Finite Group Databasebeta

← C2×#20275947 #27411758

Subgroups — C2×#20275947 #27411758

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×#16810519C211yesmaximal, characteristic, complemented
6#20275947C218yesmaximal, 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⋊Dic53C2211yescharacteristic
18C22×#11719919—21no—
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133139—24no—
21C2×#13133139C2214yescomplemented
22C2×#13133138—24no—
23C2×#13133138C2214yescomplemented
24C2×#13133137—24no—
25C2×#13133137C2214yescomplemented
26#16810519—22no—
27#16810519C2214yescomplemented
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—22no—
64C2×C4⋊Dic53D414yescomplemented
65C2×C4⋊Dic53C2312yes—
66C2×#11719919—24no—
67C2×#11719919—44no—
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×C53—22no—
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—22no—
127C22×C4×C53—42no—
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—24no—
151C22×Dic53—22no—
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—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—42no—
178C4⋊Dic53—24no—
179C4⋊Dic53—22no—
180C4⋊Dic53C2×D414yescomplemented
181#11719919—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—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—84no—
234C2×C4×C53#12012yes—
235C2×C4×C53#12114yes—
236C2×C4×C53#12114yes—
237C2×C4×C53#12112yes—
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—24no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—48no—
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—82no—
300C4×C53#39414yes—
301C4×C53C2×#12111yescharacteristic
302C4×C53C2×#12111yescharacteristic
303C4×C53C2×#12311yescharacteristic
304C4×C53C2×#12311yescharacteristic
305Dic53—24no—
306Dic53—24no—
307Dic53—44no—
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—22no—
357C22×D4—22no—
358C22×D4—21no—
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—1062no—
370C22×D4—1061no—
371C22×D4—1062no—
372C22×D4—1062no—
373C22×D4—2124no—
374C22×D4—2124no—
375C22×D4—2124no—
376C22×D4—2124no—
377C22×D4C22⋊D5314yes—
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—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—534no—
399C2×C4⋊C4—532no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1064no—
403C2×C4⋊C4—2124no—
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—42no—
440C2×D4—44no—
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—2124no—
461C2×D4—2122no—
462C2×D4—2124no—
463C2×D4—2122no—
464C2×D4C2×C22⋊D5314yes—
465C22×C4—22no—
466C22×C4—42no—
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—1064no—
478C22×C4—1062no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4#1171994512yes—
482C22×C4#1171994512yes—
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5312yes—
485C22×C4D4×D5312yes—
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—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—534no—
501C4⋊C4—1064no—
502C4⋊C4—1062no—
503C4⋊C4—2128no—
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#1313318512yes—
536C23#1313319012yes—
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—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—84no—
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—1064no—
568C2×C4—2128no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4#1313318512yes—
572C2×C4#1313318914yes—
573C2×C4#1313319012yes—
574C2×C4#1313319014yes—
575C2×C4#1313319914yes—
576C2×C4#1313320014yes—
577C2×C4#1313320312yes—
578C2×C4C2×D4×D5311yescharacteristic
579C2×C4C2×D4×D5311yescharacteristic
580C2×C4C22×C22⋊D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
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#1681053814yes—
603C22#1681075514yes—
604C22#1681077214yes—
605C22#1681081714yes—
606C22#1681082214yes—
607C22#1681096814yes—
608C22#1681098614yes—
609C22C2×#1313318511yescharacteristic
610C22C2×#1313319011yescharacteristic
611C22C2×#1313320311yescharacteristic
612C22C2×D8×D5311yescharacteristic
613C22C2×SD16×D5311yescharacteristic
614C22C2×#1313324111yescharacteristic
615C22C2×#1313324411yescharacteristic
616C4—82no—
617C4—1064no—
618C4—1064no—
619C4—2124no—
620C4#1681053814yes—
621C4C2×#1313318911yescharacteristic
622C4C2×#1313319011yescharacteristic
623C4C2×#1313319911yescharacteristic
624C4C2×#1313320011yescharacteristic
625C2—84no—
626C2—4244no—
627C2#2027594718yesdirect product factor
628C2C2×#1681053811yescharacteristic
629C2C2×#1681075511yescharacteristic
630C2C2×#1681077211yescharacteristic
631C2C2×#1681081711yescharacteristic
632C2C2×#1681082211yescharacteristic
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×#16810519C2semidirect
6. #20275947C2direct (N × H)
7. C2×#16810377C2semidirect
8. C22×#102×C53C2semidirect
21. C2×#13133139C22semidirect
23. C2×#13133138C22semidirect
25. C2×#13133137C22semidirect
27. #16810519C22semidirect
29. #16810377C22semidirect
34. C2×#102×C53C22semidirect
64. C2×C4⋊Dic53D4semidirect
180. C4⋊Dic53C2×D4semidirect
312. C22×#102D53semidirect
317. C2×C53#2041semidirect
347. C2×#102C2×D53semidirect
352. C53C2×#2041semidirect
627. C2#20275947direct (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×#168105192
#202759472
C2×#168103772
C22×#102×C532
C2×#204153