D67)⋊C4C2313yes—
117C2×(C2×D67)⋊C4C2313yes—
118C2×C4×Dic67C2313yes—
119C2×C4×Dic67C2313yes—
120C2×C4×Dic67C2313yes—
121C42×D67—46no—
122C67⋊#95C2313yes—
123C67⋊#95C2313yes—
124C67⋊#95C2313yes—
125C67⋊#95C2313yes—
126C67⋊#95C2313yes—
127C67⋊#95C2313yes—
128C67⋊#95C2313yes—
129C67⋊#95C2313yes—
130C23×C4×C67C2311yescharacteristic
131#126×C67C2×C418yescomplemented
132C2×C4⋊C4×C67C2313yes—
133C2×C4⋊C4×C67C2313yes—
134C2×C22⋊C4×C67C2313yes—
135C2×C22⋊C4×C67C2313yes—
136C2×C42×C67C2313yes—
137C2×C42×C67C2313yes—
138C2×C42×C67C2313yes—
139C2×C42×C67C2311yescharacteristic
140#95×C67C2313yes—
141#95×C67C2313yes—
142#95×C67C2313yes—
143#95×C67C2313yes—
144C23×D67—43no—
145C23×D67—43no—
146C23×D67C22×C414yes—
147C23×D67C2411yescharacteristic
148C23×D67C2411yescharacteristic
149C23×D67C2413yes—
150C2×D268—43no—
151C2×D268—43no—
152C2×D268—43no—
153C2×D268—43no—
154C2×D268—43no—
155C2×D268—43no—
156C2×D268C22×C4112yes—
157C2×D268C22×C4112yes—
158C2×D268C22×C4112yescomplemented
159C2×C22⋊D67—43no—
160C2×C22⋊D67—43no—
161C2×C22⋊D67C22×C4112yes—
162C22×Dic67—43no—
163C22×Dic67C2413yes—
164C22×Dic67C2411yescharacteristic
165C22×Dic67C2411yescharacteristic
166C22×Dic67C2413yes—
167C22×Dic67C2413yes—
168C2×C4×D67—26no—
169C2×C4×D67—26no—
170C2×C4×D67—26no—
171C2×C4×D67—26no—
172C2×C4×D67—26no—
173C2×C4×D67—43no—
174C2×C4×D67—43no—
175C2×C4×D67—43no—
176C2×C4×D67—43no—
177C2×C4×D67—43no—
178C2×C4×D67—43no—
179C2×C4×D67—43no—
180C2×C4×D67—416no—
181C2×C4×D67C22×C4112yescomplemented
182C4⋊Dic67—43no—
183C4⋊Dic67—43no—
184C4⋊Dic67—43no—
185Dic67⋊C4—43no—
186Dic67⋊C4—43no—
187Dic67⋊C4C22×C4112yescomplemented
188C22⋊Dic67—43no—
189(C2×D67)⋊C4—43no—
190(C2×D67)⋊C4—43no—
191(C2×D67)⋊C4—43no—
192(C2×D67)⋊C4—43no—
193(C2×D67)⋊C4—43no—
194(C2×D67)⋊C4—43no—
195(C2×D67)⋊C4—43no—
196(C2×D67)⋊C4—43no—
197(C2×D67)⋊C4—43no—
198(C2×D67)⋊C4—43no—
199(C2×D67)⋊C4C22×C4112yes—
200(C2×D67)⋊C4C22×C4112yes—
201(C2×D67)⋊C4C22×C4112yescomplemented
202C4×Dic67—26no—
203C4×Dic67—43no—
204C4×Dic67—43no—
205C24×C67C2411yescharacteristic
206C22×C4×C67—26no—
207C22×C4×C67C2413yes—
208C22×C4×C67C2413yes—
209C22×C4×C67C2411yescharacteristic
210C22×C4×C67C2413yes—
211C22×C4×C67C2413yes—
212C22×C4×C67C2411yescharacteristic
213C4⋊C4×C67—43no—
214C4⋊C4×C67C22×C4112yescomplemented
215C22⋊C4×C67—43no—
216C22⋊C4×C67C22×C4112yes—
217C42×C67—26no—
218C42×C67—26no—
219C42×C67—43no—
220C42×C67C22×C414yescomplemented
221C23×C67—23no—
222C23×C67—23no—
