Finite Group Databasebeta

← C2×#20275931 #27411684

Subgroups — C2×#20275931 #27411684

Order 13568.

1434 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
1C2×#16811194C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic
4C22×#13133135C211yesmaximal, characteristic, complemented
5C2×#16810529C211yesmaximal, characteristic, complemented
6#20275931C218yesmaximal, direct product factor
7C2×#16810383C211yesmaximal, characteristic, complemented
8C22×#104×C53C211yesmaximal, characteristic
9C23×Dic106C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11#16811194—22no—
12#16811194—22no—
13#16811194—22no—
14#16811194C2214yes—
15C2×#13133215—21no—
16C2×#13133204—21no—
17C2×#13133198—22no—
18C2×#13133196—21no—
19C2×#13133196—21no—
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810665—22no—
23#16810665C2214yescomplemented
24C2×#13133151—21no—
25C2×#13133149—21no—
26#16810624—22no—
27#16810624C2214yes—
28C2×#13133141—21no—
29C2×#13133138—21no—
30C2×#13133135—24no—
31C2×#13133135C2214yescomplemented
32#16810529—22no—
33#16810529C2214yescomplemented
34C2×#13133124—22no—
35#16810383C2214yescomplemented
36C22×#41×C53C2211yescharacteristic
37C22×M16×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C2×#104×C53—24no—
40C2×#104×C53C2214yes—
41C22×Dic106—22no—
42C22×Dic106—21no—
43C22×Dic106—21no—
44C22×Dic106—21no—
45C22×Dic106—21no—
46C22×Dic106—22no—
47C22×Dic106—22no—
48C22×Dic106—22no—
49C22×Dic106D412yes—
50C22×Dic106C2312yes—
51C23×Dic53—21no—
52C23×Dic53D412yes—
53C2×#11719917—21no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917C2312yes—
57C22×#88322—21no—
58#13133215—24no—
59#13133215—24no—
60#13133204—24no—
61#13133204—24no—
62#13133198—28no—
63#13133198—28no—
64#13133196—24no—
65#13133196—24no—
66#13133196—24no—
67#13133196—24no—
68#13133176—21no—
69#13133176—21no—
70#13133176D412yes—
71#13133176C2312yes—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53D411yescharacteristic
78C2×C4⋊Dic53D411yescharacteristic
79C2×C4⋊Dic53C2312yes—
80C2×#11719919—22no—
81C2×#11719919—22no—
82C2×C22⋊Dic53—22no—
83C2×C22⋊Dic53—22no—
84C2×C22⋊Dic53—21no—
85C2×C4×Dic53—21no—
86#13133151—24no—
87#13133149—24no—
88#13133141—24no—
89#13133138—24no—
90#13133135—28no—
91#13133124—28no—
92C2×#41×C53—21no—
93C2×#41×C53—21no—
94C2×#41×C53—22no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C2×#41×C53—22no—
98C2×#41×C53D412yes—
99C2×#41×C53C2312yes—
100C22×Q8×C53D411yescharacteristic
101C22×D4×C53—21no—
102C22×D4×C53D411yescharacteristic
103C23×C4×C53—21no—
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—21no—
107C2×M16×C53—22no—
108C2×M16×C53D414yes—
109C2×M16×C53C2312yes—
110C22×C8×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53—22no—
113C2×C42×C53—22no—
114C2×C42×C53C2312yes—
115#104×C53—28no—
116C24×C53—21no—
117C24×C53C2×D411yescharacteristic
118#41×C53—24no—
119#41×C53—24no—
120#41×C53—22no—
121#41×C53—22no—
122#41×C53—22no—
123#41×C53—22no—
124#41×C53—42no—
125#41×C53—42no—
126#41×C53—42no—
127#41×C53—42no—
128C2×Q8×C53—21no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53C2×D412yes—
132C2×D4×C53—22no—
133C2×D4×C53—22no—
134C2×D4×C53—21no—
135C2×D4×C53—22no—
136C2×D4×C53—22no—
137C2×D4×C53—42no—
138C2×D4×C53—42no—
