Finite Group Databasebeta

← #27411569

Subgroups — #27411569

Order 13568.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810659C211yesmaximal, characteristic, complemented
2C2×#16810619C211yesmaximal, characteristic, complemented
3C2×#16810619C211yesmaximal, characteristic, complemented
4C22×C22⋊#88322C211yesmaximal, characteristic, complemented
5#20277098C211yesmaximal, characteristic, complemented
6#20275915C212yesmaximal, complemented
7#20275908C212yesmaximal, complemented
8#20275901C212yesmaximal, complemented
9#20275894C212yesmaximal, complemented
10#20274431C211yesmaximal, characteristic, complemented
11C2×C53×#525C211yesmaximal, characteristic
12C23×#88322C2211yescharacteristic
13C23×#88322C2211yescharacteristic
14C2×#13133176C2211yescharacteristic
15C2×#13133176C2211yescharacteristic
16#16810659—24no—
17C2×#13133150C2212yescomplemented
18C2×#13133148C2212yescomplemented
19#16810619—24no—
20#16810619—24no—
21C2×#13133141C2211yescharacteristic, complemented
22C2×#13133141C2212yescomplemented
23C2×#13133141C2211yescharacteristic, complemented
24C2×#13133138C2211yescharacteristic, complemented
25C2×#13133138C2212yescomplemented
26C2×#13133138C2211yescharacteristic, complemented
27C2×C22⋊#88322—24no—
28C2×C22⋊#88322C2212yes—
29C2×C22⋊#88322C2212yes—
30#16810486—21no—
31#16810483C2212yes—
32#16810483C2212yes—
33#16810479—21no—
34C4×C22⋊Dic53—22no—
35#16810385C2212yescomplemented
36#16810385C2212yescomplemented
37C22×#41×C53C2211yescharacteristic
38C24×C4×C53C2211yescharacteristic
39C53×#525—24no—
40C53×#525—24no—
41C2×#123×C53—21no—
42C2×C22⋊Q8×C53C2212yes—
43C2×#121×C53C2212yes—
44C2×#120×C53—21no—
45C2×C4×D4×C53—22no—
46C2×#117×C53C2211yescharacteristic
47C22×#88322—22no—
48C22×#88322—22no—
49C22×#88322—22no—
50C22×#88322—22no—
51C22×#88322—22no—
52C22×#88322D414yescomplemented
53C22×#88322C2312yes—
54C22×#88322C2311yescharacteristic
55C22×#88322C2311yescharacteristic
56#13133176—22no—
57#13133176—22no—
58#13133176—22no—
59#13133176—22no—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2312yes—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53—22no—
66C2×C22⋊Dic53—22no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69#13133150—24no—
70#13133148—24no—
71#13133141—22no—
72#13133141—22no—
73#13133141—24no—
74#13133138—22no—
75#13133138—22no—
76#13133138—24no—
77C22⋊#88322—24no—
78C22⋊#88322—24no—
79#13133124—22no—
80#13133124—22no—
81C25×C53D411yescharacteristic
82C2×#41×C53—22no—
83C2×#41×C53—22no—
84C2×#41×C53—22no—
85C2×#41×C53—22no—
86C2×#41×C53—22no—
87C2×#41×C53D414yes—
88C22×Q8×C53C2311yescharacteristic
89C22×D4×C53—22no—
90C22×D4×C53—21no—
91C22×D4×C53C2311yescharacteristic
92C23×C4×C53—21no—
93C23×C4×C53—22no—
94C23×C4×C53—22no—
95C23×C4×C53—22no—
96C23×C4×C53—22no—
97C23×C4×C53—22no—
98C23×C4×C53—22no—
99C23×C4×C53D412yes—
100C23×C4×C53D411yescharacteristic
101C23×C4×C53C2312yes—
102C23×C4×C53C2311yescharacteristic
103#123×C53—28no—
104C22⋊Q8×C53—24no—
105C22⋊Q8×C53—24no—
106#121×C53—24no—
107#121×C53—24no—
108#120×C53—28no—
109C4×D4×C53—216no—
110#117×C53—22no—
111#117×C53D414yes—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53C2312yes—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—22no—
116C2×C22⋊C4×C53—22no—
117C2×C42×C53—21no—
118C24×C53—22no—
119C24×C53—24no—
120C24×C53—22no—
121C24×C53—22no—
122C24×C53—21no—
123C24×C53—42no—
124C24×C53C2×D412yes—
125C24×C53C2×D411yescharacteristic
126#41×C53—28no—
127#41×C53—28no—
128#41×C53—44no—
129#41×C53—44no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53C2×D412yes—
136C2×D4×C53—28no—
137C2×D4×C53—24no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—44no—
144C2×D4×C53—42no—
145C2×D4×C53—42no—
146C2×D4×C53—44no—
147C2×D4×C53C2×D412yes—
148C22×C4×C53—22no—
149C22×C4×C53—22no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—21no—
153C22×C4×C53—24no—
154C22×C4×C53—24no—
155C22×C4×C53—24no—
156C22×C4×C53—24no—
157C22×C4×C53—22no—
158C22×C4×C53—22no—
159C22×C4×C53—22no—
160C22×C4×C53—22no—
161C22×C4×C53—22no—
162C22×C4×C53—22no—
163C22×C4×C53—44no—
164C22×C4×C53—44no—
165C22×C4×C53—42no—
166C22×C4×C53—44no—
167C22×C4×C53—42no—
168C22×C4×C53—44no—
169C22×C4×C53C2×D412yes—
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2×D412yes—
172C22×C4×C53C2×D412yes—
173C22×C4×C53C2×D412yes—