223C23×C67C23×C412yes—
224C23×C67C2511yescharacteristic
225C22×D67—23no—
226C22×D67—23no—
227C22×D67—23no—
228C22×D67—23no—
229C22×D67—23no—
230C22×D67—23no—
231C22×D67—23no—
232C22×D67—23no—
233C22×D67—23no—
234C22×D67—23no—
235C22×D67—412no—
236C22×D67—44no—
237C22×D67—412no—
238C22×D67C23×C412yes—
239C22×D67C23×C412yes—
240C22×D67C23×C416yes—
241C2×C4×C67—23no—
242C2×C4×C67—23no—
243C2×C4×C67—23no—
244C2×C4×C67—23no—
245C2×C4×C67—23no—
246C2×C4×C67—23no—
247C2×C4×C67—23no—
248C2×C4×C67—23no—
249C2×C4×C67—23no—
250C2×C4×C67—23no—
251C2×C4×C67—23no—
252C2×C4×C67—44no—
253C2×C4×C67C23×C416yes—
254C2×C4×C67C23×C416yes—
255C2×C4×C67H(5,𝔽2)16yes—
256C22⋊D67—412no—
257C2×Dic67—23no—
258C2×Dic67—23no—
259C2×Dic67—23no—
260C2×Dic67—23no—
261C2×Dic67—23no—
262C2×Dic67—23no—
263C2×Dic67—23no—
264C2×Dic67—23no—
265C2×Dic67—23no—
266C2×Dic67—23no—
267C2×Dic67—44no—
268C2×Dic67C23×C416yes—
269D268—412no—
270D268—412no—
271D268—412no—
272C4×D67—412no—
273C4×D67—412no—
274C4×D67—44no—
275C4×D67—44no—
276C4×D67—412no—
277C4×D67—44no—
278C4×D67—44no—
279C22×C67—26no—
280C22×C67—41no—
281C22×C67—43no—
282C22×C67C24×C411yescharacteristic
283C22×C67C2×H(5,𝔽2)13yes—
284C22×C67#58513yes—
285C2×D67—26no—
286C2×D67—26no—
287C2×D67—26no—
288C2×D67—26no—
289C2×D67—26no—
290C2×D67—416no—
291C2×D67—41no—
292C2×D67—41no—
293C2×D67—43no—
294C2×D67—43no—
295C2×D67—43no—
296C2×D67—43no—
297C2×D67—43no—
298C2×D67—43no—
299C4×C67—26no—
300C4×C67—26no—
301C4×C67—41no—
302C4×C67—43no—
303C4×C67—43no—
304C4×C67#56014yescomplemented
305Dic67—26no—
306Dic67—41no—
307Dic67—41no—
308Dic67—43no—
309Dic67—43no—
310C4×#560—671nomaximal, Sylow
311C2×C67—42no—
312C2×C67C4×H(5,𝔽2)13yes—
313C2×C67C2×#56011yescharacteristic
314C2×C67#352913yes—
315D67—46no—
316D67—42no—
317D67—42no—
318C2×#560—671no—
319#2383—671no—
320#2383—671no—
321#2380—671no—
322#2377—673no—
323#2377—673no—
324#2376—673no—
325#2375—673no—
326C4×#127—671no—
327C4×#126—671no—
328C4×#126D6711yescharacteristic, complemented
329C4×#123—673no—
330C4×#121—673no—
331C4×#121—673no—
332C4×#120—673no—
333C67C4×#56011yescharacteristic, complemented, Sylow
334C24×C4—671no—
335#560—6716no—
336C2×#127—671no—
337C2×#126—671no—
338C2×#126C2×D6711yescharacteristic
339C2×#123—673no—
340C2×#121—673no—
341C2×#121—673no—
342C2×#120—673no—
343C2×C4×D4—673no—
344C2×C4×D4—673no—
345C2×C4×D4—673no—
346C2×C4×D4—673no—
347C22×C42—673no—
348#390—673no—
349#390—673no—
350#390—673no—
351#390—673no—
352#389—673no—
