Finite Group Databasebeta

← #27466384

Subgroups — #27466384

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279203C211yesmaximal, characteristic
2#20278440C211yesmaximal, characteristic, complemented
3#20276288C211yesmaximal, characteristic, complemented
4#20276287C211yesmaximal, characteristic, complemented
5#20276132C211yesmaximal, characteristic, complemented
6#20275860C211yesmaximal, characteristic, complemented
7#20275678C211yesmaximal, characteristic, complemented
8#20275399C211yesmaximal, characteristic, complemented
9#20275396C211yesmaximal, characteristic, complemented
10#20275141C211yesmaximal, characteristic
11#20275030C211yesmaximal, characteristic
12#20274844C211yesmaximal, characteristic, complemented
13#20274489C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#1824C211yesmaximal, characteristic, complemented
16C23×#88322C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133191C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133139C2211yescharacteristic
21C2×#13133139C2211yescharacteristic
22C2×#13133137C2211yescharacteristic
23C2×#13133137C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810565—21no—
27#16810555—21no—
28#16810549—21no—
29#16810549—21no—
30#16810524C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810502—21no—
35#16810502—21no—
36#16810498—21no—
37#16810488C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810488C2211yescharacteristic
40#16810481—21no—
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—41no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133208—22no—
68#13133208—41no—
69#13133191—22no—
70#13133191—22no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×C22⋊Dic53D411yescharacteristic
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133139—22no—
91#13133139—41no—
92#13133137—22no—
93#13133137—41no—
94C8×Dic53—22no—
95C8×Dic53—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C23×C4×C53—21no—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C22⋊C8×C53—22no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C23×D53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—42no—
131C22×C4×C53—81no—
132C22×C4×C53—81no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—42no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—42no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—81no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C22⋊C4×C53—42no—
172C22⋊Dic53—22no—
173C22⋊Dic53—22no—
174#11719921—42no—
175#11719921—81no—
176#11719921—81no—
177#11719921—81no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—42no—
185C4⋊Dic53—42no—
186C4⋊Dic53—81no—
187C4⋊Dic53#4112yes—
188#11719919—42no—
189#11719919—81no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—44no—
200C2×#88322#4112yes—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53#12412yes—
206C23×C53#13512yes—
207C23×C53#13512yes—
208C23×C53C22×D411yescharacteristic
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—84no—
235C2×C4×C53#12412yes—
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C22⋊D53—84no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—84no—
265D212—41no—
266D212—41no—
267D212—41no—
268D212—41no—
269C8×C53—41no—
270C8×C53—41no—
271#88322—22no—
272#88322—41no—
273#27701—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53#48814yes—
281C22×C53C2×#12411yescharacteristic
282C22×C53#54611yescharacteristic
283C22×C53#54711yescharacteristic
284C22×C53C2×#13511yescharacteristic
285C22×C53C2×#13511yescharacteristic
286C22×C53#57611yescharacteristic
287C22×C53#57711yescharacteristic
288C2×D53—81no—
289C2×D53—81no—
290C2×D53—81no—
291C2×D53—81no—
292C2×D53—81no—
293C2×D53—81no—
294C2×D53—81no—
295C4×C53—22no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#53911yescharacteristic
302C4×C53#55111yescharacteristic
303C4×C53#55211yescharacteristic
304Dic53—41no—
305Dic53—41no—
306Dic53—41no—
307Dic53—42no—
308Dic53—41no—
309Dic53—81no—
310Dic53—81no—
311#2504—531no—
312#2442—531no—
313#2101—531no—
314#2101—531no—
315#2073—531no—
316#2025—531no—
317#1991—531no—
318#1934—531no—
319#1934—531no—
320#1893—531no—
321#1874—531no—
322#1835—531no—
323#1824D5311yescharacteristic, complemented
324C8×C22⋊C4—531no—
325C2×#340—531no—
326C2×C53—24no—
327C2×C53#248811yescharacteristic
328C2×C53C2×#48811yescharacteristic
329C2×C53#313611yescharacteristic
330C2×C53#329711yescharacteristic
331C2×C53#330111yescharacteristic
332C2×C53#333711yescharacteristic
333C2×C53#333911yescharacteristic
334D53—161no—
335C23×C8—531no—
336C2×#125—531no—
337C2×#121—531no—
338C22×C4⋊C4—531no—
339C22×C4⋊C4C2×D5311yescharacteristic
340C2×#102—531no—
341C2×#102—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×C22⋊C8C2×D5311yescharacteristic