139C2×D4×C53C2×D412yes—
140C22×C4×C53—24no—
141C22×C4×C53—21no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53#4112yes—
156C22×C4×C53#4112yes—
157C22×C4×C53#4112yes—
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—24no—
164C22×Dic53—24no—
165C22×Dic53—22no—
166C22×Dic53—44no—
167C22×Dic53—41no—
168C22×Dic53—41no—
169C22×Dic53C2×D414yes—
170C2×Dic106—22no—
171C2×Dic106—22no—
172C2×Dic106—24no—
173C2×Dic106—24no—
174C2×Dic106—22no—
175C2×Dic106—22no—
176C2×Dic106—22no—
177C2×Dic106—22no—
178C2×Dic106—22no—
179C2×Dic106—22no—
180C2×Dic106—42no—
181C2×Dic106—42no—
182C2×Dic106—42no—
183C2×Dic106—42no—
184C2×Dic106—42no—
185C2×Dic106—42no—
186C2×Dic106—42no—
187C2×Dic106—42no—
188M16×C53—24no—
189M16×C53—22no—
190M16×C53C2×D414yes—
191C2×C8×C53—24no—
192C2×C8×C53—22no—
193C42×C53—22no—
194C42×C53—22no—
195C42×C53—42no—
196C22⋊Dic53—28no—
197C22⋊Dic53—28no—
198C22⋊Dic53—22no—
199C22⋊Dic53—22no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—22no—
203C4⋊Dic53—22no—
204C4⋊Dic53—41no—
205C4⋊Dic53—41no—
206C4⋊Dic53C2×D412yes—
207C4⋊Dic53C2×D412yes—
208#11719919—28no—
209#11719919—28no—
210C4×Dic53—22no—
211C4×Dic53—22no—
212#11719917—22no—
213#11719917—22no—
214#11719917—42no—
215C2×#88322—22no—
216C2×#88322—42no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53—42no—
223C23×C53#12011yescharacteristic
224C23×C53#12011yescharacteristic
225C23×C53#12112yes—
226C23×C53#12312yes—
227C23×C53C22×D411yescharacteristic
228Q8×C53—22no—
229Q8×C53—22no—
230Q8×C53—42no—
231D4×C53—22no—
232D4×C53—22no—
233D4×C53—42no—
234D4×C53—42no—
235D4×C53—44no—
236D4×C53—42no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—42no—
240C2×C4×C53—48no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—44no—
246C2×C4×C53#12011yescharacteristic
247C2×C4×C53#12011yescharacteristic
248C2×C4×C53#12112yes—
249C2×C4×C53#12112yes—
250C2×C4×C53#12112yes—
251C2×C4×C53#12112yes—
252C2×C4×C53#12112yes—
253C2×C4×C53#12312yes—
254C2×C4×C53#12312yes—
255C2×C4×C53#12312yes—
256C2×C4×C53#12412yes—
257C2×C4×C53#12412yes—
258C2×C4×C53C22×D411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×Dic53—24no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—44no—
270C2×Dic53—44no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—44no—
274C2×Dic53—42no—
275C2×Dic53—82no—
276C2×Dic53—82no—
277Dic106—42no—
278Dic106—42no—
279Dic106—42no—
280Dic106—42no—
281Dic106—44no—
282Dic106—44no—
283C8×C53—24no—
284#88322—42no—
285C2×#2038—531nomaximal, Sylow
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—21no—
290C22×C53—24no—
291C22×C53—24no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—84no—
296C22×C53#39412yes—
297C22×C53#39412yes—
298C22×C53#45412yes—
299C22×C53#45612yes—
300C22×C53C2×#12011yescharacteristic
301C22×C53C2×#12111yescharacteristic
302C22×C53C2×#12311yescharacteristic
303C4×C53—44no—
304C4×C53—42no—
305C4×C53#39412yes—
306C4×C53#39412yes—
307C4×C53C2×#12111yescharacteristic
308C4×C53C2×#12111yescharacteristic
309C4×C53C2×#12311yescharacteristic
310C4×C53C2×#12411yescharacteristic
311Dic53—44no—
312Dic53—42no—
313Dic53—42no—
314Dic53—82no—
315C2×#536—531no—
316C2×#428—531no—
317C22×#104D5311yescharacteristic
318C2×#417—531no—
319C22×#101—531no—