353#389—673no—
354#388—673no—
355#388—673no—
356#388—673no—
357#388—673no—
358#388—673no—
359#388—673no—
360#388—673no—
361#388C2×D6713yes—
362#387—673no—
363#387—673no—
364#387—673no—
365#387C2×D6713yescomplemented
366#386—673no—
367#386—673no—
368#386—673no—
369#386—673no—
370#386—673no—
371#386—673no—
372#386—673no—
373#386—673no—
374#385—673no—
375#385—673no—
376#385—673no—
377#385—673no—
378#384—671no—
379#383—671no—
380#383C2×D6711yescharacteristic, complemented
381#381—673no—
382#380—673no—
383#380—673no—
384#379—673no—
385C4×C4⋊C4—673no—
386C4×C4⋊C4C2×D6713yescomplemented
387C4×C22⋊C4—673no—
388C4×C22⋊C4—673no—
389C4×C22⋊C4—673no—
390C4×C22⋊C4C2×D6713yes—
391C43C2×D6711yescharacteristic, complemented
392C25—671no—
393C22×D4—673no—
394C22×D4—673no—
395C22×D4—673no—
396C22×D4—673no—
397C23×C4—2686no—
398C23×C4—671no—
399C23×C4—671no—
400C23×C4—671no—
401C23×C4—673no—
402C23×C4—673no—
403C23×C4—673no—
404C23×C4—671no—
405C23×C4—671no—
406C23×C4C22×D6711yescharacteristic
407#127—678no—
408#126—678no—
409#126C4×D6718yescomplemented
410#123—6724no—
411#121—6724no—
412#121—6724no—
413#120—6724no—
414C4×D4—2686no—
415C4×D4—2686no—
416C4×D4—2686no—
417C4×D4—2686no—
418C2×C4⋊C4—673no—
419C2×C4⋊C4—673no—
420C2×C4⋊C4—673no—
421C2×C4⋊C4—673no—
422C2×C4⋊C4—673no—
423C2×C4⋊C4—673no—
424C2×C4⋊C4C22×D6713yes—
425C2×C4⋊C4C22×D6713yes—
426C2×C22⋊C4—673no—
427C2×C22⋊C4—673no—
428C2×C22⋊C4—673no—
429C2×C22⋊C4—673no—
430C2×C22⋊C4—673no—
431C2×C22⋊C4—673no—
432C2×C22⋊C4—673no—
433C2×C22⋊C4—673no—
434C2×C22⋊C4—673no—
435C2×C22⋊C4—673no—
436C2×C22⋊C4—673no—
437C2×C22⋊C4—673no—
438C2×C22⋊C4—673no—
439C2×C22⋊C4—673no—
440C2×C22⋊C4C22×D6713yes—
441C2×C22⋊C4C22×D6713yes—
442C2×C42—2686no—
443C2×C42—673no—
444C2×C42—673no—
445C2×C42—673no—
446C2×C42C22×D6713yes—
447C2×C42C22×D6713yes—
448C2×C42C22×D6713yes—
449C2×C42C22×D6711yescharacteristic
450#95—673no—
451#95—673no—
452#95—673no—
453#95—673no—
454#95—673no—
455#95—673no—
456#95—673no—
457#95—673no—
458#95C22×D6713yes—
459#95C22×D6713yes—
460#95C22×D6713yes—
461#95C22×D6713yes—
462C24—2683no—
463C24—2683no—
464C24—674no—
465C24—671no—
466C24—671no—
467C24—673no—
468C24C23×D6711yescharacteristic
469C2×D4—2683no—
470C2×D4—2683no—
471C2×D4—2683no—
472C2×D4—2683no—
473C2×D4—2683no—
474C2×D4—2683no—
475C2×D4—2683no—
476C2×D4—2683no—
477C2×D4—6712no—
478C2×D4—6712no—
479C2×D4—6712no—
480C2×D4—6712no—
481C22×C4—26no—
482C22×C4—2683no—
483C22×C4—2683no—
484C22×C4—2683no—
