Finite Group Databasebeta

← C2×#20275905 #27411557

Subgroups — C2×#20275905 #27411557

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×#13133191C211yesmaximal, characteristic
2C22×#13133149C211yesmaximal, characteristic, complemented
3C22×#13133139C211yesmaximal, characteristic, complemented
4C22×#13133139C211yesmaximal, characteristic, complemented
5C2×#16810483C211yesmaximal, characteristic, complemented
6#20275905C218yesmaximal, direct product factor
7C2×#16810386C211yesmaximal, characteristic, complemented
8C22×C22⋊C8×C53C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C23×C22⋊D53—21no—
11C2×#13133191—24no—
12C2×#13133191—24no—
13C2×#13133191—22no—
14C2×#13133191C2214yes—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53—21no—
19C22×#11719921—21no—
20C2×#13133149—24no—
21C2×#13133149C2214yescomplemented
22C2×#13133139—24no—
23C2×#13133139—24no—
24C2×#13133139C2214yescomplemented
25C2×#13133139C2214yescomplemented
26#16810483—22no—
27#16810483C2214yescomplemented
28C2×#13133124—21no—
29#16810386C2214yescomplemented
30C24×C4×C53C2211yescharacteristic
31C23×C8×C53C2211yescharacteristic
32C23×C8×C53C2211yescharacteristic
33C2×C22⋊C8×C53—24no—
34C2×C22⋊C8×C53C2214yescomplemented
35C24×D53—21no—
36C22×D212—22no—
37C22×D212—22no—
38C22×D212—22no—
39C22×D212—22no—
40C22×D212—21no—
41C22×D212—22no—
42C22×D212D414yes—
43C22×D212C2312yes—
44C22×C22⋊D53—28no—
45C22×C22⋊D53—24no—
46C22×C22⋊D53—44no—
47C22×C22⋊D53—44no—
48C23×Dic53—21no—
49C23×Dic53—21no—
50C23×Dic53—21no—
51#13133191—216no—
52#13133191—28no—
53#13133191—28no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53D414yes—
63C2×C4⋊Dic53C2312yes—
64C2×C4⋊Dic53C2312yes—
65C2×C4⋊Dic53C2312yes—
66C2×C22⋊Dic53—24no—
67C2×C22⋊Dic53—44no—
68C2×#11719921—28no—
69C2×#11719921—24no—
70#13133149—28no—
71#13133139—28no—
72#13133139—28no—
73#13133124—24no—
74C25×C53D411yescharacteristic
75C23×C4×C53—22no—
76C23×C4×C53—22no—
77C23×C4×C53—22no—
78C23×C4×C53—22no—
79C23×C4×C53—22no—
80C23×C4×C53—22no—
81C23×C4×C53D411yescharacteristic
82C23×C4×C53D412yes—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C22×C8×C53—22no—
86C22×C8×C53—22no—
87C22×C8×C53—22no—
88C22×C8×C53—22no—
89C22×C8×C53—22no—
90C22×C8×C53D414yescomplemented
91C22×C8×C53C2312yes—
92C22×C8×C53C2312yes—
93C22⋊C8×C53—28no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53SD1614yes—
101C24×C53C2×D412yes—
102C24×C53C2×D411yescharacteristic
103C23×D53—24no—
104C23×D53—22no—
105C23×D53—42no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—24no—
116C22×C4×C53—24no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—44no—
124C22×C4×C53—44no—
125C22×C4×C53—44no—
126C22×C4×C53—44no—
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4112yes—
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C2×C22⋊D53—216no—
137C2×C22⋊D53—48no—
138C2×C22⋊D53—432no—
139C2×C22⋊D53—48no—
140C22×Dic53—24no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C2×D212—28no—
153C2×D212—28no—
154C2×D212—22no—
155C2×D212—22no—
156C2×D212—22no—
157C2×D212—22no—
158C2×D212—24no—
159C2×D212—22no—
160C2×D212—44no—
161C2×D212—42no—
162C2×D212—42no—
163C2×D212—44no—
164C2×D212C2×D414yes—
165C2×C8×C53—28no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—44no—
172C2×C8×C53C2×D414yescomplemented
173C22⋊Dic53—48no—
174#11719921—216no—
175C4⋊Dic53—24no—
176C4⋊Dic53—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—42no—
182C4⋊Dic53—42no—
183C4⋊Dic53C2×D414yes—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—28no—
191C23×C53—28no—
192C23×C53—28no—
193C23×C53—44no—
194C23×C53—44no—
195C23×C53—44no—
196C23×C53—44no—
197C23×C53#12012yes—
198C23×C53#12112yes—
199C23×C53#12312yes—
200C23×C53C2×SD1614yes—
201C23×C53C2×SD1612yes—
202C23×C53#13512yes—
203C23×C53AGL(1,ℤ/8)12yes—
204C23×C53AGL(1,ℤ/8)12yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—24no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—48no—
214C22×D53—48no—
