Finite Group Databasebeta

← #27452642

Subgroups — #27452642

Order 13568.

804 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279264C211yesmaximal, characteristic, complemented
2#20276128C211yesmaximal, characteristic, complemented
3#20276089C211yesmaximal, characteristic, complemented
4#20275946C211yesmaximal, characteristic
5#20275946C211yesmaximal, characteristic
6#20275401C211yesmaximal, characteristic, complemented
7#20275339C211yesmaximal, characteristic, complemented
8#20275307C211yesmaximal, characteristic, complemented
9#20275293C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20274962C211yesmaximal, characteristic, complemented
12#20274962C211yesmaximal, characteristic, complemented
13#20274725C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#2366C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133138C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810564—21no—
29#16810561—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810518—21no—
35#16810505—21no—
36#16810489C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#127×C53C2211yescharacteristic
49C2×C4×D4×C53—21no—
50C2×C4×D4×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#390—21no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#386—21no—
56C53×#386—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C4×Dic53—21no—
72#13133150—22no—
73#13133148—22no—
74C4⋊#88322—41no—
75#13133138—22no—
76#13133138—22no—
77#13133138—22no—
78#13133138—22no—
79#13133138—22no—
80#13133138—22no—
81#13133138—41no—
82#13133138—41no—
83C4×#88322—41no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53—21no—
92C22×D4×C53—21no—
93C22×D4×C53C2311yescharacteristic
94C22×D4×C53C2311yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53—21no—
97#127×C53—22no—
98#127×C53—22no—
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—41no—
102C4×D4×C53—41no—
103C4×D4×C53—44no—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53—21no—
122C24×C53—21no—
123C2×D4×C53—24no—
124C2×D4×C53—24no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53—81no—
138C2×D4×C53—81no—
139C2×D4×C53#4112yes—
140C2×D4×C53#4112yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—81no—
148C22×C4×C53—81no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53#4112yes—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—81no—
171C22⋊C4×C53—81no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—42no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53#4112yes—
187#11719919—42no—
188#11719919—42no—
189#11719919—81no—
190#11719919—81no—
191C4×Dic53—81no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53C22×D411yescharacteristic
219D4×C53—22no—
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—22no—
223D4×C53—48no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—41no—
228D4×C53—84no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—84no—
252C2×C4×C53—84no—
253C2×C4×C53#13512yes—
254C2×C4×C53AGL(1,ℤ/8)12yes—
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×C4×C532−1+411yescharacteristic
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#88322—41no—
277#26755—531nomaximal, Sylow
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53#52511yescharacteristic
293C22×C53#54611yescharacteristic
294C22×C53#54811yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53#57611yescharacteristic
298C22×C53#57711yescharacteristic
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53—42no—
306C4×C53—81no—
307C4×C53#52911yescharacteristic
308C4×C53#53911yescharacteristic
309C4×C53#55211yescharacteristic
310C4×C53#55311yescharacteristic
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315#2509—531no—
316#2366D5311yescharacteristic
317#2073—531no—
318#2065—531no—
319#2041—531no—
320#2041—531no—
321#1934—531no—
322#1926—531no—
323#1921—531no—
324#1919—531no—
325#1916—531no—
326#1862—531no—
327#1862—531no—
328#1804—531no—
329C4×#102—531no—
330C2×C53—82no—
331C2×C53—82no—
332C2×C53#246011yescharacteristic
333C2×C53#310511yescharacteristic
334C2×C53#311811yescharacteristic
335C2×C53#330411yescharacteristic
336C2×C53#331811yescharacteristic
337C2×C53#336111yescharacteristic
338C2×C53#336811yescharacteristic
339C2×#127C2×D5311yescharacteristic
340C2×C4×D4—21no—
341C2×C4×D4C2×D5311yescharacteristic
342C2×C4×D4C2×D5311yescharacteristic
343C2×#107—531no—
344C2×#106—531no—
345C2×C4⋊C8—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×#102—531no—
353C2×#102—531no—
354C2×C4×C8—531no—
355#400—1061no—
356#400—1061no—
357#390—21no—
358#390C2×D5311yescharacteristic
359#390C2×D5311yescharacteristic
360#389—531no—
361#389—531no—
362#389—531no—
363#389—531no—
364#389—1061no—
365#387—1061no—
366#386—21no—
367#386—21no—
368#382—531no—
369C4×C4⋊C4—531no—
370C4×C4⋊C4C2×D5311yescharacteristic
371C4×C4⋊C4C2×D5311yescharacteristic
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381#336—531no—
382C53#2675511yescharacteristic, complemented, Sylow
383C22×D4—21no—
384C22×D4—21no—
385C22×D4C22×D5311yescharacteristic
386C22×D4C22×D5311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C22×C8—531no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8—531no—
393#127—22no—
394#127—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—41no—
398C4×D4—41no—
399C4×D4—44no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4C22⋊D5311yescharacteristic
409C2×C4⋊C4C22⋊D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C42—21no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422C2×C42C22×D5311yescharacteristic
423#107—1062no—
424#106—1062no—
425C4⋊C8—2121no—
426#102—1062no—
427#102—1062no—
428#102—1062no—
429#102—1062no—
430#102—1062no—
431#102—1062no—
432#102—2121no—
433#102—2121no—
434C4×C8—2121no—
435#95—21no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95C22⋊D5311yescharacteristic
444#95C22⋊D5311yescharacteristic
445C24—21no—
446C24—21no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4#1171994512yes—
464C2×D4#1171994512yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—532no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—4241no—
510C4⋊C4—4241no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4#1171994712yes—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—42no—
527C42—4241no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23C22×C22⋊D5311yescharacteristic
543D4—22no—
544D4—22no—
545D4—22no—
546D4—22no—
547D4—48no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—84no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—1062no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4#1313323712yes—
590C2×C4#1313324312yes—
591C2×C4#41×D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994511yescharacteristic
594C2×C4C2×#1171994511yescharacteristic
595C2×C4#1313326211yescharacteristic
596C2×C4#1313326411yescharacteristic
597C8—2121no—
598C8—2121no—
599C8—2121no—
600C8—2121no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1681113411yescharacteristic
616C22#1681126711yescharacteristic
617C22#1681128011yescharacteristic
618C22C2×#1313323711yescharacteristic
619C22C2×#1313324311yescharacteristic
620C22#1681142611yescharacteristic
621C22#1681143411yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—2122no—
634C4#1681115311yescharacteristic
635C4#1681121511yescharacteristic
636C4#1681131311yescharacteristic
637C4#1681132211yescharacteristic
638C2—82no—
639C2—82no—
640C2#2027866111yescharacteristic
641C2#2028476211yescharacteristic
642C2#2028489911yescharacteristic
643C2#2028665411yescharacteristic
644C2#2028685611yescharacteristic
645C2#2028737111yescharacteristic
646C2#2028744011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2366C2
socleC23×C53C22×D4
prime radicalC2×#127×C53C22
hypercenter#2366D53
hypocenterC53#26755
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
1. #20279264C2semidirect
2. #20276128C2semidirect
3. #20276089C2semidirect
6. #20275401C2semidirect
7. #20275339C2semidirect
8. #20275307C2semidirect
9. #20275293C2semidirect
10. #20275274C2semidirect
11. #20274962C2semidirect
12. #20274962C2semidirect
13. #20274725C2semidirect
14. C4×#13133138C2semidirect
382. C53#26755semidirect

Maximal subgroups

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

subgroupindex
#202792642
#202761282
#202760892
#202759462
#202754012
#202753392
#202753072
#202752932
#202752742
#202749622
#202747252
C4×#131331382
C53×#23662
#2675553