485C22×C4—2683no—
486C22×C4—2683no—
487C22×C4—2683no—
488C22×C4—2683no—
489C22×C4—26816no—
490C22×C4—2683no—
491C22×C4—6712no—
492C22×C4—673no—
493C22×C4—673no—
494C22×C4—671no—
495C22×C4—671no—
496C22×C4—673no—
497C22×C4—1346no—
498C22×C4—1346no—
499C22×C4—1346no—
500C22×C4—1346no—
501C22×C4—1346no—
502C22×C4C23×D6713yes—
503C22×C4C23×D6713yes—
504C22×C4C23×D6711yescharacteristic
505C22×C4C23×D6713yes—
506C22×C4C23×D6713yes—
507C22×C4C23×D6711yescharacteristic
508C4⋊C4—43no—
509C4⋊C4—2683no—
510C4⋊C4—2683no—
511C4⋊C4—2683no—
512C4⋊C4—2683no—
513C4⋊C4—2683no—
514C4⋊C4—6712no—
515C4⋊C4C2×C4×D67112yescomplemented
516C22⋊C4—43no—
517C22⋊C4—2683no—
518C22⋊C4—2683no—
519C22⋊C4—2683no—
520C22⋊C4—2683no—
521C22⋊C4—2683no—
522C22⋊C4—2683no—
523C22⋊C4—2683no—
524C22⋊C4—2683no—
525C22⋊C4—2683no—
526C22⋊C4—2683no—
527C22⋊C4—2683no—
528C22⋊C4—6712no—
529C22⋊C4—6712no—
530C22⋊C4—6712no—
531C22⋊C4C2×C4×D67112yes—
532C42—26no—
533C42—26no—
534C42—43no—
535C42—2683no—
536C42—2683no—
537C42—1346no—
538C42C2×C4×D6714yescomplemented
539C23—23no—
540C23—23no—
541C23—26812no—
542C23—2684no—
543C23—26812no—
544C23—672no—
545C23—672no—
546C23—676no—
547C23—1343no—
548C23—1343no—
549C23—1343no—
550C23—1343no—
551C23—1343no—
552C23—1343no—
553C23—1343no—
554C23—1343no—
555C23—1343no—
556C23—1343no—
557C23C22×C4×D6712yes—
558C23C24×D6711yescharacteristic
559D4—26812no—
560D4—26812no—
561D4—26812no—
562D4—26812no—
563C2×C4—23no—
564C2×C4—23no—
565C2×C4—23no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—44no—
575C2×C4—26812no—
576C2×C4—2684no—
577C2×C4—2684no—
578C2×C4—26812no—
579C2×C4—2684no—
580C2×C4—2684no—
581C2×C4—2684no—
582C2×C4—26812no—
583C2×C4—676no—
584C2×C4—1343no—
585C2×C4—1343no—
586C2×C4—1343no—
587C2×C4—1343no—
588C2×C4—1343no—
589C2×C4—1343no—
590C2×C4—1343no—
591C2×C4—1343no—
592C2×C4—1343no—
593C2×C4—1343no—
594C2×C4C22×C4×D6716yes—
595C2×C4C22×C4×D6716yes—
596C2×C4#1452408916yes—
597C22—26no—
598C22—43no—
599C22—41no—
600C22—26816no—
601C22—2683no—
602C22—2683no—
603C22—2683no—
604C22—2683no—
605C22—2683no—
606C22—2681no—
607C22—2681no—
608C22—2683no—
609C22—1346no—
610C22—1346no—
611C22—1346no—
612C22—1346no—
613C22—1346no—
614C22C23×C4×D6711yescharacteristic
615C22C2×#1452408913yes—
616C22#1686423213yes—
617C4—26no—
618C4—26no—
619C4—43no—
620C4—43no—
621C4—41no—
622C4—2683no—
623C4—2681no—
624C4—2681no—
625C4—2683no—
626C4—1346no—
627C4#1686413714yesdirect product factor
628C2—42no—
