Finite Group Databasebeta

← C2×#20275031 #27385291

Subgroups — C2×#20275031 #27385291

Order 13568.

2614 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133192C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3C22×#13133139C211yesmaximal, characteristic, complemented
4C22×#13133137C211yesmaximal, characteristic, complemented
5C22×#13133129C211yesmaximal, characteristic, complemented
6#20275031C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C2×C53×#340C211yesmaximal, characteristic, complemented
9C23×D212—21no—
10C23×D212C2211yescharacteristic
11C23×C4×D53—21no—
12C23×C4×D53C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133192—24no—
15C2×#13133192—22no—
16C2×#13133192C418yescomplemented
17C2×#13133192C2214yes—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×#11719919—21no—
20C2×C4⋊C4×D53—22no—
21C2×C4⋊C4×D53—24no—
22C2×C4⋊C4×D53—24no—
23C2×C4⋊C4×D53C2214yescomplemented
24C22×#11719921—21no—
25C2×#13133139—24no—
26C2×#13133139C2214yescomplemented
27C2×#13133137—24no—
28C2×#13133137C2214yescomplemented
29C2×#13133129—24no—
30C2×#13133129C2214yescomplemented
31#16810385C2214yescomplemented
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C53×#340C2214yescomplemented
36C24×D53—21no—
37C24×D53D411yescharacteristic
38C22×D212—22no—
39C22×D212—22no—
40C22×D212—22no—
41C22×D212—22no—
42C22×D212—28no—
43C22×D212—24no—
44C22×D212—21no—
45C22×D212—22no—
46C22×D212—44no—
47C22×D212—44no—
48C22×D212C2×C414yescomplemented
49C22×D212C2312yes—
50C23×Dic53—21no—
51C23×Dic53D411yescharacteristic
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53—28no—
57C22×C4×D53—24no—
58C22×C4×D53—44no—
59C22×C4×D53—44no—
60C22×C4×D53C2×C414yes—
61C22×C4×D53D412yes—
62C22×C4×D53D414yes—
63C22×C4×D53C2312yes—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322C2312yes—
68#13133192—216no—
69#13133192—28no—
70#13133192C2×C4116yescomplemented
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53D414yescomplemented
74C2×C4⋊Dic53C2312yes—
75C2×#11719919—28no—
76C2×#11719919—24no—
77C4⋊C4×D53—216no—
78C4⋊C4×D53—28no—
79C4⋊C4×D53—28no—
80C2×#11719921—28no—
81C2×#11719921—24no—
82#13133139—28no—
83#13133137—28no—
84#13133129—28no—
85C23×C4×C53—21no—
86C23×C4×C53—21no—
87C23×C4×C53C2311yescharacteristic
88C22×C8×C53—22no—
89C22×C8×C53—22no—
90C22×C8×C53—22no—
91C22×C8×C53C2312yes—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2×C414yescomplemented
97C2×C4⋊C4×C53D414yescomplemented
98C2×C4⋊C4×C53C2312yes—
99C2×C4⋊C4×C53C2312yes—
100C24×C53C2×D411yescharacteristic
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—24no—
106C23×D53—22no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C23×D53—42no—
113C23×D53C22⋊C414yes—
114C23×D53D814yes—
115C23×D53SD1614yes—
116C23×D53C2×D412yes—
117C22×C4×C53—24no—
118C22×C4×C53—24no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D412yes—
128C22×C4×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53#4112yes—
131C22×C4×C53#4112yes—
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—24no—
136C22×Dic53—22no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53C22⋊C414yes—
140C22×Dic53C2×D412yes—
141C2×D212—28no—
142C2×D212—28no—
143C2×D212—22no—
144C2×D212—22no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—216no—
148C2×D212—24no—
149C2×D212—22no—
150C2×D212—432no—
151C2×D212—48no—
152C2×D212—48no—
153C2×D212—42no—
154C2×D212—44no—
155C2×D212—44no—
156C2×D212—42no—
157C2×D212C22×C414yescomplemented
158C2×C4×D53—28no—
159C2×C4×D53—28no—
160C2×C4×D53—24no—
161C2×C4×D53—28no—
162C2×C4×D53—28no—