320C2×#391—531no—
321#2038—538no—
322C2×#339—531no—
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—22no—
326C2×C53—82no—
327C2×C53#203814yescomplemented
328C2×C53C2×#39411yescharacteristic
329C2×C53C2×#45411yescharacteristic
330C2×C53C2×#45611yescharacteristic
331C22×#41C2×D5311yescharacteristic
332C23×Q8—531no—
333C22×M16—531no—
334C22×M16C2×D5311yescharacteristic
335#536—534no—
336#536—1062no—
337#536—1062no—
338#536—1062no—
339C2×C4⋊Q8—1061no—
340C2×#124—1061no—
341C2×#123—1062no—
342C2×C22⋊Q8—1061no—
343C2×C22⋊Q8—1061no—
344C2×#117—531no—
345C22×C4⋊C4—531no—
346C22×C42C2×D5311yescharacteristic
347#428—534no—
348#428—1062no—
349C2×#107—1061no—
350C2×#106—1061no—
351C2×#104—24no—
352C2×#104C2×D5314yes—
353#417—534no—
354#417—1062no—
355C2×#103—1061no—
356C2×#102—1061no—
357C2×#101—534no—
358C2×#101—1064no—
359#391—534no—
360#391—1062no—
361C2×#95—1062no—
362#339—534no—
363C53C2×#203811yescharacteristic, complemented, Sylow
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—22no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C2×#41D21212yes—
371C2×#41C22×D5312yes—
372C22×Q8—532no—
373C22×Q8—532no—
374C22×Q8—1061no—
375C22×Q8—1061no—
376C22×Q8—1062no—
377C22×Q8—1061no—
378C22×Q8—1061no—
379C22×Q8—1062no—
380C22×Q8—1062no—
381C22×Q8—1062no—
382C22×Q8D21211yescharacteristic
383C22×D4—21no—
384C22×D4D21211yescharacteristic
385C23×C4—21no—
386C23×C4—21no—
387C23×C4—532no—
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C2×M16—21no—
391C2×M16—22no—
392C2×M16—532no—
393C2×M16—1061no—
394C2×M16—1062no—
395C2×M16—1062no—
396C2×M16C22⋊D5314yes—
397C2×M16C22×D5312yes—
398C22×C8—21no—
399C22×C8—1061no—
400C4⋊Q8—1064no—
401C4⋊Q8—1064no—
402#124—1064no—
403#124—1064no—
404#123—1068no—
405#123—1068no—
406C22⋊Q8—1064no—
407C22⋊Q8—1064no—
408C22⋊Q8—1064no—
409C22⋊Q8—1064no—
410#117—532no—
411#117—532no—
412#117—1061no—
413#117—1061no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—532no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1061no—
427C2×C42—22no—
428C2×C42—22no—
429C2×C42—21no—
430C2×C42—1061no—
431C2×C42C22×D5312yes—
432#107—1064no—
433#106—1064no—
434#104—28no—
435#103—1064no—
436#102—1064no—
437#101—1068no—
438#95—1068no—
439C24—21no—
440C24D4×D5311yescharacteristic
441#41—24no—
442#41—24no—
443#41—22no—
444#41—22no—
445#41—22no—
446#41—22no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—42no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—1062no—
455C2×Q8—1062no—
456C2×Q8—1064no—
457C2×Q8—1064no—
458C2×Q8—1062no—
459C2×Q8—1062no—
460C2×Q8—1062no—
461C2×Q8—1062no—
462C2×Q8—1062no—
463C2×Q8—1062no—
464C2×Q8—2122no—
465C2×Q8—2122no—
466C2×Q8—2122no—
467C2×Q8—2122no—
468C2×Q8—2122no—
469C2×Q8—2122no—
470C2×Q8—2122no—
471C2×Q8—2122no—
472C2×Q8C2×D21212yes—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—21no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4C2×D21212yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—534no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1064no—
496C22×C4—1064no—
497C22×C4—1062no—
498C22×C4—2124no—
499C22×C4—2121no—
500C22×C4—2121no—
501C22×C4C2×D21211yescharacteristic
502C22×C4C2×D21211yescharacteristic
503C22×C4#1171994512yes—
504C22×C4D4×D5311yescharacteristic
