Finite Group Databasebeta

← C2×#20275948 #27411759

Subgroups — C2×#20275948 #27411759

Order 13568.

2230 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133213C211yesmaximal, characteristic
2C22×#13133139C211yesmaximal, characteristic, complemented
3C22×#13133138C211yesmaximal, characteristic, complemented
4C22×#13133137C211yesmaximal, characteristic, complemented
5C2×#16810522C211yesmaximal, characteristic, complemented
6#20275948C218yesmaximal, direct product factor
7C2×#16810377C211yesmaximal, characteristic, complemented
8C22×#102×C53C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C23×C22⋊D53—21no—
11C23×C22⋊D53—21no—
12C23×#88322C2211yescharacteristic
13C2×#13133213—24no—
14C2×#13133213—22no—
15C2×#13133213—24no—
16C2×#13133213C2214yes—
17C22×C4⋊Dic53—21no—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133139—24no—
21C2×#13133139C2214yescomplemented
22C2×#13133138—24no—
23C2×#13133138C2214yescomplemented
24C2×#13133137—24no—
25C2×#13133137C2214yescomplemented
26#16810522—22no—
27#16810522C2214yescomplemented
28C2×#13133124—21no—
29#16810377C2214yescomplemented
30C23×D4×C53C2211yescharacteristic
31C23×C8×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C2×#102×C53—24no—
34C2×#102×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—28no—
46C22×C22⋊D53—24no—
47C22×C22⋊D53—24no—
48C22×C22⋊D53—44no—
49C22×C22⋊D53—44no—
50C22×C22⋊D53—44no—
51C22×C22⋊D53—44no—
52C23×Dic53—21no—
53C23×Dic53D411yescharacteristic
54C23×Dic53D411yescharacteristic
55C22×#88322—22no—
56C22×#88322—22no—
57C22×#88322—22no—
58C22×#88322C2312yes—
59#13133213—216no—
60#13133213—28no—
61#13133213—28no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—24no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—44no—
67C2×C4⋊Dic53C2312yes—
68C2×C4×Dic53—22no—
69C2×C4×Dic53—22no—
70C2×C4×Dic53D412yes—
71C2×C4×Dic53C2312yes—
72#13133139—28no—
73#13133138—28no—
74#13133137—28no—
75#13133124—24no—
76C25×C53—21no—
77C22×D4×C53—22no—
78C22×D4×C53—21no—
79C22×D4×C53—22no—
80C22×D4×C53—22no—
81C22×D4×C53—22no—
82C22×D4×C53—22no—
83C22×D4×C53D414yes—
84C22×D4×C53C2312yes—
85C23×C4×C53—21no—
86C23×C4×C53C2311yescharacteristic
87C22×C8×C53—22no—
88C22×C8×C53—22no—
89C22×C8×C53—22no—
90C22×C8×C53C2312yes—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53D414yescomplemented
94C2×C4⋊C4×C53C2312yes—
95#102×C53—28no—
96C24×C53—24no—
97C24×C53—22no—
98C24×C53—42no—
99C24×C53—42no—
100C24×C53—42no—
101C24×C53—42no—
102C24×C53—42no—
103C24×C53—42no—
104C24×C53C2×D411yescharacteristic
105C23×D53—24no—
106C23×D53—22no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C23×D53—42no—
113C2×D4×C53—24no—
114C2×D4×C53—28no—
115C2×D4×C53—28no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—44no—
122C2×D4×C53—42no—
123C2×D4×C53—44no—
124C2×D4×C53—42no—
125C2×D4×C53C2×D414yes—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53#4112yes—
135C22×C4×C53#4112yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—216no—
139C2×C22⋊D53—216no—
140C2×C22⋊D53—48no—
141C2×C22⋊D53—48no—
142C2×C22⋊D53—432no—
143C2×C22⋊D53—432no—
144C2×C22⋊D53—48no—
145C2×C22⋊D53—48no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53D814yes—
155C22×Dic53SD1614yes—