163C2×C4×D53—24no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×C4×D53—22no—
168C2×C4×D53—216no—
169C2×C4×D53—48no—
170C2×C4×D53—432no—
171C2×C4×D53—48no—
172C2×C4×D53C22⋊C418yes—
173C2×C4×D53C22×C414yes—
174C2×C4×D53C2×D414yes—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—44no—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—24no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53C22×C414yescomplemented
184C4⋊C4×C53C2×D414yescomplemented
185#11719921—216no—
186C4⋊Dic53—24no—
187C4⋊Dic53—22no—
188C4⋊Dic53C2×D414yescomplemented
189#11719919—216no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—44no—
194C23×C53C2×C22⋊C412yes—
195C23×C53#12012yes—
196C23×C53#12312yes—
197C23×C53C2×D812yes—
198C23×C53C2×SD1612yes—
199C23×C53AGL(1,ℤ/8)12yes—
200C23×C53#13712yes—
201C23×C53C22×D411yescharacteristic
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—28no—
207C22×D53—28no—
208C22×D53—28no—
209C22×D53—28no—
210C22×D53—28no—
211C22×D53—28no—
212C22×D53—24no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—48no—
220C22×D53—48no—
221C22×D53—48no—
222C22×D53—84no—
223C22×D53—84no—
224C22×D53—84no—
225C22×D53—84no—
226C22×D53#102116yescomplemented
227C22×D53C2×C22⋊C414yes—
228C22×D53C2×D814yes—
229C22×D53C2×SD1614yes—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—48no—
237C2×C4×C53—48no—
238C2×C4×C53C2×C22⋊C412yes—
239C2×C4×C53C4×D414yes—
240C2×C4×C53C4×D414yes—
241C2×C4×C53#12012yes—
242C2×C4×C53#12114yes—
243C2×C4×C53#12114yes—
244C2×C4×C53#12312yes—
245C2×C4×C53C23×C411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—28no—
254C2×Dic53—24no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—48no—
258C2×Dic53C2×C22⋊C414yes—
259D212—24no—
260D212—24no—
261D212—432no—
262D212—42no—
263D212—42no—
264D212—48no—
265C4×D53—24no—
266C4×D53—24no—
267C4×D53—24no—
268C4×D53—24no—
269C4×D53—216no—
270C4×D53—432no—
271C8×C53—42no—
272#88322—42no—
273C2×#1874—531nomaximal, Sylow
274C22×C53#38614yes—
275C22×C53C2×#10214yes—
276C22×C53#41714yes—
277C22×C53#44714yes—
278C22×C53#44814yes—
279C22×C53#48014yes—
280C22×C53#48114yes—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53C2×#12011yescharacteristic
283C22×C53C2×#12311yescharacteristic
284C22×C53C22×D811yescharacteristic
285C22×C53C22×SD1611yescharacteristic
286C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C53C2×#13711yescharacteristic
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—24no—
291C2×D53—232no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53—24no—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—44no—
298C2×D53—82no—
299C2×D53—82no—
300C2×D53—816no—
301C2×D53—82no—
302C2×D53—82no—
303C2×D53C2×#10218yescomplemented
304C4×C53—44no—
305C4×C53—44no—
306C4×C53#38614yes—
307C4×C53C2×C4×D411yescharacteristic
308C4×C53C2×C4×D411yescharacteristic
309C4×C53C2×#12111yescharacteristic
310C4×C53C2×#12111yescharacteristic
311Dic53—24no—
312Dic53—44no—
313C22×#121—531no—
314C23×C4⋊C4—531no—
315C22×#102—531no—
316C22×#102—531no—
317C22×C22⋊C8—531no—
318#1874—538no—
319C2×#340—531no—
320C2×#340D5311yescharacteristic, complemented
321C2×C53#187418yescomplemented
322C2×C53C2×#38611yescharacteristic
323C2×C53C22×#10211yescharacteristic
324C2×C53C2×#41711yescharacteristic
325C2×C53C2×#44711yescharacteristic
326C2×C53C2×#44811yescharacteristic
327C2×C53C2×#48011yescharacteristic
328C2×C53C2×#48111yescharacteristic
329D53—28no—
330D53—84no—
331C23×D4—531no—
332C23×D4—1061no—
333C24×C4—531no—
334C24×C4—1061no—
335C23×C8—531no—
336C23×C8C2×D5311yescharacteristic
