Finite Group Databasebeta

← C4×#16810981 #27392150

Subgroups — C4×#16810981 #27392150

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810981C211yesmaximal, characteristic
2C4×#13133216C211yesmaximal, characteristic, complemented
3C4×#13133204C211yesmaximal, characteristic
4#20275686C211yesmaximal, characteristic, complemented
5#20275530C211yesmaximal, characteristic
6#20275414C211yesmaximal, characteristic
7#20275409C211yesmaximal, characteristic
8#20275406C211yesmaximal, characteristic, complemented
9#20275403C211yesmaximal, characteristic, complemented
10#20274847C211yesmaximal, characteristic, complemented
11C4×#13133140C211yesmaximal, characteristic, complemented
12C4×#13133142C211yesmaximal, characteristic, complemented
13C4×#13133138C211yesmaximal, characteristic
14C4×#13133127C211yesmaximal, characteristic, complemented
15C4×#102×C53C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133204C2211yescharacteristic
18C2×C4×Dic106—21no—
19C2×C4×Dic106C2211yescharacteristic
20#16810981C418yesdirect product factor
21C2×#13133142C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810528C2211yescharacteristic
33#16810527—21no—
34#16810526C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810520—21no—
37#16810519C2211yescharacteristic
38#16810517—21no—
39#16810517—21no—
40#16810492C2211yescharacteristic
41#16810488C2211yescharacteristic
42C4×#11719919—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C22⋊Dic53—21no—
45C4×C22⋊Dic53—21no—
46C42×Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106—21no—
61C22×Dic106C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216C2×C414yescomplemented
66#13133204—22no—
67#13133204C2×C414yes—
68C4×Dic106—22no—
69C4×Dic106—22no—
70C4×Dic106—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133142—22no—
93#13133142C2×C414yescomplemented
94#13133140—22no—
95#13133140C2×C414yescomplemented
96#13133138—22no—
97#13133138C2×C414yes—
98#13133127—22no—
99#13133127—22no—
100#13133127—22no—
101#13133127C2×C414yescomplemented
102C4×#88322—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×D4×C53C2311yescharacteristic
111C23×C4×C53—21no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C4×D4×C53—22no—
115C4×D4×C53—22no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122#102×C53—22no—
123#102×C53C2×C414yes—
124C4×C8×C53—22no—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C22×C412yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—24no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×Dic106C22×C412yes—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C2×C8×C53#4112yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53C22×C412yes—
171C22⋊C4×C53—42no—
172C42×C53—42no—
173C42×C53#4112yes—
174C42×C53#4112yes—
175C22⋊Dic53—24no—
176C22⋊Dic53—24no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53C22×C412yes—
186#11719919—24no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—41no—
196C4×Dic53—41no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53C22×C412yes—
200C4×Dic53#4112yes—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322C22×C412yes—
206C2×#88322#4112yes—
207C23×C53—22no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53C22×D411yescharacteristic
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—41no—
218D4×C53—41no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
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—84no—
231C2×C4×C53#12412yes—
232C2×C4×C53#12412yes—
233C2×C4×C53AGL(1,ℤ/8)12yes—
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—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53C4×D412yes—
264C2×Dic53C4×D412yes—
265Dic106—22no—
266Dic106—22no—
267Dic106—48no—
268Dic106—41no—
269Dic106—41no—
270C8×C53—22no—
271C8×C53—22no—
272#88322—22no—
273#88322—22no—
274C4×#489—531nomaximal, Sylow
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53C2×C4×D411yescharacteristic
283C22×C53C2×#12411yescharacteristic
284C22×C53#53211yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53C2×#13711yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57511yescharacteristic
289C4×C53—22no—
290C4×C53—42no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53#48914yescomplemented
294C4×C53C4×#4111yescharacteristic
295C4×C53C4×#4111yescharacteristic
296C4×C53#52911yescharacteristic
297C4×C53#52911yescharacteristic
298Dic53—22no—
299Dic53—22no—
300Dic53—41no—
301Dic53—41no—
302Dic53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53—81no—
306C2×#489—531no—
307C4×C4⋊Q8—531no—
308C4×#124—531no—
309#1993—531no—
310#1959—531no—
311#1936—531no—
312#1936—531no—
313#1935—531no—
314#1935—531no—
315#1836—531no—
316C4×#103—531no—
317C4×#103—531no—
318C4×#102—531no—
319C4×#102D5311yescharacteristic
320C4×#97—531no—
321C2×C53—82no—