174C22×C4×C53#4112yes—
175C22×C4×C53#4112yes—
176C22×C4×C53#4112yes—
177C22×C4×C53C2411yescharacteristic
178C22×Dic53—22no—
179C22×Dic53—21no—
180C22×Dic53—21no—
181C4⋊C4×C53—28no—
182C4⋊C4×C53—22no—
183C4⋊C4×C53C2×D414yes—
184C22⋊C4×C53—28no—
185C22⋊C4×C53—28no—
186C22⋊C4×C53—24no—
187C42×C53—24no—
188C22⋊Dic53—44no—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—44no—
192C4⋊Dic53—22no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—22no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4×Dic53—42no—
199C4×Dic53—42no—
200C2×#88322—28no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—22no—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—44no—
209C2×#88322C2×D412yes—
210C2×#88322C2×D412yes—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—28no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—22no—
220C23×C53—22no—
221C23×C53—48no—
222C23×C53—48no—
223C23×C53—44no—
224C23×C53—44no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—42no—
228C23×C53—42no—
229C23×C53#12012yes—
230C23×C53#12112yes—
231C23×C53#12312yes—
232C23×C53#13514yes—
233C23×C53C22×D411yescharacteristic
234Q8×C53—22no—
235Q8×C53—22no—
236Q8×C53—41no—
237Q8×C53—41no—
238D4×C53—22no—
239D4×C53—22no—
240D4×C53—416no—
241D4×C53—48no—
242D4×C53—41no—
243D4×C53—41no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—44no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—48no—
259C2×C4×C53—48no—
260C2×C4×C53#12012yes—
261C2×C4×C53#12114yes—
262C2×C4×C53#12114yes—
263C2×C4×C53#12112yes—
264C2×C4×C53#12314yes—
265C2×C4×C53#12312yes—
266C2×C4×C53#12414yes—
267C2×C4×C53C22×D411yescharacteristic
268C2×C4×C53C22×D411yescharacteristic
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×Dic53—42no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280#88322—22no—
281#88322—22no—
282#88322—42no—
283#23479—531nomaximal, Sylow
284C22×C53—24no—
285C22×C53—24no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—24no—
289C22×C53—24no—
290C22×C53—24no—
291C22×C53—416no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—44no—
298C22×C53—82no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53#39414yes—
302C22×C53C2×#12011yescharacteristic
303C22×C53C2×#12111yescharacteristic
304C22×C53C2×#12311yescharacteristic
305C22×C53C2×#13512yes—
306C22×C53#57512yes—
307C4×C53—42no—
308C4×C53—42no—
309C4×C53—44no—
310C4×C53#39414yes—
311C4×C53C2×#12111yescharacteristic
312C4×C53C2×#12111yescharacteristic
313C4×C53C2×#12311yescharacteristic
314C4×C53C2×#12411yescharacteristic
315Dic53—82no—
316Dic53—81no—
317Dic53—81no—
318C2×#525D5311yescharacteristic
319C2×#427—531no—
320C2×#416—531no—
321C2×#416—531no—
322C22×C22⋊C8—531no—
323#2280—531no—
324#2036—532no—
325#2035—532no—
326#2034—532no—
327#2033—532no—
328#1735—531no—
329C2×C53—24no—
330C2×C53—44no—
331C2×C53—82no—
332C2×C53C2×#39411yescharacteristic
333C2×C53#300212yes—
334C2×C53#305011yescharacteristic
335C2×C53#305411yescharacteristic
336C2×C53#309112yes—
337C22×#41C2×D5311yescharacteristic
338C24×C4C2×D5311yescharacteristic
339C23×C8—531no—
340C23×C8—531no—
341#525—24no—
342#525—24no—
343C2×#123—21no—
344C2×C22⋊Q8C2×D5312yes—
345C2×#121C2×D5312yes—
346C2×#120—21no—
347C2×C4×D4—22no—
348C2×#117—531no—
349C2×#117—531no—
350C2×#117C2×D5311yescharacteristic
351#427—1064no—
352C2×#107—532no—
353C2×#106—532no—
354#416—1064no—
355#416—1064no—
356C2×#103—531no—
357C2×#103—532no—
358C2×#103—531no—
359C2×#102—531no—
360C2×#102—532no—
361C2×#102—531no—
362C2×C22⋊C8—532no—
363C2×C22⋊C8—532no—
364C2×C22⋊C8—1064no—
365#381—1061no—
366#380—532no—
367#380—532no—
368#379—1061no—
369C4×C22⋊C4—1062no—
370#340—532no—
371#340—532no—
372C53#2347911yescharacteristic, complemented, Sylow
373C25C22⋊D5311yescharacteristic
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41C22⋊D5314yes—
380C22×Q8C22×D5311yescharacteristic
381C22×D4—22no—
382C22×D4—21no—
383C22×D4C22×D5311yescharacteristic
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—21no—
391C23×C4C22⋊D5311yescharacteristic
392C23×C4C22⋊D5312yes—
393C23×C4C22×D5312yes—
394C23×C4C22×D5311yescharacteristic
395C22×C8—534no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—532no—
399C22×C8—1062no—
400C22×C8—1062no—
401C22×C8—1062no—
402C22×C8—1062no—
403C22×C8—1062no—
404#123—28no—
405C22⋊Q8—24no—
