Finite Group Databasebeta

← C4×#16810962 #27391256

Subgroups — C4×#16810962 #27391256

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810962C211yesmaximal, characteristic
2C4×#13133193C211yesmaximal, characteristic
3C4×#13133175C211yesmaximal, characteristic
4#20275359C211yesmaximal, characteristic, complemented
5#20275351C211yesmaximal, characteristic, complemented
6#20275292C211yesmaximal, characteristic, complemented
7#20275284C211yesmaximal, characteristic, complemented
8#20275024C211yesmaximal, characteristic
9#20274915C211yesmaximal, characteristic
10#20274808C211yesmaximal, characteristic
11C4×#13133150C211yesmaximal, characteristic, complemented
12C4×#13133149C211yesmaximal, characteristic, complemented
13C4×#13133141C211yesmaximal, characteristic, complemented
14C4×#13133129C211yesmaximal, characteristic
15C4×#103×C53C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×#13133175C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×C42×D53C2211yescharacteristic
20#16810962C418yesdirect product factor
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810530C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810504—21no—
33#16810494—21no—
34#16810480C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4×#11719919—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×#11719921—21no—
41C4×#11719921—21no—
42#16810470—21no—
43#16810468—21no—
44C42×Dic53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133193—22no—
66#13133193C2×C414yes—
67#13133175—22no—
68#13133175—22no—
69#13133175—22no—
70#13133175C2×C414yes—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C42×D53—22no—
91C42×D53—22no—
92#13133150—22no—
93#13133150C2×C414yescomplemented
94#13133149—22no—
95#13133149C2×C414yescomplemented
96#13133141—22no—
97#13133141C2×C414yescomplemented
98#13133129—22no—
99#13133129C2×C414yes—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×Q8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C4×Q8×C53—22no—
113C4×Q8×C53—22no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#103×C53—22no—
122#103×C53C2×C414yescomplemented
123C4×C8×C53—22no—
124#95×C53—21no—
125C23×D53C2×D411yescharacteristic
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53C22×C412yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—21no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—41no—
156C2×C4×D53C22×C412yes—
157C2×C4×D53#4112yes—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53C22×C412yes—
170C4⋊C4×C53#4112yes—
171C42×C53—42no—
172C42×C53—42no—
173C42×C53#4112yes—
174C42×C53#4112yes—
175#11719921—24no—
176#11719921—24no—
177#11719921—24no—
178#11719921—42no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53C22×C412yes—
186C4⋊Dic53#4112yes—
187#11719919—24no—
188#11719919—42no—
189C4×Dic53—24no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—24no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C412yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53C4×D412yes—
213Q8×C53—22no—
214Q8×C53—22no—
215Q8×C53—41no—
216Q8×C53—41no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53#12312yes—
232C2×C4×C53#12312yes—
233C2×C4×C53#13512yes—
234C2×C4×C53#13712yes—
235C2×C4×C53C23×C411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53C4×D412yes—
261C2×Dic53#13512yes—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—22no—
266C4×D53—44no—
267C4×D53—44no—
268C8×C53—22no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272C4×#485—531nomaximal, Sylow
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×#12311yescharacteristic
275C22×C53#52511yescharacteristic
276C22×C53C2×#13511yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×#13711yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—41no—
286C2×D53—41no—
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—42no—
290C4×C53—42no—
291C4×C53—81no—
292C4×C53#48514yescomplemented
293C4×C53C4×#4111yescharacteristic
294C4×C53C4×#4111yescharacteristic
295C4×C53#52911yescharacteristic
296C4×C53#52911yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—41no—
300Dic53—42no—
301Dic53—42no—
302Dic53—42no—
303Dic53—81no—
304C2×#485—531no—
305C4×C22⋊Q8—531no—
306C4×#117—531no—
307#1928—531no—
308#1927—531no—
309#1919—531no—
310#1918—531no—
311#1873—531no—
312#1854—531no—
313#1825—531no—
314C4×#107—531no—
315C4×#106—531no—
316C4×#103—531no—
317C4×#103D5311yescharacteristic, complemented
318C4×C22⋊C8—531no—
319C2×C53C4×#12311yescharacteristic
320C2×C53C4×#13511yescharacteristic
321C2×C53C4×#13711yescharacteristic
322C2×C53C2×#48511yescharacteristic
323C2×C53#309111yescharacteristic
324C2×C53#310211yescharacteristic
325C2×C53#310711yescharacteristic
326D53—42no—
327C2×C22⋊Q8—531no—
328C2×C4×Q8C2×D5311yescharacteristic
329C2×#117—531no—
330C2×#117—531no—