337C2×#121—538no—
338C2×#121—534no—
339C2×#121—1062no—
340C2×#121—1064no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—534no—
343C22×C4⋊C4—1064no—
344C22×C4⋊C4—1062no—
345C22×C4⋊C4—1061no—
346C22×C4⋊C4—1064no—
347C22×C4⋊C4C2×D5311yescharacteristic
348C22×C4⋊C4C2×D5311yescharacteristic
349C22×C22⋊C4—1061no—
350C2×#102—534no—
351C2×#102—534no—
352C2×#102—1064no—
353C2×#102—1064no—
354C2×C22⋊C8—534no—
355C2×C22⋊C8—1064no—
356#340—534no—
357#340C2×D5314yescomplemented
358C53C2×#187411yescharacteristic, complemented, Sylow
359C25—531no—
360C25—1061no—
361C22×D4—534no—
362C22×D4—532no—
363C22×D4—1068no—
364C22×D4—1064no—
365C22×D4—1062no—
366C22×D4—1062no—
367C22×D4—1062no—
368C22×D4—1061no—
369C22×D4—1062no—
370C22×D4—1062no—
371C22×D4—2124no—
372C22×D4—2124no—
373C23×C4—21no—
374C23×C4—21no—
375C23×C4—531no—
376C23×C4—534no—
377C23×C4—532no—
378C23×C4—534no—
379C23×C4—532no—
380C23×C4—1064no—
381C23×C4—1068no—
382C23×C4—1062no—
383C23×C4—1062no—
384C23×C4—1062no—
385C23×C4—1062no—
386C23×C4—1061no—
387C23×C4—2124no—
388C23×C4—2124no—
389C23×C4C22×D5311yescharacteristic
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—532no—
394C22×C8—1062no—
395C22×C8—1062no—
396C22×C8—1062no—
397C22×C8C22×D5312yes—
398#121—5316no—
399#121—10616no—
400#121—1068no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—532no—
407C2×C4⋊C4—1068no—
408C2×C4⋊C4—10616no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1068no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1068no—
414C2×C4⋊C4C4×D5314yescomplemented
415C2×C4⋊C4D21214yescomplemented
416C2×C4⋊C4C22×D5312yes—
417C2×C4⋊C4C22×D5312yes—
418C2×C22⋊C4—1068no—
419C2×C22⋊C4—1064no—
420#102—1068no—
421#102—1068no—
422C22⋊C8—1068no—
423C24—534no—
424C24—534no—
425C24—534no—
426C24—532no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1064no—
432C24—1062no—
433C24—2122no—
434C24—2122no—
435C24—2122no—
436C24—2122no—
437C24—2122no—
438C24—2122no—
439C24D4×D5311yescharacteristic
440C2×D4—534no—
441C2×D4—10616no—
442C2×D4—1064no—
443C2×D4—1068no—
444C2×D4—1068no—
445C2×D4—1062no—
446C2×D4—1062no—
447C2×D4—1062no—
448C2×D4—1062no—
449C2×D4—1062no—
450C2×D4—21232no—
451C2×D4—2128no—
452C2×D4—2128no—
453C2×D4—2122no—
454C2×D4—2124no—
455C2×D4—2124no—
456C2×D4—2122no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—534no—
466C22×C4—532no—
467C22×C4—538no—
468C22×C4—534no—
469C22×C4—534no—
470C22×C4—10616no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1068no—
474C22×C4—1068no—
475C22×C4—1064no—
476C22×C4—1068no—
477C22×C4—1068no—
478C22×C4—1064no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1064no—
484C22×C4—1062no—
485C22×C4—2128no—
486C22×C4—21232no—
487C22×C4—2128no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4C2×C4×D5312yes—
491C22×C4C2×D21212yes—
492C22×C4#1171994512yes—
493C22×C4D4×D5311yescharacteristic
494C22×C4D4×D5312yes—
495C22×C4D4×D5312yes—
496C22×C4#1171994912yes—
497C22×C4C23×D5311yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—2124no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—534no—
511C4⋊C4—10616no—
512C4⋊C4—1064no—
513C4⋊C4—1062no—
514C4⋊C4C2×C4×D5314yescomplemented
515C4⋊C4C2×D21214yescomplemented
516C22⋊C4—10616no—
517C23—534no—
518C23—5316no—
519C23—534no—
520C23—534no—
521C23—1068no—
522C23—1068no—
523C23—1068no—
524C23—1062no—
525C23—1062no—
526C23—1068no—
527C23—1062no—
528C23—1062no—
529C23—1068no—
530C23—1068no—
531C23—1064no—
532C23—4244no—
533C23—4244no—
534C23—4244no—
535C23—4244no—
536C23—2128no—
537C23—2122no—
538C23—2128no—
539C23—2122no—
