Finite Group Databasebeta

← C2×#20275906 #27411558

Subgroups — C2×#20275906 #27411558

Order 13568.

2006 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
1C22×#13133192C211yesmaximal, characteristic
2C22×#13133139C211yesmaximal, characteristic, complemented
3C22×#13133137C211yesmaximal, characteristic, complemented
4C22×#13133129C211yesmaximal, characteristic, complemented
5C2×#16810480C211yesmaximal, characteristic, complemented
6#20275906C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#106×C53C211yesmaximal, characteristic, complemented
9C23×D212—21no—
10C23×D212C2211yescharacteristic
11C23×C4×D53C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13C2×#13133192—24no—
14C2×#13133192—22no—
15C2×#13133192—24no—
16C2×#13133192C2214yes—
17C22×C4⋊Dic53C2211yescharacteristic
18C22×#11719921—21no—
19C22×#11719921—21no—
20C2×#13133139—24no—
21C2×#13133139C2214yescomplemented
22C2×#13133137—24no—
23C2×#13133137C2214yescomplemented
24C2×#13133129—24no—
25C2×#13133129C2214yescomplemented
26#16810480—22no—
27#16810480C2214yescomplemented
28C2×#13133124—21no—
29#16810385C2214yescomplemented
30C23×C8×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C2×#106×C53—24no—
34C2×#106×C53C2214yescomplemented
35C24×D53—21no—
36C24×D53D411yescharacteristic
37C22×D212—22no—
38C22×D212—22no—
39C22×D212—22no—
40C22×D212—22no—
41C22×D212—28no—
42C22×D212—24no—
43C22×D212—21no—
44C22×D212—22no—
45C22×D212—44no—
46C22×D212—44no—
47C22×D212D414yes—
48C22×D212C2312yes—
49C23×Dic53—21no—
50C23×Dic53D411yescharacteristic
51C22×C4×D53—22no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53D412yes—
58C22×C4×D53C2312yes—
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322C2312yes—
63#13133192—216no—
64#13133192—28no—
65#13133192—28no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53—22no—
69C2×C4⋊Dic53C2312yes—
70C2×#11719921—28no—
71C2×#11719921—24no—
72C2×#11719921—24no—
73C2×#11719921—44no—
74#13133139—28no—
75#13133137—28no—
76#13133129—28no—
77#13133124—24no—
78C23×C4×C53—21no—
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C22×C8×C53—22no—
82C22×C8×C53—22no—
83C22×C8×C53D414yescomplemented
84C22×C8×C53C2312yes—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53D414yes—
91C2×C4⋊C4×C53C2312yes—
92C2×C4⋊C4×C53C2312yes—
93#106×C53—28no—
94C24×C53C2×D411yescharacteristic
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—24no—
102C23×D53—22no—
103C23×D53—42no—
104C23×D53—42no—
105C23×D53—42no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53SD1614yes—
110C23×D53C2×D412yes—
111C22×C4×C53—24no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—42no—
115C22×C4×C53—42no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×D412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—42no—
132C22×Dic53—42no—
133C22×Dic53—42no—
134C22×Dic53C2×D412yes—
135C2×D212—28no—
136C2×D212—28no—
137C2×D212—22no—
138C2×D212—22no—
139C2×D212—22no—
140C2×D212—22no—
141C2×D212—216no—
142C2×D212—24no—
143C2×D212—22no—
144C2×D212—48no—
145C2×D212—432no—
146C2×D212—48no—
147C2×D212—44no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212—44no—
151C2×D212C2×D414yes—
152C2×C4×D53—24no—
153C2×C4×D53—24no—
154C2×C4×D53—24no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—44no—
162C2×C4×D53—44no—
163C2×C4×D53—44no—
164C2×C4×D53—44no—
165C2×C8×C53—28no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53C2×D414yescomplemented
169C4⋊C4×C53—24no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53C2×D414yes—
175#11719921—216no—
176#11719921—48no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—42no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—44no—
184C23×C53#12012yes—
185C23×C53#12112yes—
186C23×C53#12312yes—
187C23×C53C2×SD1612yes—
188C23×C53#13512yes—
189C23×C53AGL(1,ℤ/8)12yes—
190C23×C53AGL(1,ℤ/8)12yes—
191C23×C53C22×D411yescharacteristic
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—22no—
198C22×D53—28no—
199C22×D53—28no—
200C22×D53—28no—
201C22×D53—24no—
202C22×D53—44no—
203C22×D53—44no—
204C22×D53—44no—
205C22×D53—44no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—48no—
213C22×D53—48no—
214C22×D53—48no—
215C22×D53—84no—
216C22×D53—84no—
217C22×D53—84no—
