Finite Group Databasebeta

← C2×#20275894 #27411537

Subgroups — C2×#20275894 #27411537

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×#13133150C211yesmaximal, characteristic, complemented
2C22×#13133138C211yesmaximal, characteristic, complemented
3C22×#13133138C211yesmaximal, characteristic, complemented
4C22×C22⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810483C211yesmaximal, characteristic, complemented
6#20275894C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#121×C53C211yesmaximal, characteristic
9C23×#88322C2211yescharacteristic
10C23×#88322C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C22⋊Dic53—21no—
14C2×#13133150—24no—
15C2×#13133150C2214yescomplemented
16C2×#13133138—24no—
17C2×#13133138—24no—
18C2×#13133138C2214yescomplemented
19C2×#13133138C2214yescomplemented
20C2×C22⋊#88322—24no—
21C2×C22⋊#88322C2214yescomplemented
22#16810483—22no—
23#16810483C2214yescomplemented
24C2×#13133124—21no—
25#16810385C2214yescomplemented
26C23×D4×C53—21no—
27C23×D4×C53C2211yescharacteristic
28C24×C4×C53C2211yescharacteristic
29C2×#121×C53—24no—
30C2×#121×C53—22no—
31C2×#121×C53—24no—
32C2×#121×C53C2214yes—
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53—21no—
35C23×Dic53—21no—
36C23×Dic53—21no—
37C22×#88322—22no—
38C22×#88322—22no—
39C22×#88322—22no—
40C22×#88322—22no—
41C22×#88322—22no—
42C22×#88322D414yescomplemented
43C22×#88322C2312yes—
44C22×#88322C2312yes—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53C2312yes—
52C2×C4⋊Dic53C2312yes—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—44no—
55#13133150—28no—
56#13133138—28no—
57#13133138—28no—
58C22⋊#88322—28no—
59#13133124—24no—
60C25×C53—21no—
61C25×C53D411yescharacteristic
62C22×D4×C53—28no—
63C22×D4×C53—24no—
64C22×D4×C53—22no—
65C22×D4×C53—21no—
66C22×D4×C53—22no—
67C22×D4×C53—22no—
68C22×D4×C53—22no—
69C22×D4×C53—22no—
70C22×D4×C53—44no—
71C22×D4×C53—44no—
72C22×D4×C53D414yes—
73C22×D4×C53C2312yes—
74C23×C4×C53—21no—
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
85#121×C53—216no—
86#121×C53—28no—
87#121×C53—28no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53D414yes—
91C2×C4⋊C4×C53C2312yes—
92C2×C22⋊C4×C53—28no—
93C2×C22⋊C4×C53—24no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—24no—
101C24×C53—22no—
102C24×C53—42no—
103C24×C53—42no—
104C24×C53—42no—
105C24×C53—42no—
106C24×C53—42no—
107C24×C53—42no—
108C24×C53D814yes—
109C24×C53C2×D412yes—
110C24×C53C2×D411yescharacteristic
111C2×D4×C53—216no—
112C2×D4×C53—24no—
113C2×D4×C53—28no—
114C2×D4×C53—28no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—48no—
121C2×D4×C53—432no—
122C2×D4×C53—48no—
123C2×D4×C53—44no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—44no—
127C2×D4×C53C2×D414yes—
128C22×C4×C53—22no—
129C22×C4×C53—24no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—44no—
145C22×C4×C53—44no—
146C22×C4×C53—44no—
147C22×C4×C53—44no—
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53#4112yes—
154C22×C4×C53#4112yes—
155C22×C4×C53#4112yes—
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C22×Dic53—42no—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53C2×D414yes—
168C22⋊C4×C53—216no—
169C22⋊Dic53—48no—
170C4⋊Dic53—22no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—42no—
175C4⋊Dic53—42no—
176C2×#88322—28no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—44no—
183C2×#88322C2×D414yescomplemented
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—24no—
194C23×C53—44no—
195C23×C53—44no—
196C23×C53—44no—
197C23×C53—44no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—48no—
205C23×C53—48no—
206C23×C53—48no—
207C23×C53—84no—
208C23×C53—84no—
209C23×C53—84no—
210C23×C53—84no—
211C23×C53#12012yes—