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D412yes—
158C2×D212—28no—
159C2×D212—28no—
160C2×D212—22no—
161C2×D212—22no—
162C2×D212—22no—
163C2×D212—22no—
164C2×D212—24no—
165C2×D212—22no—
166C2×D212—44no—
167C2×D212—42no—
168C2×D212—44no—
169C2×D212—42no—
170C2×D212C2×D414yes—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—44no—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53C2×D414yescomplemented
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—42no—
181C4⋊Dic53—48no—
182C4×Dic53—24no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—44no—
189C23×C53—24no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—48no—
197C23×C53—48no—
198C23×C53—48no—
199C23×C53—84no—
200C23×C53—84no—
201C23×C53—84no—
202C23×C53—84no—
203C23×C53#12012yes—
204C23×C53#12112yes—
205C23×C53#12412yes—
206C23×C53C2×D812yes—
207C23×C53C2×SD1612yes—
208C23×C53AGL(1,ℤ/8)12yes—
209C23×C53AGL(1,ℤ/8)12yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—24no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—48no—
219C22×D53—48no—
220C22×D53—48no—
221C22×D53—84no—
222C22×D53—84no—
223C22×D53—84no—
224C22×D53—84no—
225D4×C53—24no—
226D4×C53—24no—
227D4×C53—42no—
228D4×C53—42no—
229D4×C53—48no—
230C2×C4×C53—24no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—48no—
234C2×C4×C53#12012yes—
235C2×C4×C53#12114yes—
236C2×C4×C53#12112yes—
237C2×C4×C53#12114yes—
238C2×C4×C53#12314yes—
239C2×C4×C53#12314yes—
240C2×C4×C53#12412yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C22⋊D53—432no—
248C22⋊D53—432no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—28no—
254C2×Dic53—28no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—84no—
259C2×Dic53C2×D814yes—
260C2×Dic53C2×SD1614yes—
261D212—24no—
262D212—24no—
263D212—42no—
264D212—42no—
265D212—48no—
266C8×C53—42no—
267#88322—42no—
268C2×#2041—531nomaximal, Sylow
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—44no—
272C22×C53—82no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—816no—
277C22×C53#39414yes—
278C22×C53#44714yes—
279C22×C53#45014yes—
280C22×C53#45914yes—
281C22×C53#46014yes—
282C22×C53#48614yes—
283C22×C53#49014yes—
284C22×C53C2×#12011yescharacteristic
285C22×C53C2×#12111yescharacteristic
286C22×C53C2×#12411yescharacteristic
287C22×C53C22×D811yescharacteristic
288C22×C53C22×SD1611yescharacteristic
289C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
291C2×D53—44no—
292C2×D53—44no—
293C2×D53—44no—
294C2×D53—82no—
295C2×D53—82no—
296C2×D53—82no—
297C2×D53—82no—
298C2×D53—816no—
299C4×C53—44no—
300C4×C53#39414yes—
301C4×C53C2×#12111yescharacteristic
302C4×C53C2×#12111yescharacteristic
303C4×C53C2×#12311yescharacteristic
304C4×C53C2×#12311yescharacteristic
305Dic53—24no—
306Dic53—24no—
307Dic53—82no—
308C22×#127—531no—
309C22×#102—531no—
310C22×#102—531no—
311C22×#102—531no—
312C22×#102D5311yescharacteristic, complemented
313C2×#389—531no—
314#2041—538no—
315C2×#336—531no—
316C2×C53—84no—
317C2×C53#204118yescomplemented
318C2×C53C2×#39411yescharacteristic
319C2×C53C2×#44711yescharacteristic
320C2×C53C2×#45011yescharacteristic
321C2×C53C2×#45911yescharacteristic
322C2×C53C2×#46011yescharacteristic
323C2×C53C2×#48611yescharacteristic
324C2×C53C2×#49011yescharacteristic
325D53—84no—
