Finite Group Databasebeta

← C2×#20275938 #27411726

Subgroups — C2×#20275938 #27411726

Order 13568.

1586 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811201C211yesmaximal, characteristic
2C22×#13133145C211yesmaximal, characteristic, complemented
3C2×#16810625C211yesmaximal, characteristic, complemented
4C2×#16810624C211yesmaximal, characteristic, complemented
5C2×#16810493C211yesmaximal, characteristic, complemented
6#20275938C218yesmaximal, direct product factor
7C2×#16810391C211yesmaximal, characteristic, complemented
8C22×#104×C53C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811201—22no—
12#16811201—22no—
13#16811201—22no—
14#16811201C2214yes—
15C2×#13133196—21no—
16C2×#13133193—21no—
17C2×#13133191—21no—
18C2×#13133189—22no—
19C2×#13133188—21no—
20C22×C4⋊Dic53—22no—
21C22×C4⋊Dic53C2211yescharacteristic
22C22×C4⋊Dic53C2211yescharacteristic
23C2×#13133145—24no—
24C2×#13133145C2214yescomplemented
25#16810625—22no—
26#16810625C2214yescomplemented
27#16810624—22no—
28#16810624C2214yescomplemented
29C2×#13133142—21no—
30C2×#13133141—21no—
31C2×#13133139—21no—
32C2×#13133138—21no—
33#16810493—22no—
34#16810493C2214yescomplemented
35#16810391C2214yescomplemented
36C22×#41×C53C2211yescharacteristic
37C22×M16×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C2×#104×C53—24no—
40C2×#104×C53C2214yescomplemented
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945D412yes—
48C2×#11719945C2312yes—
49C22×Dic106D411yescharacteristic
50C22×D212D411yescharacteristic
51C22×C22⋊D53—21no—
52C22×C22⋊D53—22no—
53C23×Dic53—21no—
54C23×Dic53D412yes—
55C22×C4×D53—21no—
56C2×#11719917—21no—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C2×#11719917C2312yes—
60C22×#88322—21no—
61#13133196—24no—
62#13133196—24no—
63#13133193—24no—
64#13133193—24no—
65#13133191—24no—
66#13133191—24no—
67#13133189—28no—
68#13133189—28no—
69#13133188—24no—
70#13133188—24no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—28no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—48no—
82C2×C4⋊Dic53D412yes—
83C2×C4⋊Dic53C2312yes—
84C2×C4⋊Dic53C2312yes—
85C2×#11719919—22no—
86C2×C22⋊Dic53—22no—
87C2×#11719921—22no—
88#13133145—28no—
89#13133142—24no—
90#13133141—24no—
91#13133139—24no—
92#13133138—24no—
93C2×#41×C53—21no—
94C2×#41×C53—21no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C2×#41×C53—22no—
98C2×#41×C53—22no—
99C2×#41×C53D412yes—
100C2×#41×C53C2312yes—
101C22×Q8×C53D411yescharacteristic
102C22×D4×C53—21no—
103C22×D4×C53D411yescharacteristic
104C23×C4×C53—21no—
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107C2×M16×C53—21no—
108C2×M16×C53—22no—
109C2×M16×C53—22no—
110C2×M16×C53C2312yes—
111C22×C8×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53—22no—
114C2×C42×C53D414yescomplemented
115C2×C42×C53C2312yes—
116#104×C53—28no—
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C23×D53—21no—
120#41×C53—24no—
121#41×C53—24no—
122#41×C53—22no—
123#41×C53—22no—
124#41×C53—22no—
125#41×C53—22no—
126#41×C53—42no—
127#41×C53—42no—
128#41×C53—42no—
129#41×C53—42no—
130C2×Q8×C53—21no—
131C2×Q8×C53—22no—
132C2×Q8×C53—22no—
133C2×Q8×C53C2×D412yes—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—21no—
137C2×D4×C53—22no—
138C2×D4×C53—22no—
139C2×D4×C53—42no—
140C2×D4×C53—42no—
141C2×D4×C53C2×D412yes—
142C22×C4×C53—24no—
143C22×C4×C53—24no—
144C22×C4×C53—24no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—42no—
150C22×C4×C53—42no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D412yes—
157C22×C4×C53#4112yes—
158C22×C4×C53#4112yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C2×C22⋊D53—22no—