212C23×C53#12112yes—
213C23×C53#12312yes—
214C23×C53C2×D814yes—
215C23×C53C2×D812yes—
216C23×C53#13512yes—
217C23×C53AGL(1,ℤ/8)12yes—
218C23×C53#13712yes—
219C23×C53C22×D411yescharacteristic
220D4×C53—24no—
221D4×C53—24no—
222D4×C53—432no—
223D4×C53—42no—
224D4×C53—42no—
225D4×C53—48no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—44no—
233C2×C4×C53—48no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—48no—
239C2×C4×C53#12012yes—
240C2×C4×C53#12114yes—
241C2×C4×C53#12114yes—
242C2×C4×C53#12112yes—
243C2×C4×C53#12314yes—
244C2×C4×C53#12312yes—
245C2×C4×C53#12414yes—
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—84no—
259C2×Dic53—84no—
260#88322—24no—
261#88322—42no—
262C2×#2033—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—44no—
272C22×C53—44no—
273C22×C53—44no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53—816no—
279C22×C53#39414yes—
280C22×C53#44714yes—
281C22×C53#45214yes—
282C22×C53#46614yes—
283C22×C53#46814yes—
284C22×C53#48014yes—
285C22×C53#48214yes—
286C22×C53C2×#12011yescharacteristic
287C22×C53C2×#12111yescharacteristic
288C22×C53C2×#12311yescharacteristic
289C22×C53C22×D811yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C53C2×#13711yescharacteristic
293C4×C53—42no—
294C4×C53—44no—
295C4×C53#39414yes—
296C4×C53C2×#12111yescharacteristic
297C4×C53C2×#12111yescharacteristic
298C4×C53C2×#12311yescharacteristic
299C4×C53C2×#12411yescharacteristic
300Dic53—82no—
301Dic53—82no—
302C22×#121D5311yescharacteristic
303C22×#107—531no—
304C22×#102—531no—
305C22×#102—531no—
306C22×C22⋊C8—531no—
307C2×#380—531no—
308#2033—538no—
309C2×#340—531no—
310C2×C53—44no—
311C2×C53—84no—
312C2×C53#203318yescomplemented
313C2×C53C2×#39411yescharacteristic
314C2×C53C2×#44711yescharacteristic
315C2×C53C2×#45211yescharacteristic
316C2×C53C2×#46611yescharacteristic
317C2×C53C2×#46811yescharacteristic
318C2×C53C2×#48011yescharacteristic
319C2×C53C2×#48211yescharacteristic
320C23×D4—21no—
321C23×D4C2×D5311yescharacteristic
322C24×C4C2×D5311yescharacteristic
323C23×C8—531no—
324C23×C8—531no—
325C2×#121—24no—
326C2×#121—22no—
327C2×#121—24no—
328C2×#121C2×D5314yes—
329C22×C4⋊C4—531no—
330C22×C4⋊C4—531no—
331C22×C4⋊C4C2×D5311yescharacteristic
332C22×C22⋊C4—21no—
333C22×C22⋊C4—1061no—
334C2×#107—534no—
335C2×#107—1064no—
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×#203311yescharacteristic, complemented, Sylow
347C25—21no—
348C25C22⋊D5311yescharacteristic
349C22×D4—28no—
350C22×D4—24no—
351C22×D4—22no—
352C22×D4—22no—
353C22×D4—22no—
354C22×D4—22no—
355C22×D4—21no—
356C22×D4—22no—
357C22×D4—44no—
358C22×D4—44no—
359C22×D4C22⋊D5314yes—
360C22×D4C22×D5312yes—
361C23×C4—21no—
362C23×C4—22no—
363C23×C4—22no—
364C23×C4—22no—
365C23×C4—22no—
366C23×C4—22no—
367C23×C4—22no—
368C23×C4—1061no—
369C23×C4—1061no—
370C23×C4C22⋊D5312yes—
371C23×C4C22⋊D5311yescharacteristic
372C23×C4C22×D5312yes—
373C23×C4C22×D5311yescharacteristic
374C22×C8—534no—
375C22×C8—532no—
376C22×C8—532no—
377C22×C8—1062no—
378C22×C8—1062no—
379C22×C8—1062no—
380C22×C8—1062no—
381C22×C8—1062no—
382#121—216no—
383#121—28no—
384#121—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—532no—
388C2×C4⋊C4—532no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22⋊D5314yes—
396C2×C4⋊C4C22×D5312yes—
397C2×C22⋊C4—28no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—2124no—
401#107—1068no—
402#102—1068no—
403#102—1068no—
404C22⋊C8—1068no—
405#95—1064no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—24no—
413C24—22no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24#1171992214yes—
421C24C2×C22⋊D5312yes—
422C24C2×C22⋊D5311yescharacteristic