345C2×C4×C8—531no—
346C2×C4×C8—531no—
347#400—1061no—
348#398—1061no—
349#397—1061no—
350#397—1061no—
351#390—531no—
352#390—531no—
353#389—531no—
354#389—531no—
355#386—1061no—
356#386—1061no—
357#385—1061no—
358#382—531no—
359#382—531no—
360#382—531no—
361#380—1061no—
362#380C2×D5311yescharacteristic
363C4×C22⋊C4—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340C2×D5311yescharacteristic
373#340C2×D5311yescharacteristic
374#340C2×D5311yescharacteristic
375#340C2×D5311yescharacteristic
376#336—531no—
377#336—531no—
378C53#2770111yescharacteristic, complemented, Sylow
379C22×D4—531no—
380C22×D4—1061no—
381C23×C4—21no—
382C23×C4—1061no—
383C23×C4C22×D5311yescharacteristic
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—1062no—
387C22×C8—1062no—
388C22×C8—2121no—
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391#125—1062no—
392#125—2121no—
393#121—1062no—
394#121—1062no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—2121no—
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C42—531no—
421C2×C42—531no—
422#102—1062no—
423#102—1062no—
424#102—2121no—
425#102—2121no—
426C22⋊C8—22no—
427C4×C8—1062no—
428C4×C8—2121no—
429#95—21no—
430#95—531no—
431#95—531no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—1061no—
441C24D4×D5311yescharacteristic
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—2122no—
449C2×D4—2122no—
450C2×D4—2122no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—4241no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4#1171994512yes—
473C22×C4D4×D5311yescharacteristic
474C22×C4#1171994712yes—
475C22×C4C23×D5311yescharacteristic
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—532no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—2124no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—532no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—4241no—
505C4⋊C4—4241no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C22⋊C4—42no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—4241no—
513C22⋊C4—4241no—
514C22⋊C4—4241no—
515C22⋊C4—4241no—
516C22⋊C4—2122no—
517C42—1061no—
518C42—1061no—
519C42—1061no—
520C42—1061no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—2121no—
526C23—2121no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23—2121no—
532C23#1313320312yes—
533C23#1313323812yes—
534C23#1313324012yes—
535C23C2×D4×D5311yescharacteristic
536D4—4244no—
537D4—2121no—
538D4—2121no—
539D4—2121no—
540D4—2121no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—84no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1062no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—4244no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4#1313320312yes—
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994511yescharacteristic
587C2×C4#1313326311yescharacteristic
588C2×C4#1313326411yescharacteristic
589C8—41no—
590C8—41no—
591C8—1062no—
592C8—2121no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—4241no—
600C22—4241no—
601C22—4241no—
602C22—4241no—
603C22—4241no—
604C22—4241no—
605C22—4241no—
606C22#1681097914yes—
607C22C2×#1313320311yescharacteristic
608C22#1681126511yescharacteristic
609C22#1681127111yescharacteristic
610C22C2×#1313323811yescharacteristic
611C22C2×#1313324011yescharacteristic
612C22#1681142711yescharacteristic
613C22#1681143311yescharacteristic
614C4—22no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—4241no—
620C4—4241no—
621C4—2121no—
622C4—2122no—
623C4—2121no—
624C4—2121no—
625C4—2121no—
626C4#1681115011yescharacteristic
627C4#1681121711yescharacteristic
628C4#1681129711yescharacteristic
629C4#1681130611yescharacteristic
630C2—24no—
631C2—8481no—
632C2#2027903711yescharacteristic
633C2C2×#1681097911yescharacteristic
634C2#2028506511yescharacteristic
635C2#2028656311yescharacteristic
636C2#2028661511yescharacteristic
637C2#2028709211yescharacteristic
638C2#2028711511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1824D53
hypocenterC53#27701
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. #20278440C2semidirect
3. #20276288C2semidirect
4. #20276287C2semidirect
5. #20276132C2semidirect
6. #20275860C2semidirect
7. #20275678C2semidirect
8. #20275399C2semidirect
9. #20275396C2semidirect
12. #20274844C2semidirect
13. #20274489C2semidirect
14. C2×#16810385C2semidirect
15. C53×#1824C2semidirect
323. #1824D53semidirect
378. C53#27701semidirect

Maximal subgroups

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

subgroupindex
#202792032
#202784402
#202762882
#202762872
#202761322
#202758602
#202756782
#202753992
#202753962
#202751412
#202750302
#202748442
#202744892
C2×#168103852
C53×#18242
#2770153