540C23—2128no—
541C23—2122no—
542C23—2122no—
543C23—2122no—
544C23—2122no—
545C23C22⋊C4×D5312yes—
546C23#1313318712yes—
547C23#1313320212yes—
548C23D8×D5312yes—
549C23SD16×D5312yes—
550C23#1313324412yes—
551C23#1313324912yes—
552C23C2×D4×D5311yescharacteristic
553D4—1064no—
554D4—1064no—
555D4—21232no—
556D4—2128no—
557D4—2122no—
558D4—2122no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—534no—
568C2×C4—1068no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—10616no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—21232no—
578C2×C4—2128no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4C22⋊C4×D5312yes—
582C2×C4#1313317914yes—
583C2×C4C4×D21214yes—
584C2×C4#1313318712yes—
585C2×C4#1313319014yes—
586C2×C4#1313319214yes—
587C2×C4#1313320212yes—
588C2×C4C22×C4×D5311yescharacteristic
589C2×C4C2×D4×D5311yescharacteristic
590C2×C4C2×D4×D5311yescharacteristic
591C2×C4C22×D21211yescharacteristic
592C2×C4C2×#1171994511yescharacteristic
593C2×C4C2×#1171994911yescharacteristic
594C8—42no—
595C8—2122no—
596C22—538no—
597C22—1064no—
598C22—10632no—
599C22—1064no—
600C22—1064no—
601C22—1064no—
602C22—1064no—
603C22—1064no—
604C22—4242no—
605C22—42416no—
606C22—4242no—
607C22—4242no—
608C22—4242no—
609C22—2124no—
610C22—2124no—
611C22—2124no—
612C22#1681050114yes—
613C22#102×D5314yes—
614C22#1681062314yes—
615C22#1681075814yes—
616C22#1681076314yes—
617C22#1681093814yes—
618C22#1681094314yes—
619C22C2×C22⋊C4×D5311yescharacteristic
620C22C2×#1313318711yescharacteristic
621C22C2×#1313320211yescharacteristic
622C22C2×D8×D5311yescharacteristic
623C22C2×SD16×D5311yescharacteristic
624C22C2×#1313324411yescharacteristic
625C22C2×#1313324911yescharacteristic
626C4—44no—
627C4—44no—
628C4—1064no—
629C4—2124no—
630C4#1681050114yes—
631C4C2×#1313317911yescharacteristic
632C4C2×C4×D21211yescharacteristic
633C4C2×#1313319011yescharacteristic
634C4C2×#1313319211yescharacteristic
635C2—1068no—
636C2—4244no—
637C2#2027503118yesdirect product factor
638C2C2×#1681050111yescharacteristic
639C2C2×#102×D5311yescharacteristic
640C2C2×#1681062311yescharacteristic
641C2C2×#1681075811yescharacteristic
642C2C2×#1681076311yescharacteristic
643C2C2×#1681093811yescharacteristic
644C2C2×#1681094311yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC24×C53C2×D4
prime radicalC22×#13133192C2
cosocleC22×C4×C53C24
hypercenterC2×#340D53
hypocenterC53C2×#1874
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. C22×#13133139C2semidirect
4. C22×#13133137C2semidirect
5. C22×#13133129C2semidirect
6. #20275031C2direct (N × H)
7. C2×#16810385C2semidirect
8. C2×C53×#340C2semidirect
16. C2×#13133192C4semidirect
23. C2×C4⋊C4×D53C22semidirect
26. C2×#13133139C22semidirect
28. C2×#13133137C22semidirect
30. C2×#13133129C22semidirect
31. #16810385C22semidirect
35. C53×#340C22semidirect
48. C22×D212C2×C4semidirect
70. #13133192C2×C4semidirect
73. C2×C4⋊Dic53D4semidirect
96. C2×C4⋊C4×C53C2×C4semidirect
97. C2×C4⋊C4×C53D4semidirect
157. C2×D212C22×C4semidirect
183. C4⋊C4×C53C22×C4semidirect
184. C4⋊C4×C53C2×D4semidirect
188. C4⋊Dic53C2×D4semidirect
226. C22×D53#102semidirect
303. C2×D53C2×#102semidirect
320. C2×#340D53semidirect
321. C2×C53#1874semidirect
357. #340C2×D53semidirect
358. C53C2×#1874semidirect
414. C2×C4⋊C4C4×D53semidirect
415. C2×C4⋊C4D212semidirect
514. C4⋊C4C2×C4×D53semidirect
515. C4⋊C4C2×D212semidirect
637. C2#20275031direct (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×C4⋊C4×D532
C22×#131331392
C22×#131331372
C22×#131331292
#202750312
C2×#168103852
C2×C53×#3402
C2×#187453