423C2×D4—216no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—28no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—432no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4C2×C22⋊D5314yes—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4D4×D5312yes—
469C22×C4#1171994712yes—
470C22×C4#1171994712yes—
471C22×C4#1171994712yes—
472C22×C4C2×C22⋊D5312yes—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5312yes—
475C22×C4C2×C22⋊D5312yes—
476C22×C4C23×D5311yescharacteristic
477C2×C8—534no—
478C2×C8—1068no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—2124no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—1062no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4—1062no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4C2×C22⋊D5314yes—
494C22⋊C4—216no—
495C22⋊C4—2128no—
496C23—28no—
497C23—22no—
498C23—28no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—24no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23#1313318612yes—
524C23#1313318912yes—
525C23#1313319812yes—
526C23C2×#1171992214yes—
527C23C2×#1171992212yes—
528C23#1313323712yes—
529C23#1313324312yes—
530C23#1313324712yes—
531C23C22×C22⋊D5311yescharacteristic
532D4—24no—
533D4—24no—
534D4—432no—
535D4—48no—
536D4—42no—
537D4—42no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—4244no—
552C2×C4—4244no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4#1313318612yes—
560C2×C4#1313318814yes—
561C2×C4#1313318912yes—
562C2×C4#1313318914yes—
563C2×C4#1313319812yes—
564C2×C4#1313319814yes—
565C2×C4#1313320414yes—
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C22×C22⋊D5311yescharacteristic
568C2×C4C22×C22⋊D5311yescharacteristic
569C2×C4C2×#1171994711yescharacteristic
570C2×C4C2×#1171994711yescharacteristic
571C2×C4C2×#1171994711yescharacteristic
572C8—1064no—
573C8—2122no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—416no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—816no—
590C22#1681053714yes—
591C22#1681075714yes—
592C22#1681078514yes—
593C22#1681085714yes—
594C22#1681086714yes—
595C22#1681093414yes—
596C22#1681094414yes—
597C22C2×#1313318611yescharacteristic
598C22C2×#1313318911yescharacteristic
599C22C2×#1313319811yescharacteristic
600C22C22×#1171992211yescharacteristic
601C22C2×#1313323711yescharacteristic
602C22C2×#1313324311yescharacteristic
603C22C2×#1313324711yescharacteristic
604C4—42no—
605C4—44no—
606C4—4242no—
607C4—4242no—
608C4#1681053714yes—
609C4C2×#1313318811yescharacteristic
610C4C2×#1313318911yescharacteristic
611C4C2×#1313319811yescharacteristic
612C4C2×#1313320411yescharacteristic
613C2—44no—
614C2—84no—
615C2#2027589418yesdirect product factor
616C2C2×#1681053711yescharacteristic
617C2C2×#1681075711yescharacteristic
618C2C2×#1681078511yescharacteristic
619C2C2×#1681085711yescharacteristic
620C2C2×#1681086711yescharacteristic
621C2C2×#1681093411yescharacteristic
622C2C2×#1681094411yescharacteristic
subgroupG/H
centerC24C2×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC22×#121×C53C2
socleC24×C53C2×D4
hypercenterC22×#121D53
hypocenterC53C2×#2033
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
1. C22×#13133150C2semidirect
2. C22×#13133138C2semidirect
3. C22×#13133138C2semidirect
4. C22×C22⋊#88322C2semidirect
5. C2×#16810483C2semidirect
6. #20275894C2direct (N × H)
7. C2×#16810385C2semidirect
15. C2×#13133150C22semidirect
18. C2×#13133138C22semidirect
19. C2×#13133138C22semidirect
21. C2×C22⋊#88322C22semidirect
23. #16810483C22semidirect
25. #16810385C22semidirect
42. C22×#88322D4semidirect
183. C2×#88322C2×D4semidirect
312. C2×C53#2033semidirect
346. C53C2×#2033semidirect
615. C2#20275894direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331502
C22×#131331382
C22×C22⋊#883222
C2×#168104832
#202758942
C2×#168103852
C22×#121×C532
C2×#203353