Finite Group Databasebeta

← #27452644

Subgroups — #27452644

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#20277779C211yesmaximal, characteristic
2#20276129C211yesmaximal, characteristic, complemented
3#20276086C211yesmaximal, characteristic, complemented
4#20275949C211yesmaximal, characteristic
5#20275949C211yesmaximal, characteristic
6#20275397C211yesmaximal, characteristic, complemented
7#20275342C211yesmaximal, characteristic, complemented
8#20275304C211yesmaximal, characteristic, complemented
9#20275290C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20274961C211yesmaximal, characteristic, complemented
12#20274961C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic, complemented
14C4×#13133137C211yesmaximal, characteristic, complemented
15C53×#2509C211yesmaximal, characteristic, complemented
16C2×#13133214C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133179—21no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133137C2211yescharacteristic
24C2×#13133137C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810525C2211yescharacteristic
34#16810524—21no—
35#16810501—21no—
36#16810501—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#400—21no—
49C53×#400—21no—
50C53×#389—21no—
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×D212—21no—
60C22×D212—21no—
61C22×D212C2311yescharacteristic
62C22×D212C2311yescharacteristic
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133214—22no—
70#13133214—22no—
71C4×D212—22no—
72C4×D212—22no—
73C4×D212—41no—
74C4×D212—41no—
75#13133179—44no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85#13133150—22no—
86#13133148—22no—
87C4⋊#88322—41no—
88#13133137—22no—
89#13133137—22no—
90#13133137—22no—
91#13133137—22no—
92#13133137—22no—
93#13133137—22no—
94#13133137—41no—
95#13133137—41no—
96C4×#88322—41no—
97#13133124—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C23×D53—21no—
122C23×D53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—24no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—24no—
148C2×D212—42no—
149C2×D212—42no—
150C2×D212—81no—
151C2×D212—81no—
152C2×D212#4112yes—
153C2×D212#4112yes—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—81no—
159C2×C4×D53—81no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
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—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—81no—
172C4⋊C4×C53—81no—
173C4⋊C4×C53#4112yes—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—81no—
181#11719921—42no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185#11719921—81no—
186#11719921—81no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53#4112yes—
191C4×Dic53—42no—
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×C53C22×D411yescharacteristic
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53#13512yes—
248C2×C4×C53AGL(1,ℤ/8)12yes—
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—22no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261D212—22no—
262D212—22no—
263D212—22no—
264D212—22no—
265D212—48no—
266D212—41no—
267D212—41no—
268D212—41no—
269D212—41no—
270D212—84no—
271C4×D53—84no—
272C4×D53—84no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#88322—41no—
277#26755—531nomaximal, Sylow
278C22×C53#52511yescharacteristic
279C22×C53#54611yescharacteristic
280C22×C53#54811yescharacteristic
281C22×C53C2×#13511yescharacteristic
282C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C53#57611yescharacteristic
284C22×C53#57711yescharacteristic
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C2×D53—81no—
292C2×D53—81no—
293C2×D53—81no—
294C2×D53—81no—
295C2×D53—81no—
296C2×D53—81no—
297C2×D53—81no—
298C2×D53—81no—
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—42no—
302C4×C53—41no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53—41no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53#52911yescharacteristic
310C4×C53#53911yescharacteristic
311C4×C53#55211yescharacteristic
312C4×C53#55311yescharacteristic
313Dic53—42no—
314Dic53—81no—
315#2509D5311yescharacteristic, complemented
316#2366—531no—
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#246011yescharacteristic
331C2×C53#310511yescharacteristic
332C2×C53#311811yescharacteristic
333C2×C53#330411yescharacteristic
334C2×C53#331811yescharacteristic
335C2×C53#336111yescharacteristic
336C2×C53#336811yescharacteristic
337D53—82no—
338D53—82no—
339C2×#127—531no—
340C2×C4×D4—531no—
341C2×C4×D4—531no—
342C2×C4×D4—1061no—
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—21no—