215C22×D53—48no—
216C22×D53—84no—
217C22×D53—84no—
218C22×D53—84no—
219C22×D53—84no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—44no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—48no—
229C2×C4×C53#12012yes—
230C2×C4×C53#12114yes—
231C2×C4×C53#12114yes—
232C2×C4×C53#12112yes—
233C2×C4×C53#12314yes—
234C2×C4×C53#12312yes—
235C2×C4×C53#12414yes—
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C22⋊D53—432no—
243C2×Dic53—24no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—48no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—84no—
254C2×Dic53—84no—
255D212—24no—
256D212—24no—
257D212—42no—
258D212—42no—
259D212—48no—
260C8×C53—24no—
261C8×C53—42no—
262C2×#2034—531nomaximal, Sylow
263C22×C53—24no—
264C22×C53—24no—
265C22×C53—24no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—416no—
271C22×C53#39414yes—
272C22×C53#44914yes—
273C22×C53#45014yes—
274C22×C53#46514yes—
275C22×C53#46914yes—
276C22×C53#48114yes—
277C22×C53#48214yes—
278C22×C53C2×#12011yescharacteristic
279C22×C53C2×#12111yescharacteristic
280C22×C53C2×#12311yescharacteristic
281C22×C53C22×SD1611yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
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—42no—
294C4×C53#39414yes—
295C4×C53C2×#12111yescharacteristic
296C4×C53C2×#12111yescharacteristic
297C4×C53C2×#12311yescharacteristic
298C4×C53C2×#12411yescharacteristic
299Dic53—44no—
300Dic53—82no—
301Dic53—82no—
302C22×#121—531no—
303C22×#106—531no—
304C22×#102—531no—
305C22×#102—531no—
306C22×C22⋊C8D5311yescharacteristic, complemented
307C2×#380—531no—
308#2034—538no—
309C2×#340—531no—
310C2×C53—44no—
311C2×C53#203418yescomplemented
312C2×C53C2×#39411yescharacteristic
313C2×C53C2×#45011yescharacteristic
314C2×C53C2×#44911yescharacteristic
315C2×C53C2×#46511yescharacteristic
316C2×C53C2×#46911yescharacteristic
317C2×C53C2×#48211yescharacteristic
318C2×C53C2×#48111yescharacteristic
319D53—84no—
320C23×D4—531no—
321C23×D4—1061no—
322C24×C4C2×D5311yescharacteristic
323C23×C8C2×D5311yescharacteristic
324C23×C8C2×D5311yescharacteristic
325C2×#121—534no—
326C2×#121—1064no—
327C2×#121—1064no—
328C2×#121—1062no—
329C22×C4⋊C4—531no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4—531no—
332C22×C22⋊C4—1061no—
333C22×C22⋊C4—1061no—
334C2×#106—534no—
335C2×#106—1064no—
336C2×#102—534no—
337C2×#102—534no—
338C2×#102—1064no—
339C2×#102—1064no—
340C2×C22⋊C8—24no—
341C2×C22⋊C8C2×D5314yescomplemented
342#380—534no—
343#380—1062no—
344C2×#95—1061no—
345#340—534no—
346C53C2×#203411yescharacteristic, complemented, Sylow
347C25—1061no—
348C25D21211yescharacteristic
349C22×D4—534no—
350C22×D4—532no—
351C22×D4—1068no—
352C22×D4—1064no—
353C22×D4—1061no—
354C22×D4—1062no—
355C22×D4—1062no—
356C22×D4—1062no—
357C22×D4—1062no—
358C22×D4—1062no—
359C22×D4—2124no—
360C22×D4—2124no—
361C23×C4—22no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—1061no—
368C23×C4—1061no—
369C23×C4—1061no—
370C23×C4D21212yes—
371C23×C4D21211yescharacteristic
372C23×C4C22×D5312yes—
373C23×C4C22×D5311yescharacteristic
374C22×C8—22no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—22no—
378C22×C8—22no—
379C22×C8C22⋊D5314yescomplemented
380C22×C8C22×D5312yes—
381C22×C8C22×D5312yes—
382#121—1068no—
383#121—1068no—
384#121—10616no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—532no—
388C2×C4⋊C4—532no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C22⋊C4—1068no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—2124no—
401#106—1068no—
402#102—1068no—
403#102—1068no—
404C22⋊C8—28no—
405#95—1064no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—1064no—
413C24—1062no—
414C24—2122no—
415C24—2122no—
416C24—2122no—
417C24—2122no—
418C24—2122no—
419C24—2122no—
420C24#1171991314yes—
421C24C2×D21212yes—
422C24C2×D21211yescharacteristic
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—21232no—
434C2×D4—2128no—
435C2×D4—2128no—
436C2×D4—2124no—
437C2×D4—2122no—