162C2×C22⋊D53—22no—
163C2×C22⋊D53—28no—
164C2×C22⋊D53—42no—
165C2×C22⋊D53—42no—
166C2×C22⋊D53—84no—
167C22×Dic53—22no—
168C22×Dic53—21no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—44no—
172C22×Dic53—44no—
173C22×Dic53—41no—
174C22×Dic53—41no—
175C22×Dic53C2×D414yes—
176#11719945—24no—
177#11719945—24no—
178#11719945—22no—
179#11719945—22no—
180#11719945—22no—
181#11719945—22no—
182#11719945—42no—
183#11719945—42no—
184#11719945—42no—
185#11719945—42no—
186C2×D212—22no—
187C2×D212—22no—
188C2×D212—21no—
189C2×D212C2×D412yes—
190C2×C4×D53—22no—
191C2×C4×D53—22no—
192C2×C4×D53—42no—
193C2×C4×D53—42no—
194C2×Dic106—22no—
195C2×Dic106—22no—
196C2×Dic106—21no—
197C2×Dic106C2×D412yes—
198M16×C53—22no—
199M16×C53—22no—
200M16×C53—42no—
201C2×C8×C53—22no—
202C2×C8×C53—42no—
203C42×C53—24no—
204C42×C53—22no—
205C42×C53C2×D414yescomplemented
206C22⋊Dic53—28no—
207#11719921—28no—
208C4⋊Dic53—22no—
209C4⋊Dic53—22no—
210C4⋊Dic53—22no—
211C4⋊Dic53—22no—
212C4⋊Dic53—22no—
213C4⋊Dic53—22no—
214C4⋊Dic53—416no—
215C4⋊Dic53—41no—
216C4⋊Dic53—41no—
217#11719919—28no—
218#11719917—22no—
219#11719917—22no—
220#11719917—42no—
221C2×#88322—22no—
222C2×#88322—42no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—42no—
227C23×C53—42no—
228C23×C53—42no—
229C23×C53#12011yescharacteristic
230C23×C53#12011yescharacteristic
231C23×C53#12112yes—
232C23×C53#12412yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—22no—
235C22×D53—42no—
236C22×D53—42no—
237C22×D53—42no—
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—42no—
241D4×C53—22no—
242D4×C53—22no—
243D4×C53—42no—
244D4×C53—42no—
245D4×C53—44no—
246D4×C53—42no—
247C2×C4×C53—24no—
248C2×C4×C53—216no—
249C2×C4×C53—24no—
250C2×C4×C53—24no—
251C2×C4×C53—22no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—44no—
256C2×C4×C53#12011yescharacteristic
257C2×C4×C53#12011yescharacteristic
258C2×C4×C53#12112yes—
259C2×C4×C53#12112yes—
260C2×C4×C53#12112yes—
261C2×C4×C53#12112yes—
262C2×C4×C53#12112yes—
263C2×C4×C53#12312yes—
264C2×C4×C53#12312yes—
265C2×C4×C53#12312yes—
266C2×C4×C53#12312yes—
267C2×C4×C53#12412yes—
268C2×C4×C53C22×D411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C22⋊D53—42no—
275C22⋊D53—42no—
276C22⋊D53—44no—
277C22⋊D53—88no—
278C2×Dic53—22no—
279C2×Dic53—24no—
280C2×Dic53—42no—
281C2×Dic53—44no—
282C2×Dic53—44no—
283C2×Dic53—44no—
284C2×Dic53—42no—
285C2×Dic53—82no—
286C2×Dic53—82no—
287C2×Dic53—82no—
288D212—22no—
289D212—22no—
290D212—42no—
291C4×D53—42no—
292C4×D53—42no—
293C4×D53—44no—
294Dic106—22no—
295Dic106—22no—
296Dic106—42no—
297C8×C53—42no—
298#88322—42no—
299C2×#2039—531nomaximal, Sylow
300C22×C53—21no—
301C22×C53—21no—
302C22×C53—21no—
303C22×C53—21no—
304C22×C53—24no—
305C22×C53—24no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53—42no—
309C22×C53—84no—
310C22×C53#39412yes—
311C22×C53#39412yes—
312C22×C53#45312yes—
313C22×C53#45612yes—
314C22×C53C2×#12011yescharacteristic
315C22×C53C2×#12111yescharacteristic
316C22×C53C2×#12411yescharacteristic
317C2×D53—42no—
318C2×D53—42no—
319C2×D53—42no—
320C2×D53—84no—
321C4×C53—28no—
322C4×C53—42no—
323C4×C53#39412yes—
324C4×C53#39412yes—
325C4×C53C2×#12111yescharacteristic
326C4×C53C2×#12111yescharacteristic
327C4×C53C2×#12311yescharacteristic
328C4×C53C2×#12311yescharacteristic
329Dic53—42no—
330Dic53—44no—
331Dic53—82no—
332C2×#537—531no—
333C22×#104—531no—
334C22×#104D5311yescharacteristic, complemented
335C2×#417—531no—
336C2×#417—531no—
337C2×#384—531no—
338#2039—538no—
339C2×#343—531no—
340C2×C53—21no—
341C2×C53—21no—
342C2×C53—22no—
343C2×C53—82no—
344C2×C53#203914yescomplemented