406C22⋊Q8—24no—
407#121—24no—
408#121—24no—
409#120—28no—
410C4×D4—216no—
411#117—22no—
412#117—1062no—
413#117—1062no—
414#117—1062no—
415#117—1062no—
416#117C22⋊D5314yes—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—532no—
421C2×C4⋊C4C22×D5312yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4—1062no—
429C2×C42—21no—
430C2×C42—1061no—
431C2×C42—1061no—
432#107—1064no—
433#106—1064no—
434#103—1062no—
435#103—1062no—
436#103—1064no—
437#102—1062no—
438#102—1062no—
439#102—1064no—
440C22⋊C8—1064no—
441C22⋊C8—1064no—
442#95—1062no—
443#95—1062no—
444C24—22no—
445C24—24no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—42no—
450C24C2×C22⋊D5312yes—
451C24C2×C22⋊D5311yescharacteristic
452#41—28no—
453#41—28no—
454#41—44no—
455#41—44no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×C22⋊D5312yes—
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4C2×C22⋊D5312yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—1062no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4D4×D5312yes—
499C22×C4#1171994712yes—
500C22×C4#1171994712yes—
501C22×C4#1171994712yes—
502C22×C4C2×C22⋊D5312yes—
503C22×C4C2×C22⋊D5312yes—
504C22×C4C2×C22⋊D5312yes—
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507C2×C8—532no—
508C2×C8—532no—
509C2×C8—1068no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1062no—
515C2×C8—1062no—
516C2×C8—1062no—
517C2×C8—2124no—
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1062no—
523C4⋊C4—1062no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4C2×C22⋊D5314yes—
527C22⋊C4—28no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—2124no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C22⋊C4—2124no—
534C42—24no—
535C42—2122no—
536C42—2122no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—48no—
548C23—48no—
549C23—44no—
550C23—44no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23#1313318612yes—
556C23#1313318912yes—
557C23#1313319812yes—
558C23#1313323714yes—
559C23C22×C22⋊D5311yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—416no—
567D4—48no—
568D4—41no—
569D4—41no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—2122no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4#1313318612yes—
594C2×C4#1313318814yes—
595C2×C4#1313318912yes—
596C2×C4#1313318914yes—
597C2×C4#1313319812yes—
598C2×C4#1313319814yes—
599C2×C4#1313320414yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C22×C22⋊D5311yescharacteristic
602C2×C4C22×C22⋊D5311yescharacteristic
603C2×C4C2×#1171994711yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994711yescharacteristic
606C8—1062no—
607C8—1062no—
608C8—2122no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22#1681053714yes—
627C22C2×#1313318611yescharacteristic
628C22C2×#1313318911yescharacteristic
629C22C2×#1313319811yescharacteristic
630C22C2×#1313323712yes—
631C22#1681142112yes—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—4242no—
636C4—4241no—
637C4—4241no—
638C4#1681053714yes—
639C4C2×#1313318811yescharacteristic
640C4C2×#1313318911yescharacteristic
641C4C2×#1313319811yescharacteristic
642C4C2×#1313320411yescharacteristic
643C2—24no—
644C2—44no—
645C2—82no—
646C2C2×#1681053711yescharacteristic
647C2#2028393312yes—
648C2#2028429311yescharacteristic
649C2#2028432111yescharacteristic
650C2#2028462412yes—
subgroupG/H
centerC22×C4C2×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#525C2
socleC23×C53C22×D4
hypercenterC2×#525D53
hypocenterC53#23479
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C2×#16810659C2semidirect
2. C2×#16810619C2semidirect
3. C2×#16810619C2semidirect
4. C22×C22⋊#88322C2semidirect
5. #20277098C2semidirect
6. #20275915C2semidirect
7. #20275908C2semidirect
8. #20275901C2semidirect
9. #20275894C2semidirect
10. #20274431C2semidirect
17. C2×#13133150C22semidirect
18. C2×#13133148C22semidirect
21. C2×#13133141C22semidirect
22. C2×#13133141C22semidirect
23. C2×#13133141C22semidirect
24. C2×#13133138C22semidirect
25. C2×#13133138C22semidirect
26. C2×#13133138C22semidirect
35. #16810385C22semidirect
36. #16810385C22semidirect
52. C22×#88322D4semidirect
372. C53#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#168106592
C2×#168106192
C22×C22⋊#883222
#202770982
#202759152
#202759082
#202759012
#202758942
#202744312
C2×C53×#5252
#2347953