331C22×C42—531no—
332#485—538no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#106—531no—
336C2×#106—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#103C2×D5311yescharacteristic
340C2×#103C2×D5311yescharacteristic
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C4×C8—531no—
344C2×C4×C8C2×D5311yescharacteristic
345#391—531no—
346#389—531no—
347#387—1061no—
348#385—1061no—
349#380—531no—
350C4×C4⋊C4—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—1061no—
355C4×C4⋊C4C2×D5311yescharacteristic
356C4×C22⋊C4—1061no—
357C4×C22⋊C4—1061no—
358C4×C22⋊C4—1061no—
359#376—1061no—
360C43—1061no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371C53C4×#48511yescharacteristic, complemented, Sylow
372C22×Q8C22×D5311yescharacteristic
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C22⋊Q8—534no—
381C22⋊Q8—1062no—
382C4×Q8—22no—
383C4×Q8—22no—
384#117—534no—
385#117—1062no—
386#117—1062no—
387#117—1062no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42—1062no—
410C2×C42—1062no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42C22×D5311yescharacteristic
416#107—534no—
417#107—1062no—
418#106—534no—
419#106—1062no—
420#103—22no—
421#103—534no—
422#103—1062no—
423#103C4×D5314yescomplemented
424C22⋊C8—534no—
425C22⋊C8—1062no—
426C4×C8—22no—
427C4×C8—1062no—
428#95—21no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24—531no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D5312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—531no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1061no—
466C22×C4—2121no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8C2×C4×D5312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—532no—
490C4⋊C4—1064no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—2122no—
497C4⋊C4C2×C4×D5312yes—
498C4⋊C4#1171994512yes—
499C22⋊C4—1064no—
500C22⋊C4—1064no—
501C22⋊C4—1064no—
502C22⋊C4—2122no—
503C42—42no—
504C42—42no—
505C42—1064no—
506C42—1064no—
507C42—1062no—
508C42—1062no—
509C42—1061no—
510C42—1061no—
511C42—2122no—
512C42—2122no—
513C42—2122no—
514C42—2122no—
515C42#1171994512yes—
516C42#1171994712yes—
517C23—532no—
518C23—1061no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23C2×D4×D5311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—532no—
544C2×C4—532no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1061no—
556C2×C4—1062no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—2124no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2124no—
570C2×C4#1313319912yes—
571C2×C4#1313319912yes—
572C2×C4#1313323812yes—
573C2×C4#1313324912yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4#41×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4C2×#1171994711yescharacteristic
578C2×C4C2×#1171994511yescharacteristic
579C2×C4C2×#1171994511yescharacteristic
580C8—22no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—1062no—
585C22—1062no—
586C22—1062no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22—2121no—
591C22C4×D4×D5311yescharacteristic
592C22C2×#1313319911yescharacteristic
593C22#1681113211yescharacteristic
594C22#135×D5311yescharacteristic
595C22C2×#1313323811yescharacteristic
596C22C2×#1313324911yescharacteristic
597C22#1681142011yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—1062no—
604C4—1062no—
605C4—4241no—
606C4—2122no—
607C4—2122no—
608C4—2122no—
609C4—2121no—
610C4#1681096214yesdirect product factor
611C4C4×#1171994711yescharacteristic
612C4C4×#1171994511yescharacteristic
613C4#1681115111yescharacteristic
614C4#1681115511yescharacteristic
615C2—2122no—
616C2C4×#1313319911yescharacteristic
617C2C4×#1313323811yescharacteristic
618C2C4×#1313324911yescharacteristic
619C2C2×#1681096211yescharacteristic
620C2#2028462611yescharacteristic
621C2#2028473711yescharacteristic
622C2#2028478411yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#103×C53C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenterC4×#103D53
hypocenterC53C4×#485
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
4. #20275359C2semidirect
5. #20275351C2semidirect
6. #20275292C2semidirect
7. #20275284C2semidirect
11. C4×#13133150C2semidirect
12. C4×#13133149C2semidirect
13. C4×#13133141C2semidirect
15. C4×#103×C53C2semidirect
20. #16810962C4direct (N × H)
93. #13133150C2×C4semidirect
95. #13133149C2×C4semidirect
97. #13133141C2×C4semidirect
122. #103×C53C2×C4semidirect
292. C4×C53#485semidirect
317. C4×#103D53semidirect
371. C53C4×#485semidirect
423. #103C4×D53semidirect
610. C4#16810962direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109622
C4×#131331932
C4×#131331752
#202753592
#202753512
#202752922
#202752842
#202750242
#202749152
#202748082
C4×#131331502
C4×#131331492
C4×#131331412
C4×#131331292
C4×#103×C532
C4×#48553