345C2×C53C2×#39411yescharacteristic
346C2×C53C2×#45311yescharacteristic
347C2×C53C2×#45611yescharacteristic
348D53—82no—
349C22×#41—531no—
350C22×#41C2×D5311yescharacteristic
351C22×M16—531no—
352C22×M16C2×D5311yescharacteristic
353#537—534no—
354#537—1062no—
355#537—1062no—
356#537—1062no—
357C2×C22⋊Q8—1061no—
358C2×C22⋊Q8—1061no—
359C2×#121—1061no—
360C2×#121—1062no—
361C2×#121—1061no—
362C22×C4⋊C4—531no—
363C22×C4⋊C4—531no—
364C22×C4⋊C4—1062no—
365C22×C42C2×D5311yescharacteristic
366C2×#104—24no—
367C2×#104—534no—
368C2×#104—1064no—
369C2×#104C2×D5314yescomplemented
370#417—534no—
371#417—534no—
372#417—1062no—
373#417—1062no—
374C2×#103—1061no—
375C2×#103—1061no—
376C2×#102—1061no—
377C2×#102—1061no—
378#384—534no—
379#384—1062no—
380#343—534no—
381C53C2×#203911yescharacteristic, complemented, Sylow
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—22no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—532no—
389C2×#41—532no—
390C2×#41—1061no—
391C2×#41—1061no—
392C2×#41—1062no—
393C2×#41—1062no—
394C2×#41—1062no—
395C2×#41—1062no—
396C2×#41D21212yes—
397C2×#41C22×D5312yes—
398C22×Q8—531no—
399C22×Q8D21211yescharacteristic
400C22×D4—21no—
401C22×D4—531no—
402C22×D4—1062no—
403C22×D4—1061no—
404C22×D4D21211yescharacteristic
405C23×C4—21no—
406C23×C4—21no—
407C23×C4—532no—
408C23×C4—1061no—
409C23×C4—1061no—
410C23×C4C22×D5311yescharacteristic
411C2×M16—21no—
412C2×M16—22no—
413C2×M16—22no—
414C2×M16—532no—
415C2×M16—1061no—
416C2×M16—1062no—
417C2×M16—1062no—
418C2×M16C22×D5312yes—
419C22×C8—21no—
420C22×C8—1061no—
421C22⋊Q8—1064no—
422C22⋊Q8—1064no—
423C22⋊Q8—1064no—
424C22⋊Q8—1064no—
425#121—1064no—
426#121—1064no—
427#121—1068no—
428#121—1064no—
429#121—1068no—
430#121—1064no—
431C2×C4⋊C4—532no—
432C2×C4⋊C4—532no—
433C2×C4⋊C4—532no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1068no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—2128no—
446C2×C22⋊C4—1062no—
447C2×C22⋊C4—1062no—
448C2×C42—21no—
449C2×C42—22no—
450C2×C42C22⋊D5314yescomplemented
451C2×C42C22×D5312yes—
452#104—28no—
453#104—1068no—
454#103—1064no—
455#103—1064no—
456#102—1064no—
457#102—1064no—
458C24—21no—
459C24—1061no—
460C24D4×D5311yescharacteristic
461#41—24no—
462#41—24no—
463#41—22no—
464#41—22no—
465#41—22no—
466#41—22no—
467#41—42no—
468#41—42no—
469#41—42no—
470#41—42no—
471#41—1064no—
472#41—1064no—
473#41—1062no—
474#41—1062no—
475#41—1062no—
476#41—1062no—
477#41—2122no—
478#41—2122no—
479#41—2122no—
480#41—2122no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—532no—
485C2×Q8—1061no—
486C2×Q8—1062no—
487C2×Q8—1062no—
488C2×Q8C2×D21212yes—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—532no—
497C2×D4—1068no—
498C2×D4—1062no—
499C2×D4—1062no—
500C2×D4—1061no—
501C2×D4—1062no—
502C2×D4—1062no—
503C2×D4—4244no—
504C2×D4—2122no—
505C2×D4—2122no—
506C2×D4C2×D21212yes—
507C22×C4—24no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—534no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1061no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—2124no—
524C22×C4—2121no—
525C22×C4—2121no—
526C22×C4—2124no—
527C22×C4—2122no—
528C22×C4—2122no—
529C22×C4C2×D21211yescharacteristic
530C22×C4C2×D21211yescharacteristic
531C22×C4#1171994512yes—
532C22×C4D4×D5311yescharacteristic
533C22×C4D4×D5311yescharacteristic
534C22×C4D4×D5311yescharacteristic
535C22×C4#1171994712yes—
536C22×C4#1171994712yes—
537C22×C4C2×C22⋊D5312yes—
538C22×C4C23×D5311yescharacteristic
539M16—22no—
540M16—22no—
541M16—42no—
542M16—1062no—
543M16—1062no—
544M16—2122no—
545C2×C8—22no—
546C2×C8—42no—
547C2×C8—1062no—
548C2×C8—2122no—