438C2×D4—2122no—
439C2×D4—2124no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—1064no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4C2×D21212yes—
469C22×C4C2×D21211yescharacteristic
470C22×C4#1171994512yes—
471C22×C4D4×D5312yes—
472C22×C4D4×D5312yes—
473C22×C4#1171994712yes—
474C22×C4#1171994712yes—
475C22×C4C2×C22⋊D5312yes—
476C22×C4C23×D5311yescharacteristic
477C2×C8—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8C2×C22⋊D5314yescomplemented
485C4⋊C4—534no—
486C4⋊C4—1064no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C22⋊C4—10616no—
495C22⋊C4—2128no—
496C23—28no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—1064no—
510C23—4244no—
511C23—4244no—
512C23—4244no—
513C23—4244no—
514C23—2128no—
515C23—2128no—
516C23—2128no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23#1313318712yes—
524C23#1313319112yes—
525C23#1313320112yes—
526C23C2×#1171991314yes—
527C23C2×#1171991312yes—
528C23#1313323912yes—
529C23#1313324512yes—
530C23#1313324512yes—
531C23C22×D21211yescharacteristic
532D4—1064no—
533D4—1064no—
534D4—21232no—
535D4—2128no—
536D4—2122no—
537D4—2122no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—1064no—
548C2×C4—4244no—
549C2×C4—4244no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2128no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313318712yes—
560C2×C4#1313318814yes—
561C2×C4#1313318914yes—
562C2×C4#1313319112yes—
563C2×C4#1313319914yes—
564C2×C4#1313320112yes—
565C2×C4#1313320314yes—
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C2×D4×D5311yescharacteristic
568C2×C4C22×C22⋊D5311yescharacteristic
569C2×C4C2×#1171994711yescharacteristic
570C2×C4C2×#1171994711yescharacteristic
571C2×C4C2×#1171994511yescharacteristic
572C8—24no—
573C8—42no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—416no—
582C22—4242no—
583C22—4242no—
584C22—4242no—
585C22—42416no—
586C22—4242no—
587C22—2124no—
588C22—2124no—
589C22—2124no—
590C22#1681053914yes—
591C22#1681077014yes—
592C22#1681077614yes—
593C22#1681085214yes—
594C22#1681087614yes—
595C22#1681094214yes—
596C22#1681094814yes—
597C22C2×#1313318711yescharacteristic
598C22C2×#1313319111yescharacteristic
599C22C2×#1313320111yescharacteristic
600C22C22×#1171991311yescharacteristic
601C22C2×#1313323911yescharacteristic
602C22C2×#1313324511yescharacteristic
603C22C2×#1313324511yescharacteristic
604C4—42no—
605C4—4242no—
606C4—4242no—
607C4—2124no—
608C4#1681053914yes—
609C4C2×#1313318811yescharacteristic
610C4C2×#1313318911yescharacteristic
611C4C2×#1313319911yescharacteristic
612C4C2×#1313320311yescharacteristic
613C2—44no—
614C2—4244no—
615C2#2027590518yesdirect product factor
616C2C2×#1681053911yescharacteristic
617C2C2×#1681077611yescharacteristic
618C2C2×#1681077011yescharacteristic
619C2C2×#1681085211yescharacteristic
620C2C2×#1681087611yescharacteristic
621C2C2×#1681094811yescharacteristic
622C2C2×#1681094211yescharacteristic
subgroupG/H
centerC24C2×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×C22⋊C8×C53C2
socleC24×C53C2×D4
prime radicalC22×#13133191C2
hypercenterC22×C22⋊C8D53
hypocenterC53C2×#2034
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#13133149C2semidirect
3. C22×#13133139C2semidirect
4. C22×#13133139C2semidirect
5. C2×#16810483C2semidirect
6. #20275905C2direct (N × H)
7. C2×#16810386C2semidirect
8. C22×C22⋊C8×C53C2semidirect
21. C2×#13133149C22semidirect
24. C2×#13133139C22semidirect
25. C2×#13133139C22semidirect
27. #16810483C22semidirect
29. #16810386C22semidirect
34. C2×C22⋊C8×C53C22semidirect
90. C22×C8×C53D4semidirect
172. C2×C8×C53C2×D4semidirect
306. C22×C22⋊C8D53semidirect
311. C2×C53#2034semidirect
341. C2×C22⋊C8C2×D53semidirect
346. C53C2×#2034semidirect
379. C22×C8C22⋊D53semidirect
484. C2×C8C2×C22⋊D53semidirect
615. C2#20275905direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331912
C22×#131331492
C22×#131331392
C2×#168104832
#202759052
C2×#168103862
C22×C22⋊C8×C532
C2×#203453