505C22×C4D4×D5311yescharacteristic
506C22×C4D4×D5311yescharacteristic
507C22×C4#1171994712yes—
508C22×C4#1171994712yes—
509C22×C4C2×C22⋊D5312yes—
510C22×C4C23×D5311yescharacteristic
511M16—24no—
512M16—22no—
513M16—1062no—
514M16—1062no—
515M16—2122no—
516M16C2×C22⋊D5314yes—
517C2×C8—24no—
518C2×C8—22no—
519C2×C8—1062no—
520C2×C8—2122no—
521C4⋊C4—532no—
522C4⋊C4—532no—
523C4⋊C4—1068no—
524C4⋊C4—1068no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1062no—
528C4⋊C4—1062no—
529C4⋊C4—2121no—
530C4⋊C4—2121no—
531C22⋊C4—1068no—
532C22⋊C4—1068no—
533C22⋊C4—1062no—
534C22⋊C4—1062no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—1062no—
539C42—1062no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23#1313318711yescharacteristic
547C23#1313318711yescharacteristic
548C23#1313318812yes—
549C23#1313319912yes—
550C23C2×D4×D5311yescharacteristic
551Q8—22no—
552Q8—22no—
553Q8—42no—
554Q8—2124no—
555Q8—2124no—
556Q8—2122no—
557Q8—2122no—
558Q8—2122no—
559Q8—2122no—
560D4—22no—
561D4—22no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—42no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—4242no—
581C2×C4—4242no—
582C2×C4—2124no—
583C2×C4—2122no—
584C2×C4—2124no—
585C2×C4—2124no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4#1313318711yescharacteristic
589C2×C4#1313318711yescharacteristic
590C2×C4#1313318812yes—
591C2×C4#1313318912yes—
592C2×C4#1313318912yes—
593C2×C4#1313319112yes—
594C2×C4#1313319112yes—
595C2×C4#1313319912yes—
596C2×C4#1313320112yes—
597C2×C4#1313320112yes—
598C2×C4#1313320312yes—
599C2×C4#1313320312yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C22×C22⋊D5311yescharacteristic
602C2×C4C22×D21211yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994511yescharacteristic
606C8—24no—
607C8—2122no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—84no—
618C22#1681053912yes—
619C22#1681053912yes—
620C22#1681079712yes—
621C22#1681080712yes—
622C22C2×#1313318711yescharacteristic
623C22C2×#1313318811yescharacteristic
624C22C2×#1313319911yescharacteristic
625C4—44no—
626C4—42no—
627C4—4242no—
628C4—2122no—
629C4—2122no—
630C4—2124no—
631C4#1681053912yes—
632C4#1681053912yes—
633C4C2×#1313318911yescharacteristic
634C4C2×#1313319111yescharacteristic
635C4C2×#1313320111yescharacteristic
636C4C2×#1313320311yescharacteristic
637C2—21no—
638C2—22no—
639C2—21no—
640C2—82no—
641C2#2027593114yesdirect product factor
642C2C2×#1681053911yescharacteristic
643C2C2×#1681079711yescharacteristic
644C2C2×#1681080711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×#104×C53C2
socleC23×C53#120
prime radicalC22×#41×C53C22
hypercenterC22×#104D53
hypocenterC53C2×#2038
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. C2×#16810665C2semidirect
4. C22×#13133135C2semidirect
5. C2×#16810529C2semidirect
6. #20275931C2direct (N × H)
7. C2×#16810383C2semidirect
23. #16810665C22semidirect
31. C2×#13133135C22semidirect
33. #16810529C22semidirect
35. #16810383C22semidirect
327. C2×C53#2038semidirect
363. C53C2×#2038semidirect
641. C2#20275931direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111942
C2×#168106652
C2×#168106242
C22×#131331352
C2×#168105292
#202759312
C2×#168103832
C22×#104×C532
C2×#203853