356#400—21no—
357#390—531no—
358#390—531no—
359#390—1061no—
360#389—21no—
361#389C2×D5311yescharacteristic
362#389C2×D5311yescharacteristic
363#389C2×D5311yescharacteristic
364#389C2×D5311yescharacteristic
365#387—21no—
366#386—1061no—
367#386—1061no—
368#382C2×D5311yescharacteristic
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—531no—
384C22×D4—531no—
385C22×D4—1061no—
386C22×D4—1061no—
387C23×C4—1061no—
388C23×C4—1061no—
389C22×C8—531no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8—531no—
393#127—1062no—
394#127—1062no—
395C4×D4—1062no—
396C4×D4—1062no—
397C4×D4—2121no—
398C4×D4—2121no—
399C4×D4—2124no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4C22⋊D5311yescharacteristic
407C2×C4⋊C4C22⋊D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
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—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1061no—
443#95C22⋊D5311yescharacteristic
444#95C22⋊D5311yescharacteristic
445C24—1061no—
446C24—1061no—
447C2×D4—532no—
448C2×D4—532no—
449C2×D4—1064no—
450C2×D4—1064no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—1061no—
457C2×D4—1061no—
458C2×D4—1061no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—4241no—
462C2×D4—4241no—
463C2×D4—2122no—
464C2×D4—2122no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—4241no—
472C22×C4—4241no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4—2122no—
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—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—532no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4#1171994912yes—
514C22⋊C4—4241no—
515C22⋊C4—4241no—
516C22⋊C4—2122no—
517C22⋊C4—2122no—
518C22⋊C4—2122no—
519C22⋊C4—2122no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—81no—
527C42—2122no—
528C23—1062no—
529C23—1062no—
530C23—2121no—
531C23—2121no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2121no—
539C23—2121no—
540C23—2121no—
541C23—2121no—
542C23C22×C22⋊D5311yescharacteristic
543D4—1062no—
544D4—1062no—
545D4—1062no—
546D4—1062no—
547D4—4244no—
548D4—2128no—
549D4—2121no—
550D4—2121no—
551D4—2121no—
552D4—2121no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
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—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—1062no—
582C2×C4—4244no—
583C2×C4—4244no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4#1313323712yes—
590C2×C4#1313324212yes—
591C2×C4#41×D5311yescharacteristic
592C2×C4#41×D5311yescharacteristic
593C2×C4#41×D5311yescharacteristic
594C2×C4C2×#1171994911yescharacteristic
595C2×C4#1313326311yescharacteristic
596C2×C4#1313326511yescharacteristic
597C8—2121no—
598C8—2121no—
599C8—2121no—
600C8—2121no—
601C22—4241no—
602C22—4241no—
603C22—4241no—
604C22—4241no—
605C22—4241no—
606C22—4241no—
607C22—4241no—
608C22—4241no—
609C22—2122no—
610C22—2122no—
611C22—2122no—
612C22—2122no—
613C22—2122no—
614C22—2122no—
615C22#1681113011yescharacteristic
616C22#1681126111yescharacteristic
617C22#1681128411yescharacteristic
618C22C2×#1313323711yescharacteristic
619C22C2×#1313324211yescharacteristic
620C22#1681142611yescharacteristic
621C22#1681143111yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—4241no—
633C4—2122no—
634C4#1681115811yescharacteristic
635C4#1681122111yescharacteristic
636C4#1681130511yescharacteristic
637C4#1681132611yescharacteristic
638C2—4242no—
639C2—4242no—
640C2#2027866311yescharacteristic
641C2#2028475811yescharacteristic
642C2#2028489311yescharacteristic
643C2#2028664811yescharacteristic
644C2#2028685011yescharacteristic
645C2#2028737511yescharacteristic
646C2#2028744411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2509C2
socleC23×C53C22×D4
prime radicalC2×#13133214C22
hypercenter#2509D53
hypocenterC53#26755
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2509

Complements

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

NH ≅ G/Ntype
2. #20276129C2semidirect
3. #20276086C2semidirect
6. #20275397C2semidirect
7. #20275342C2semidirect
8. #20275304C2semidirect
9. #20275290C2semidirect
10. #20275274C2semidirect
11. #20274961C2semidirect
12. #20274961C2semidirect
13. #20274722C2semidirect
14. C4×#13133137C2semidirect
15. C53×#2509C2semidirect
315. #2509D53semidirect
382. C53#26755semidirect

Maximal subgroups

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

subgroupindex
#202777792
#202761292
#202760862
#202759492
#202753972
#202753422
#202753042
#202752902
#202752742
#202749612
#202747222
C4×#131331372
C53×#25092
#2675553