326C23×D4—531no—
327C23×D4—1061no—
328C23×D4—1061no—
329C23×D4C2×D5311yescharacteristic
330C23×C8—531no—
331C23×C8C2×D5311yescharacteristic
332C2×#127—534no—
333C2×#127—1062no—
334C2×#127—1064no—
335C2×#127—1064no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—1061no—
338C22×C4⋊C4C2×D5311yescharacteristic
339C22×C42—531no—
340C2×#102—24no—
341C2×#102—534no—
342C2×#102—534no—
343C2×#102—534no—
344C2×#102—1064no—
345C2×#102—1064no—
346C2×#102—1064no—
347C2×#102C2×D5314yescomplemented
348#389—534no—
349#389—1062no—
350C2×#95—1061no—
351#336—534no—
352C53C2×#204111yescharacteristic, complemented, Sylow
353C25—21no—
354C25—1061no—
355C22×D4—22no—
356C22×D4—21no—
357C22×D4—22no—
358C22×D4—22no—
359C22×D4—22no—
360C22×D4—22no—
361C22×D4—534no—
362C22×D4—532no—
363C22×D4—1064no—
364C22×D4—1064no—
365C22×D4—1068no—
366C22×D4—1068no—
367C22×D4—1062no—
368C22×D4—1062no—
369C22×D4—1061no—
370C22×D4—1062no—
371C22×D4—1062no—
372C22×D4—1062no—
373C22×D4—2124no—
374C22×D4—2124no—
375C22×D4—2124no—
376C22×D4—2124no—
377C22×D4D21214yes—
378C22×D4C22×D5312yes—
379C23×C4—21no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—1061no—
383C23×C4C22×D5311yescharacteristic
384C22×C8—22no—
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—532no—
388C22×C8—1062no—
389C22×C8—1062no—
390C22×C8—1062no—
391C22×C8C22×D5312yes—
392#127—1068no—
393#127—10616no—
394#127—1068no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—532no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—2124no—
403C2×C4⋊C4C22⋊D5314yescomplemented
404C2×C4⋊C4C22×D5312yes—
405C2×C42—532no—
406C2×C42—532no—
407C2×C42—1062no—
408C2×C42—1062no—
409#102—28no—
410#102—1068no—
411#102—1068no—
412#102—1068no—
413#95—1064no—
414C24—24no—
415C24—22no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—1064no—
423C24—1062no—
424C24—2122no—
425C24—2122no—
426C24—2122no—
427C24—2122no—
428C24—2122no—
429C24—2122no—
430C24D4×D5311yescharacteristic
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—28no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—534no—
444C2×D4—10616no—
445C2×D4—10616no—
446C2×D4—1064no—
447C2×D4—1068no—
448C2×D4—1068no—
449C2×D4—1062no—
450C2×D4—1062no—
451C2×D4—1062no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—2128no—
455C2×D4—2128no—
456C2×D4—21232no—
457C2×D4—21232no—
458C2×D4—2128no—
459C2×D4—2128no—
460C2×D4—2122no—
461C2×D4—2124no—
462C2×D4—2124no—
463C2×D4—2122no—
464C2×D4C2×D21214yes—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4C2×D21212yes—
482C22×C4#1171994512yes—
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5312yes—
485C22×C4#1171994712yes—
486C22×C4#1171994712yes—
487C22×C4C2×C22⋊D5312yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—44no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—1062no—
496C2×C8—2124no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—1062no—
500C4⋊C4—1062no—
501C4⋊C4—2128no—
502C4⋊C4—2122no—
503C4⋊C4C2×C22⋊D5314yescomplemented
504C42—1064no—
505C42—1062no—
506C42—1062no—
507C23—24no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—1064no—
522C23—4244no—
523C23—4244no—
524C23—4244no—
525C23—4244no—
526C23—2128no—
527C23—2128no—
528C23—2128no—
529C23—2122no—
530C23—2122no—