629C2—2682no—
630C2—2682no—
631C2—2686no—
632C2C4×#1452408913yes—
633C2C2×#1686413711yescharacteristic
634C2#2257434813yes—
subgroupG/H
centerC22×C4C23×D67
commutatorC22×C67C24×C4
FrattiniC23C24×D67
FittingC4×#126×C67C2
socle, cosocleC23×C67C25
prime radicalC2×#16864137C2
hypercenterC4×#126D67
hypocenterC67C4×#560
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C4×#126

Complements

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

NH ≅ G/Ntype
2. #22563401C2semidirect
3. #22563398C2semidirect
4. #22563385C2semidirect
5. #22563364C2semidirect
6. #22563362C2semidirect
7. #22563356C2semidirect
8. #22563349C2semidirect
9. C4×#14524040C2semidirect
10. C4×#14524037C2semidirect
11. C4×#14524028C2semidirect
12. C4×#14524018C2semidirect
13. C4×#14524016C2semidirect
14. C4×#14524013C2semidirect
15. C4×#126×C67C2semidirect
17. #16864137C4direct (N × H)
26. C2×C4×D268C22semidirect
28. C2×C42×D67C22semidirect
30. #16863298C22semidirect
33. #16863295C22semidirect
34. #16863293C22semidirect
37. #16863288C22semidirect
41. #16863285C22semidirect
42. #16863283C22semidirect
43. #16863282C22semidirect
44. #16863280C22semidirect
49. #16863274C22semidirect
50. #16863274C22semidirect
56. #16863267C22semidirect
57. #16863266C22semidirect
58. #16863264C22semidirect
59. #16863258C22semidirect
60. #16863255C22semidirect
61. #16863253C22semidirect
63. C4×Dic67⋊C4C22semidirect
66. C4×(C2×D67)⋊C4C22semidirect
69. #387×C67C22semidirect
70. #383×C67C22semidirect
71. C4×C4⋊C4×C67C22semidirect
73. C43×C67C22semidirect
88. #14524040C2×C4semidirect
89. #14524037C2×C4semidirect
90. #14524028C2×C4semidirect
91. #14524018C2×C4semidirect
92. #14524016C2×C4semidirect
93. #14524013C2×C4semidirect
131. #126×C67C2×C4semidirect
158. C2×D268C22×C4semidirect
181. C2×C4×D67C22×C4semidirect
187. Dic67⋊C4C22×C4semidirect
201. (C2×D67)⋊C4C22×C4semidirect
214. C4⋊C4×C67C22×C4semidirect
220. C42×C67C22×C4semidirect
304. C4×C67#560semidirect
328. C4×#126D67semidirect
333. C67C4×#560semidirect
365. #387C2×D67semidirect
380. #383C2×D67semidirect
386. C4×C4⋊C4C2×D67semidirect
391. C43C2×D67semidirect
409. #126C4×D67semidirect
515. C4⋊C4C2×C4×D67semidirect
538. C42C2×C4×D67semidirect
627. C4#16864137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168641372
#225634012
#225633982
#225633852
#225633642
#225633622
#225633562
#225633492
C4×#145240402
C4×#145240372
C4×#145240282
C4×#145240182
C4×#145240162
C4×#145240132
C4×#126×C672
C4×#56067