218C22×D53—84no—
219C22×D53C2×SD1614yes—
220C2×C4×C53—24no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—48no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—84no—
228C2×C4×C53#12012yes—
229C2×C4×C53#12114yes—
230C2×C4×C53#12114yes—
231C2×C4×C53#12112yes—
232C2×C4×C53#12314yes—
233C2×C4×C53#12312yes—
234C2×C4×C53#12414yes—
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—44no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—84no—
249D212—24no—
250D212—24no—
251D212—432no—
252D212—42no—
253D212—42no—
254D212—48no—
255C4×D53—42no—
256C4×D53—42no—
257C4×D53—42no—
258C4×D53—42no—
259C4×D53—48no—
260C8×C53—24no—
261#88322—42no—
262C2×#2034—531nomaximal, Sylow
263C22×C53#39414yes—
264C22×C53#44914yes—
265C22×C53#45014yes—
266C22×C53#46514yes—
267C22×C53#46914yes—
268C22×C53#48114yes—
269C22×C53#48214yes—
270C22×C53C2×#12011yescharacteristic
271C22×C53C2×#12111yescharacteristic
272C22×C53C2×#12311yescharacteristic
273C22×C53C22×SD1611yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C2×D53—24no—
278C2×D53—24no—
279C2×D53—24no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—416no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—44no—
288C2×D53—82no—
289C2×D53—82no—
290C2×D53—82no—
291C2×D53—82no—
292C2×D53—816no—
293C4×C53—44no—
294C4×C53—82no—
295C4×C53#39414yes—
296C4×C53C2×#12111yescharacteristic
297C4×C53C2×#12111yescharacteristic
298C4×C53C2×#12311yescharacteristic
299C4×C53C2×#12411yescharacteristic
300Dic53—42no—
301Dic53—82no—
302C22×#121—531no—
303C22×#106D5311yescharacteristic, complemented
304C22×#102—531no—
305C22×#102—531no—
306C22×C22⋊C8—531no—
307C2×#380—531no—
308#2034—538no—
309C2×#340—531no—
310C2×C53#203418yescomplemented
311C2×C53C2×#39411yescharacteristic
312C2×C53C2×#45011yescharacteristic
313C2×C53C2×#44911yescharacteristic
314C2×C53C2×#46511yescharacteristic
315C2×C53C2×#46911yescharacteristic
316C2×C53C2×#48211yescharacteristic
317C2×C53C2×#48111yescharacteristic
318D53—44no—
319D53—84no—
320C23×D4—531no—
321C23×D4—1061no—
322C24×C4—531no—
323C23×C8—531no—
324C23×C8C2×D5311yescharacteristic
325C2×#121—534no—
326C2×#121—1062no—
327C2×#121—1064no—
328C2×#121—1064no—
329C22×C4⋊C4—531no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C22×C4⋊C4C2×D5311yescharacteristic
332C22×C22⋊C4—1061no—
333C22×C22⋊C4—1061no—
334C2×#106—24no—
335C2×#106C2×D5314yescomplemented
336C2×#102—534no—
337C2×#102—534no—
338C2×#102—1064no—
339C2×#102—1064no—
340C2×C22⋊C8—534no—
341C2×C22⋊C8—1064no—
342#380—534no—
343#380—1062no—
344C2×#95—1061no—
345#340—534no—
346C53C2×#203411yescharacteristic, complemented, Sylow
347C25—531no—
348C25—1061no—
349C22×D4—534no—
350C22×D4—532no—
351C22×D4—1064no—
352C22×D4—1068no—
353C22×D4—1062no—
354C22×D4—1062no—
355C22×D4—1061no—
356C22×D4—1062no—
357C22×D4—1062no—
358C22×D4—1062no—
359C22×D4—2124no—
360C22×D4—2124no—
361C23×C4—21no—
362C23×C4—21no—
363C23×C4—531no—
364C23×C4—532no—
365C23×C4—532no—
366C23×C4—1062no—
367C23×C4—1062no—
368C23×C4—1062no—
369C23×C4—1062no—
370C23×C4—1062no—
371C23×C4—1062no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—532no—
377C22×C8—1062no—
378C22×C8—1062no—
379C22×C8—1062no—
380C22×C8D21214yescomplemented
381C22×C8C22×D5312yes—
382#121—1068no—
383#121—10616no—
384#121—1068no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—532no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22⋊D5314yes—
395C2×C4⋊C4C22×D5312yes—
396C2×C4⋊C4C22×D5312yes—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—2124no—
401#106—28no—
402#102—1068no—
403#102—1068no—
404C22⋊C8—1068no—
405#95—1064no—
406C24—534no—
407C24—532no—
408C24—1062no—
409C24—1062no—
410C24—1062no—
411C24—1062no—
412C24—1062no—
413C24—1062no—
414C24—1064no—
415C24—1062no—
416C24—2122no—
417C24—2122no—
418C24—2122no—
419C24—2122no—
420C24—2122no—
421C24—2122no—
422C24D4×D5311yescharacteristic
423C2×D4—534no—
424C2×D4—10616no—
425C2×D4—1064no—
426C2×D4—1068no—
427C2×D4—1068no—
428C2×D4—1062no—
429C2×D4—1062no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—1062no—
433C2×D4—2128no—
434C2×D4—21232no—
435C2×D4—2128no—
436C2×D4—2124no—
437C2×D4—2122no—
438C2×D4—2122no—