531C23—2122no—
532C23—2122no—
533C23—2122no—
534C23—2122no—
535C23#1313318712yes—
536C23#1313318912yes—
537C23#1313320312yes—
538C23D8×D5312yes—
539C23SD16×D5312yes—
540C23#1313324112yes—
541C23#1313324412yes—
542C23C2×D4×D5311yescharacteristic
543D4—24no—
544D4—24no—
545D4—48no—
546D4—42no—
547D4—42no—
548D4—1064no—
549D4—1064no—
550D4—21232no—
551D4—21232no—
552D4—2128no—
553D4—2122no—
554D4—2122no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—534no—
560C2×C4—534no—
561C2×C4—1068no—
562C2×C4—1068no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—4244no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4#1313318712yes—
572C2×C4#1313318814yes—
573C2×C4#1313318912yes—
574C2×C4#1313319114yes—
575C2×C4#1313319914yes—
576C2×C4#1313320114yes—
577C2×C4#1313320312yes—
578C2×C4C2×D4×D5311yescharacteristic
579C2×C4C22×C22⋊D5311yescharacteristic
580C2×C4C22×D21211yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994711yescharacteristic
583C2×C4C2×#1171994511yescharacteristic
584C8—42no—
585C8—2122no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—816no—
593C22—82no—
594C22—4242no—
595C22—4242no—
596C22—4242no—
597C22—42416no—
598C22—4242no—
599C22—2124no—
600C22—2124no—
601C22—2124no—
602C22#1681053914yes—
603C22#1681075814yes—
604C22#1681077514yes—
605C22#1681081614yes—
606C22#1681082114yes—
607C22#1681096814yes—
608C22#1681098614yes—
609C22C2×#1313318711yescharacteristic
610C22C2×#1313318911yescharacteristic
611C22C2×#1313320311yescharacteristic
612C22C2×D8×D5311yescharacteristic
613C22C2×SD16×D5311yescharacteristic
614C22C2×#1313324111yescharacteristic
615C22C2×#1313324411yescharacteristic
616C4—44no—
617C4—1064no—
618C4—1064no—
619C4—4242no—
620C4#1681053914yes—
621C4C2×#1313318811yescharacteristic
622C4C2×#1313319111yescharacteristic
623C4C2×#1313319911yescharacteristic
624C4C2×#1313320111yescharacteristic
625C2—84no—
626C2—4244no—
627C2#2027594818yesdirect product factor
628C2C2×#1681053911yescharacteristic
629C2C2×#1681075811yescharacteristic
630C2C2×#1681077511yescharacteristic
631C2C2×#1681081611yescharacteristic
632C2C2×#1681082111yescharacteristic
633C2C2×#1681096811yescharacteristic
634C2C2×#1681098611yescharacteristic
subgroupG/H
centerC24D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#102×C53C2
socleC24×C53C2×D4
prime radicalC22×#13133213C2
hypercenterC22×#102D53
hypocenterC53C2×#2041
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×#13133139C2semidirect
3. C22×#13133138C2semidirect
4. C22×#13133137C2semidirect
5. C2×#16810522C2semidirect
6. #20275948C2direct (N × H)
7. C2×#16810377C2semidirect
8. C22×#102×C53C2semidirect
21. C2×#13133139C22semidirect
23. C2×#13133138C22semidirect
25. C2×#13133137C22semidirect
27. #16810522C22semidirect
29. #16810377C22semidirect
34. C2×#102×C53C22semidirect
93. C2×C4⋊C4×C53D4semidirect
177. C4⋊C4×C53C2×D4semidirect
312. C22×#102D53semidirect
317. C2×C53#2041semidirect
347. C2×#102C2×D53semidirect
352. C53C2×#2041semidirect
403. C2×C4⋊C4C22⋊D53semidirect
503. C4⋊C4C2×C22⋊D53semidirect
627. C2#20275948direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332132
C22×#131331392
C22×#131331382
C22×#131331372
C2×#168105222
#202759482
C2×#168103772
C22×#102×C532
C2×#204153