Finite Group Databasebeta

← #27456745

Subgroups — #27456745

Order 13568.

812 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
1#20277729C211yesmaximal, characteristic
2#20276119C211yesmaximal, characteristic, complemented
3#20276100C211yesmaximal, characteristic, complemented
4#20275897C211yesmaximal, characteristic
5#20275897C211yesmaximal, characteristic
6#20275471C211yesmaximal, characteristic, complemented
7#20275445C211yesmaximal, characteristic, complemented
8#20275323C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20275283C211yesmaximal, characteristic, complemented
11#20274961C211yesmaximal, characteristic, complemented
12#20274902C211yesmaximal, characteristic, complemented
13#20274799C211yesmaximal, characteristic, complemented
14C4×#13133137C211yesmaximal, characteristic, complemented
15C53×#2496C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133177—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810525C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810515—21no—
35#16810500—21no—
36#16810497—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#126×C53—21no—
49C53×#400—21no—
50C53×#397—21no—
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#387—21no—
54C53×#384C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133191—22no—
68#13133191—22no—
69C4×D212—22no—
70C4×D212—41no—
71#13133177—44no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53—21no—
84#13133150—22no—
85#13133150—22no—
86#13133150—22no—
87#13133150—41no—
88C4⋊#88322—22no—
89#13133137—22no—
90#13133137—22no—
91#13133137—22no—
92#13133137—41no—
93C22⋊#88322—22no—
94C22⋊#88322—41no—
95C4×#88322—41no—
96#13133124—21no—
97#13133124—21no—
98C23×C4×C53C2311yescharacteristic
99#126×C53—44no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53D411yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120#95×C53D411yescharacteristic
121#95×C53D411yescharacteristic
122C24×C53C2×D411yescharacteristic
123C23×D53—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—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—24no—
141C2×C22⋊D53—42no—
142C2×C22⋊D53—42no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—81no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212#4112yes—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—81no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C22⋊C4×C53—41no—
171C22⋊C4×C53—42no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53D814yes—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—42no—
179#11719921—24no—
180#11719921—42no—
181#11719921—42no—
182#11719921—81no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53#4112yes—
189C4⋊Dic53#4112yes—
190#11719919—42no—
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#4112yes—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53C2×D812yes—
211C23×C53C22×D411yescharacteristic
212C22×D53—22no—
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—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
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—41no—
248C2×C4×C53C2×D812yes—
249C2×C4×C53C2×D812yes—
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C22⋊D53—48no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—84no—
267D212—22no—
268D212—22no—
269D212—41no—
270D212—41no—
271C4×D53—84no—
272#88322—22no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#27030—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53#52511yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54711yescharacteristic
287C22×C53C22×D811yescharacteristic
288C22×C53#57511yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57811yescharacteristic
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53—81no—
295C2×D53—81no—
296C2×D53—81no—
297C2×D53—81no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53—81no—
307C4×C53#52911yescharacteristic
308C4×C53#53911yescharacteristic
309C4×C53#55311yescharacteristic
310C4×C53#55611yescharacteristic
311Dic53—42no—
312Dic53—42no—
313Dic53—81no—
314#2496D5311yescharacteristic, complemented
315#2361—531no—
316#2071—531no—
317#2067—531no—
318#2033—531no—
319#2033—531no—
320#1947—531no—
321#1942—531no—
322#1923—531no—
323#1918—531no—
324#1918—531no—
325#1862—531no—
326#1852—531no—
327#1823—531no—
328C4×#102—531no—
329C2×C53—42no—
330C2×C53#246611yescharacteristic
331C2×C53#310011yescharacteristic
332C2×C53#312011yescharacteristic
333C2×C53#329411yescharacteristic
334C2×C53#331311yescharacteristic
335C2×C53#333411yescharacteristic
336C2×C53#335111yescharacteristic
337D53—82no—
338C2×#126—21no—
339C2×#121—531no—
340C2×C4×D4—531no—
341C2×C4×D4—1061no—
342C22×C4⋊C4—531no—
343C2×#107—531no—
344C2×#107—531no—
345C2×#107—531no—
346C2×#107—531no—
