Finite Group Databasebeta

← #27456746

Subgroups — #27456746

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#20279139C211yesmaximal, characteristic, complemented
2#20277731C211yesmaximal, characteristic
3#20276122C211yesmaximal, characteristic, complemented
4#20276098C211yesmaximal, characteristic, complemented
5#20275898C211yesmaximal, characteristic
6#20275897C211yesmaximal, characteristic
7#20275472C211yesmaximal, characteristic, complemented
8#20275446C211yesmaximal, characteristic, complemented
9#20275324C211yesmaximal, characteristic, complemented
10#20275288C211yesmaximal, characteristic, complemented
11#20275284C211yesmaximal, characteristic, complemented
12#20274959C211yesmaximal, characteristic, complemented
13#20274801C211yesmaximal, characteristic, complemented
14#20274529C211yesmaximal, characteristic, complemented
15C53×#1852C211yesmaximal, characteristic, complemented
16C2×#13133210—21no—
17C2×#13133191C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×C4×C22⋊D53—21no—
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810565—21no—
30#16810551—21no—
31#16810524C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810513—21no—
36#16810505—21no—
37#16810502—21no—
38#16810494—21no—
39#16810492C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C22⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133210—44no—
65#13133191—22no—
66#13133191—22no—
67#13133179—22no—
68#13133179—41no—
69C4×C22⋊D53—44no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133150—22no—
91#13133146—22no—
92#13133139—22no—
93#13133139—22no—
94#13133137—22no—
95#13133137—41no—
96C22⋊#88322—41no—
97C8×Dic53—41no—
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—41no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118#107×C53—22no—
119#107×C53—41no—
120C22⋊C8×C53—22no—
121#95×C53—21no—
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×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—81no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—24no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—42no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
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—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53#4112yes—
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—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53#4112yes—
173C22⋊C4×C53—42no—
174C22⋊Dic53—41no—
175C22⋊Dic53—42no—
176C22⋊Dic53D814yes—
177#11719921—24no—
178#11719921—42no—
179#11719921—42no—
180#11719921—81no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53#4112yes—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
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—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
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—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—84no—
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C22⋊D53—48no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—22no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53C2×D812yes—
266C2×Dic53C2×D812yes—
267D212—22no—
268D212—22no—
269D212—41no—
270D212—41no—
271C4×D53—84no—
272C8×C53—22no—
273C8×C53—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—41no—
299C4×C53—42no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53#52911yescharacteristic
304C4×C53#53911yescharacteristic
305C4×C53#55311yescharacteristic
306C4×C53#55611yescharacteristic
307Dic53—22no—
308Dic53—22no—
309Dic53—41no—
310Dic53—42no—
311Dic53—41no—
312Dic53—81no—
313Dic53—81no—
314#2496—531no—
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#1852D5311yescharacteristic, complemented
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—1061no—
339C2×#121—531no—
340C2×C4×D4—531no—
341C2×C4×D4—1061no—
342C22×C4⋊C4C2×D5311yescharacteristic
343C2×#107—531no—
344C2×#107—531no—
345C2×#107C2×D5311yescharacteristic
346C2×#107C2×D5311yescharacteristic
347C2×C4⋊C8—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×#102—531no—
351C2×#102—531no—
352C2×C22⋊C8—531no—
353C2×C22⋊C8C2×D5311yescharacteristic
354C2×C4×C8—531no—
355#400—1061no—
356#397—1061no—
357#390—531no—
358#389—531no—
359#389—531no—
360#389—531no—
361#388—1061no—
362#387—1061no—
363#386—1061no—
364#385—1061no—
365#384—531no—
366#380—531no—
367#380C2×D5311yescharacteristic
368C4×C4⋊C4—531no—
369C4×C4⋊C4—531no—
370C4×C22⋊C4—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340C2×D5311yescharacteristic
378#340C2×D5311yescharacteristic
379#336—531no—
380#336—531no—
381C53#2703011yescharacteristic, complemented, Sylow
382C22×D4—531no—
383C22×D4—1061no—
384C23×C4—21no—
385C23×C4—1061no—
386C23×C4C22×D5311yescharacteristic
387C22×C8—531no—
388C22×C8—531no—
389C22×C8C22×D5311yescharacteristic
390C22×C8C22×D5311yescharacteristic
391#126—2124no—
392#121—1062no—
393#121—1062no—
394C4×D4—1062no—
395C4×D4—2121no—
396C4×D4—2124no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—1061no—
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—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—
422C2×C42—1061no—
423C2×C42—1061no—
424#107—22no—
425#107—41no—
426#107—1062no—
427#107—1062no—
428C4⋊C8—1062no—
429#102—1062no—
430#102—1062no—
431#102—1062no—
432#102—2121no—
433C22⋊C8—22no—
434C22⋊C8—2121no—
435C4×C8—2121no—
436#95—21no—
437#95—531no—
438#95—531no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24—1061no—
446C24D4×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—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—4241no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4D4×D5311yescharacteristic
480C22×C4C23×D5311yescharacteristic
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8#1171994512yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—532no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4#1171994712yes—
513C22⋊C4—42no—
514C22⋊C4—534no—
515C22⋊C4—1064no—
516C22⋊C4—4241no—
517C22⋊C4—2122no—
518C22⋊C4—2122no—
519C22⋊C4—2121no—
520C22⋊C4—2122no—
521C42—1061no—
522C42—1061no—
523C42—1061no—
524C42—1061no—
525C42—2122no—
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—
540C23D8×D5312yes—
541C23C2×D4×D5311yescharacteristic
542D4—1062no—
543D4—1062no—
544D4—2128no—
545D4—2121no—
546D4—2121no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
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—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—84no—
567C2×C4—532no—
568C2×C4—532no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—1062no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—4244no—
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#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4C2×#1171994511yescharacteristic
593C2×C4#1313326211yescharacteristic
594C2×C4#1313326511yescharacteristic
595C8—22no—
596C8—41no—
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#1681113111yescharacteristic
614C22#1681126011yescharacteristic
615C22#1681127411yescharacteristic
616C22C2×D8×D5311yescharacteristic
617C22#1681142211yescharacteristic
618C22#1681143311yescharacteristic
619C22#1681143611yescharacteristic
620C4—42no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—1062no—
626C4—1062no—
627C4—4241no—
628C4—4241no—
629C4—2122no—
630C4—2121no—
631C4—2121no—
632C4#1681115511yescharacteristic
633C4#1681121911yescharacteristic
634C4#1681132211yescharacteristic
635C4#1681134411yescharacteristic
636C2—42no—
637C2—4242no—
638C2#2027874811yescharacteristic
639C2#2028471211yescharacteristic
640C2#2028492411yescharacteristic
641C2#2028652111yescharacteristic
642C2#2028677411yescharacteristic
643C2#2028706011yescharacteristic
644C2#2028727611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1852D53
hypocenterC53#27030
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. #20279139C2semidirect
3. #20276122C2semidirect
4. #20276098C2semidirect
7. #20275472C2semidirect
8. #20275446C2semidirect
9. #20275324C2semidirect
10. #20275288C2semidirect
11. #20275284C2semidirect
12. #20274959C2semidirect
13. #20274801C2semidirect
14. #20274529C2semidirect
15. C53×#1852C2semidirect
326. #1852D53semidirect
381. C53#27030semidirect

Maximal subgroups

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

subgroupindex
#202791392
#202777312
#202761222
#202760982
#202758982
#202758972
#202754722
#202754462
#202753242
#202752882
#202752842
#202749592
#202748012
#202745292
C53×#18522
#2703053