439C2×D4—2124no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—532no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1064no—
453C22×C4—1064no—
454C22×C4—1064no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2124no—
466C22×C4—2124no—
467C22×C4—2124no—
468C22×C4—2124no—
469C22×C4C2×D21212yes—
470C22×C4#1171994512yes—
471C22×C4D4×D5311yescharacteristic
472C22×C4D4×D5312yes—
473C22×C4#1171994912yes—
474C22×C4#1171994912yes—
475C22×C4C2×C22⋊D5312yes—
476C22×C4C23×D5311yescharacteristic
477C2×C8—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—2124no—
484C2×C8C2×D21214yescomplemented
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—2122no—
493C4⋊C4C2×C22⋊D5314yes—
494C22⋊C4—10616no—
495C22⋊C4—2128no—
496C23—534no—
497C23—1068no—
498C23—1062no—
499C23—1068no—
500C23—1062no—
501C23—1062no—
502C23—1062no—
503C23—1062no—
504C23—1062no—
505C23—1068no—
506C23—1064no—
507C23—4244no—
508C23—4244no—
509C23—4244no—
510C23—4244no—
511C23—2128no—
512C23—2128no—
513C23—2128no—
514C23—2122no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2124no—
521C23—2124no—
522C23—2124no—
523C23—2124no—
524C23#1313318512yes—
525C23#1313319212yes—
526C23#1313320212yes—
527C23SD16×D5312yes—
528C23#1313323812yes—
529C23#1313324412yes—
530C23#1313324412yes—
531C23C2×D4×D5311yescharacteristic
532D4—1064no—
533D4—1064no—
534D4—21232no—
535D4—2128no—
536D4—2122no—
537D4—2122no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—4244no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2124no—
554C2×C4—2128no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313318512yes—
560C2×C4#1313319114yes—
561C2×C4#1313319212yes—
562C2×C4#1313319214yes—
563C2×C4#1313320212yes—
564C2×C4#1313320214yes—
565C2×C4#1313320614yes—
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C22×C22⋊D5311yescharacteristic
568C2×C4C22×D21211yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4C2×#1171994911yescharacteristic
571C2×C4C2×#1171994911yescharacteristic
572C8—24no—
573C8—2122no—
574C22—1064no—
575C22—1064no—
576C22—1064no—
577C22—4242no—
578C22—4242no—
579C22—4242no—
580C22—42416no—
581C22—4242no—
582C22—2124no—
583C22—2124no—
584C22—2124no—
585C22—2122no—
586C22—2122no—
587C22—2122no—
588C22—2122no—
589C22—21216no—
590C22#1681054014yes—
591C22#1681076714yes—
592C22#1681077214yes—
593C22#1681085314yes—
594C22#1681087714yes—
595C22#1681094314yes—
596C22#1681094914yes—
597C22C2×#1313318511yescharacteristic
598C22C2×#1313319211yescharacteristic
599C22C2×#1313320211yescharacteristic
600C22C2×SD16×D5311yescharacteristic
601C22C2×#1313323811yescharacteristic
602C22C2×#1313324411yescharacteristic
603C22C2×#1313324411yescharacteristic
604C4—44no—
605C4—82no—
606C4—4242no—
607C4—2122no—
608C4#1681054014yes—
609C4C2×#1313319111yescharacteristic
610C4C2×#1313319211yescharacteristic
611C4C2×#1313320211yescharacteristic
612C4C2×#1313320611yescharacteristic
613C2—4244no—
614C2—2124no—
615C2#2027590618yesdirect product factor
616C2C2×#1681054011yescharacteristic
617C2C2×#1681077211yescharacteristic
618C2C2×#1681076711yescharacteristic
619C2C2×#1681085311yescharacteristic
620C2C2×#1681087711yescharacteristic
621C2C2×#1681094911yescharacteristic
622C2C2×#1681094311yescharacteristic
subgroupG/H
centerC24D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#106×C53C2
socleC24×C53C2×D4
prime radicalC22×#13133192C2
hypercenterC22×#106D53
hypocenterC53C2×#2034
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
2. C22×#13133139C2semidirect
3. C22×#13133137C2semidirect
4. C22×#13133129C2semidirect
5. C2×#16810480C2semidirect
6. #20275906C2direct (N × H)
7. C2×#16810385C2semidirect
8. C22×#106×C53C2semidirect
21. C2×#13133139C22semidirect
23. C2×#13133137C22semidirect
25. C2×#13133129C22semidirect
27. #16810480C22semidirect
29. #16810385C22semidirect
34. C2×#106×C53C22semidirect
83. C22×C8×C53D4semidirect
168. C2×C8×C53C2×D4semidirect
303. C22×#106D53semidirect
310. C2×C53#2034semidirect
335. C2×#106C2×D53semidirect
346. C53C2×#2034semidirect
380. C22×C8D212semidirect
484. C2×C8C2×D212semidirect
615. C2#20275906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331922
C22×#131331392
C22×#131331372
C22×#131331292
C2×#168104802
#202759062
C2×#168103852
C22×#106×C532
C2×#203453