549C4⋊C4—1068no—
550C4⋊C4—1062no—
551C4⋊C4—1062no—
552C4⋊C4—1062no—
553C4⋊C4—1062no—
554C4⋊C4—1062no—
555C4⋊C4—1062no—
556C4⋊C4—21216no—
557C4⋊C4—2121no—
558C4⋊C4—2121no—
559C22⋊C4—1068no—
560C22⋊C4—1068no—
561C42—24no—
562C42—22no—
563C42C2×C22⋊D5314yescomplemented
564C23—22no—
565C23—22no—
566C23—22no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—1062no—
571C23—2122no—
572C23—2122no—
573C23—2122no—
574C23#1313318711yescharacteristic
575C23#1313318711yescharacteristic
576C23#1313318912yes—
577C23#1313320312yes—
578C23C2×D4×D5311yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—42no—
582Q8—1062no—
583Q8—1062no—
584Q8—2122no—
585D4—22no—
586D4—22no—
587D4—44no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—1062no—
592D4—1062no—
593D4—4248no—
594D4—2124no—
595D4—2122no—
596D4—2122no—
597D4—2122no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—216no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—1064no—
608C2×C4—1062no—
609C2×C4—4242no—
610C2×C4—4242no—
611C2×C4—4242no—
612C2×C4—2124no—
613C2×C4—2122no—
614C2×C4—2124no—
615C2×C4—2122no—
616C2×C4—2124no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2124no—
620C2×C4#1313318711yescharacteristic
621C2×C4#1313318711yescharacteristic
622C2×C4#1313318812yes—
623C2×C4#1313318812yes—
624C2×C4#1313318912yes—
625C2×C4#1313319112yes—
626C2×C4#1313319112yes—
627C2×C4#1313319912yes—
628C2×C4#1313319912yes—
629C2×C4#1313320112yes—
630C2×C4#1313320112yes—
631C2×C4#1313320312yes—
632C2×C4C2×D4×D5311yescharacteristic
633C2×C4C22×C22⋊D5311yescharacteristic
634C2×C4C22×D21211yescharacteristic
635C2×C4C2×#1171994711yescharacteristic
636C2×C4C2×#1171994711yescharacteristic
637C2×C4C2×#1171994511yescharacteristic
638C8—42no—
639C8—2122no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—84no—
650C22—4244no—
651C22—2122no—
652C22—2122no—
653C22—2122no—
654C22#1681053912yes—
655C22#1681053912yes—
656C22#1681079212yes—
657C22#1681080712yes—
658C22C2×#1313318711yescharacteristic
659C22C2×#1313318911yescharacteristic
660C22C2×#1313320311yescharacteristic
661C4—28no—
662C4—42no—
663C4—4242no—
664C4—2124no—
665C4—2122no—
666C4#1681053912yes—
667C4#1681053912yes—
668C4C2×#1313318811yescharacteristic
669C4C2×#1313319111yescharacteristic
670C4C2×#1313319911yescharacteristic
671C4C2×#1313320111yescharacteristic
672C2—21no—
673C2—22no—
674C2—21no—
675C2—82no—
676C2—4242no—
677C2#2027593814yesdirect product factor
678C2C2×#1681053911yescharacteristic
679C2C2×#1681079211yescharacteristic
680C2C2×#1681080711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#104×C53C2
socleC23×C53#120
prime radicalC2×#16811201C2
hypercenterC22×#104D53
hypocenterC53C2×#2039
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#104

Complements

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

NH ≅ G/Ntype
2. C22×#13133145C2semidirect
3. C2×#16810625C2semidirect
4. C2×#16810624C2semidirect
5. C2×#16810493C2semidirect
6. #20275938C2direct (N × H)
7. C2×#16810391C2semidirect
8. C22×#104×C53C2semidirect
24. C2×#13133145C22semidirect
26. #16810625C22semidirect
28. #16810624C22semidirect
34. #16810493C22semidirect
35. #16810391C22semidirect
40. C2×#104×C53C22semidirect
114. C2×C42×C53D4semidirect
205. C42×C53C2×D4semidirect
334. C22×#104D53semidirect
344. C2×C53#2039semidirect
369. C2×#104C2×D53semidirect
381. C53C2×#2039semidirect
450. C2×C42C22⋊D53semidirect
563. C42C2×C22⋊D53semidirect
677. C2#20275938direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168112012
C22×#131331452
C2×#168106252
C2×#168106242
C2×#168104932
#202759382
C2×#168103912
C22×#104×C532
C2×#203953