347C2×C4⋊C8—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×C22⋊C8—531no—
353C2×C22⋊C8—531no—
354C2×C4×C8—531no—
355#400—21no—
356#397—21no—
357#390—531no—
358#389—531no—
359#389C2×D5311yescharacteristic
360#389C2×D5311yescharacteristic
361#388—1061no—
362#387—21no—
363#386—1061no—
364#385—1061no—
365#384C2×D5311yescharacteristic
366#380C2×D5311yescharacteristic
367#380C2×D5311yescharacteristic
368C4×C4⋊C4—531no—
369C4×C4⋊C4C2×D5311yescharacteristic
370C4×C22⋊C4C2×D5311yescharacteristic
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#336—531no—
380#336—531no—
381C53#2703011yescharacteristic, complemented, Sylow
382C22×D4—531no—
383C22×D4—1061no—
384C23×C4—1061no—
385C23×C4—1061no—
386C23×C4C22×D5311yescharacteristic
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391#126—44no—
392#121—1062no—
393#121—1062no—
394C4×D4—1062no—
395C4×D4—2121no—
396C4×D4—2124no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—1062no—
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—21no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4—1061no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4C22⋊D5311yescharacteristic
419C2×C22⋊C4C22⋊D5311yescharacteristic
420C2×C42—21no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424#107—1062no—
425#107—1062no—
426#107—1062no—
427#107—2121no—
428C4⋊C8—1062no—
429#102—1062no—
430#102—1062no—
431#102—1062no—
432#102—2121no—
433C22⋊C8—1062no—
434C22⋊C8—2121no—
435C4×C8—2121no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1061no—
442#95—1061no—
443#95C22⋊D5311yescharacteristic
444#95C22⋊D5311yescharacteristic
445C24—1061no—
446C24C2×C22⋊D5311yescharacteristic
447C2×D4—532no—
448C2×D4—1064no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—2122no—
455C2×D4—2122no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—4241no—
470C22×C4—4241no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C23×D5311yescharacteristic
481C2×C8—532no—
482C2×C8—1061no—
483C2×C8—1061no—
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—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—532no—
506C4⋊C4—532no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—1064no—
517C22⋊C4—4241no—
518C22⋊C4—2122no—
519C22⋊C4—2122no—
520C22⋊C4#1171992214yes—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—42no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—1062no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2121no—
539C23—2121no—
540C23C2×#1171992212yes—
541C23C22×C22⋊D5311yescharacteristic
542D4—1062no—
543D4—1062no—
544D4—2128no—
545D4—2121no—
546D4—2121no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—4244no—
579C2×C4—4244no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4C2×#1171992212yes—
588C2×C4C2×#1171992212yes—
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4#41×D5311yescharacteristic
592C2×C4#41×D5311yescharacteristic
593C2×C4#1313326311yescharacteristic
594C2×C4#1313326411yescharacteristic
595C8—1062no—
596C8—2121no—
597C8—2121no—
598C8—2121no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—4241no—
607C22—4241no—
608C22—4241no—
609C22—4241no—
610C22—2122no—
611C22—2122no—
612C22—2122no—
613C22#1681113011yescharacteristic
614C22#1681126111yescharacteristic
615C22#1681127211yescharacteristic
616C22C22×#1171992211yescharacteristic
617C22#1681142111yescharacteristic
618C22#1681143111yescharacteristic
619C22#1681143811yescharacteristic
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—4241no—
630C4—2122no—
631C4—2122no—
632C4#1681115611yescharacteristic
633C4#1681121411yescharacteristic
634C4#1681132411yescharacteristic
635C4#1681134011yescharacteristic
636C2—42no—
637C2—4242no—
638C2#2027874711yescharacteristic
639C2#2028471111yescharacteristic
640C2#2028492311yescharacteristic
641C2#2028652211yescharacteristic
642C2#2028677511yescharacteristic
643C2#2028705811yescharacteristic
644C2#2028727511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2496C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2496D53
hypocenterC53#27030
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
2. #20276119C2semidirect
3. #20276100C2semidirect
6. #20275471C2semidirect
7. #20275445C2semidirect
8. #20275323C2semidirect
9. #20275286C2semidirect
10. #20275283C2semidirect
11. #20274961C2semidirect
12. #20274902C2semidirect
13. #20274799C2semidirect
14. C4×#13133137C2semidirect
15. C53×#2496C2semidirect
314. #2496D53semidirect
381. C53#27030semidirect

Maximal subgroups

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

subgroupindex
#202777292
#202761192
#202761002
#202758972
#202754712
#202754452
#202753232
#202752862
#202752832
#202749612
#202749022
#202747992
C4×#131331372
C53×#24962
#2703053