322C2×C53C4×#12411yescharacteristic
323C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
324C2×C53C4×#13711yescharacteristic
325C2×C53C2×#48911yescharacteristic
326C2×C53#313511yescharacteristic
327C2×C53#317111yescharacteristic
328C2×C53#317211yescharacteristic
329C2×C4⋊Q8—531no—
330C2×#124—531no—
331C2×C4×Q8—531no—
332C2×C4×Q8—1061no—
333C2×C4×D4C2×D5311yescharacteristic
334#489—538no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×#103—531no—
339C2×#102—531no—
340C2×#102—531no—
341C2×#102C2×D5311yescharacteristic
342C2×#102C2×D5311yescharacteristic
343C2×#97—531no—
344C2×#97—531no—
345C2×#97—531no—
346C2×#97—531no—
347C2×C4×C8—531no—
348C2×C4×C8C2×D5311yescharacteristic
349#391—531no—
350#391—1061no—
351#390—531no—
352#389—531no—
353#389—531no—
354#389—1061no—
355#388—1061no—
356#388—1061no—
357#384—531no—
358#382—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—1061no—
361C4×C4⋊C4C2×D5311yescharacteristic
362C4×C22⋊C4—1061no—
363C4×C22⋊C4—1061no—
364C43—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373C53C4×#48911yescharacteristic, complemented, Sylow
374C22×Q8—531no—
375C22×Q8—1061no—
376C22×D4C22×D5311yescharacteristic
377C23×C4—21no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C4⋊Q8—534no—
383C4⋊Q8—1062no—
384#124—534no—
385#124—1062no—
386C4×Q8—1062no—
387C4×Q8—1062no—
388C4×Q8—2124no—
389C4×D4—22no—
390C4×D4—22no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418#103—534no—
419#103—534no—
420#103—1062no—
421#103—1062no—
422#102—22no—
423#102—534no—
424#102—1062no—
425#102C4×D5314yes—
426#97—534no—
427#97—1062no—
428#97—1062no—
429#97—1062no—
430C4×C8—22no—
431C4×C8—1062no—
432#95—21no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—21no—
441C2×Q8—532no—
442C2×Q8—1064no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—2122no—
449C2×Q8—2122no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C4×D5312yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—532no—
478C2×C8—532no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8C2×C4×D5312yes—
484C2×C8#1171994512yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—532no—
491C4⋊C4—1064no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4C2×C4×D5312yes—
503C22⋊C4—42no—
504C22⋊C4—1064no—
505C22⋊C4—1064no—
506C22⋊C4—2122no—
507C22⋊C4—2122no—
508C42—42no—
509C42—532no—
510C42—532no—
511C42—1062no—
512C42—1062no—
513C42—1061no—
514C42—1061no—
515C42—2121no—
516C42—2121no—
517C42—2122no—
518C42—2122no—
519C42#1171994512yes—
520C42#1171994712yes—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×D4×D5311yescharacteristic
529Q8—1062no—
530Q8—1062no—
531Q8—2128no—
532Q8—2121no—
533Q8—2121no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—84no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4#1313320312yes—
575C2×C4#1313320312yes—
576C2×C4#1313324112yes—
577C2×C4#1313324612yes—
578C2×C4C22×C4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C8—22no—
585C8—22no—
586C8—1062no—
587C8—1062no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22C4×D4×D5311yescharacteristic
596C22C2×#1313320311yescharacteristic
597C22#1681117011yescharacteristic
598C22C2×#1313324111yescharacteristic
599C22C2×#1313324611yescharacteristic
600C22#1681142011yescharacteristic
601C22#1681142011yescharacteristic
602C4—22no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4—1062no—
607C4—1062no—
608C4—4241no—
609C4—4241no—
610C4—2122no—
611C4—2121no—
612C4—2122no—
613C4—2121no—
614C4#1681098114yesdirect product factor
615C4C4×#1171994711yescharacteristic
616C4C4×#1171994511yescharacteristic
617C4#1681115111yescharacteristic
618C4#1681115311yescharacteristic
619C2—82no—
620C2C4×#1313320311yescharacteristic
621C2C4×#1313324111yescharacteristic
622C2C4×#1313324611yescharacteristic
623C2C2×#1681098111yescharacteristic
624C2#2028505811yescharacteristic
625C2#2028530811yescharacteristic
626C2#2028531711yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#102×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC4×#102D53
hypocenterC53C4×#489
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C4×#13133216C2semidirect
4. #20275686C2semidirect
8. #20275406C2semidirect
9. #20275403C2semidirect
10. #20274847C2semidirect
11. C4×#13133140C2semidirect
12. C4×#13133142C2semidirect
14. C4×#13133127C2semidirect
20. #16810981C4direct (N × H)
65. #13133216C2×C4semidirect
93. #13133142C2×C4semidirect
95. #13133140C2×C4semidirect
101. #13133127C2×C4semidirect
293. C4×C53#489semidirect
373. C53C4×#489semidirect
614. C4#16810981direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109812
C4×#131332162
C4×#131332042
#202756862
#202755302
#202754142
#202754092
#202754062
#202754032
#202748472
C4×#131331402
C4×#131331422
C4×#131331382
C4×#131331272
C4×#102×C532
C4×#48953