Finite Group Databasebeta

← #27456777

Subgroups — #27456777

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279148C211yesmaximal, characteristic, complemented
2#20277729C211yesmaximal, characteristic
3#20276120C211yesmaximal, characteristic, complemented
4#20276088C211yesmaximal, characteristic, complemented
5#20275905C211yesmaximal, characteristic
6#20275905C211yesmaximal, characteristic
7#20275479C211yesmaximal, characteristic, complemented
8#20275443C211yesmaximal, characteristic, complemented
9#20275301C211yesmaximal, characteristic, complemented
10#20275294C211yesmaximal, characteristic, complemented
11#20274979C211yesmaximal, characteristic, complemented
12#20274910C211yesmaximal, characteristic, complemented
13#20274802C211yesmaximal, characteristic, complemented
14C4×#13133139C211yesmaximal, characteristic, complemented
15C53×#1923C211yesmaximal, characteristic, complemented
16C2×#13133191C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133177—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28#16810565—21no—
29#16810546—21no—
30#16810525C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810516—21no—
35#16810515—21no—
36#16810510—21no—
37#16810500—21no—
38#16810497—21no—
39#16810493C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×C4×D53—21no—
63#13133191—22no—
64#13133191—22no—
65C4×D212—22no—
66C4×D212—41no—
67#13133177—44no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91#13133149—22no—
92#13133149—22no—
93#13133149—22no—
94#13133149—41no—
95#13133139—22no—
96#13133139—22no—
97#13133139—22no—
98#13133139—41no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C4⋊C8×C53—22no—
116C22⋊C8×C53—22no—
117C22⋊C8×C53—41no—
118C4×C8×C53—41no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C23×D53—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×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C2×C22⋊D53—24no—
136C2×C22⋊D53—42no—
137C2×C22⋊D53—42no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53—81no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212#4112yes—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53—81no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53#4112yes—
169C22⋊C4×C53—41no—
170C22⋊C4×C53SD1614yes—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C22⋊Dic53—81no—
176C22⋊Dic53—81no—
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—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—42no—
194C4⋊Dic53—81no—
195C4⋊Dic53—81no—
196C4⋊Dic53#4112yes—
197C4⋊Dic53#4112yes—
198#11719919—42no—
199#11719919—42no—
200#11719919—81no—
201C4×Dic53—42no—
202C4×Dic53—81no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53C2×SD1612yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C2×C4×C53—21no—
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—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53C2×SD1612yes—
236C2×C4×C53C2×SD1612yes—
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C22⋊D53—48no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
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—41no—
265C2×Dic53—84no—
266D212—22no—
267D212—22no—
268D212—41no—
269D212—41no—
270C4×D53—84no—
271C8×C53—22no—
272C8×C53—41no—
273C8×C53—41no—
274C8×C53—41no—
275#27032—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53#52511yescharacteristic
284C22×C53#54611yescharacteristic
285C22×C53#54711yescharacteristic
286C22×C53C22×SD1611yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57611yescharacteristic
289C22×C53#57711yescharacteristic
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—42no—
293C2×D53—81no—
294C2×D53—81no—
295C2×D53—81no—
296C2×D53—81no—
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53#52911yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53#55311yescharacteristic
305C4×C53#55611yescharacteristic
306Dic53—42no—
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313#2497—531no—
314#2361—531no—
315#2071—531no—
316#2065—531no—
317#2034—531no—
318#2034—531no—
319#1948—531no—
320#1941—531no—
321#1923D5311yescharacteristic, complemented
322#1920—531no—
323#1919—531no—
324#1864—531no—
325#1853—531no—
326#1823—531no—
327C4×#102—531no—
328C2×C53—42no—
329C2×C53#246611yescharacteristic
330C2×C53#310111yescharacteristic
331C2×C53#311911yescharacteristic
332C2×C53#329511yescharacteristic
333C2×C53#332111yescharacteristic
334C2×C53#333511yescharacteristic
335C2×C53#335511yescharacteristic
336D53—82no—
337C2×#121—531no—
338C2×C4×D4—531no—
339C2×C4×D4—1061no—
340C22×C4⋊C4—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×#106—531no—
344C2×#106—531no—
345C2×C4⋊C8C2×D5311yescharacteristic
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×C22⋊C8C2×D5311yescharacteristic
351C2×C22⋊C8C2×D5311yescharacteristic
352C2×C4×C8C2×D5311yescharacteristic
353#400—1061no—
354#396—1061no—
355#390—531no—
356#389—531no—
357#389—531no—
358#389—531no—
359#388—1061no—
360#388—1061no—
361#387—1061no—
362#386—1061no—
363#385—1061no—
364#384—531no—
365#380—531no—
366#380—531no—
367C4×C4⋊C4—531no—
368C4×C4⋊C4—531no—
369C4×C22⋊C4C2×D5311yescharacteristic
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336C2×D5311yescharacteristic
379#336C2×D5311yescharacteristic
380C53#2703211yescharacteristic, complemented, Sylow
381C22×D4—531no—
382C22×D4—1061no—
383C23×C4—1061no—
384C23×C4—1061no—
385C23×C4C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390#121—1062no—
391#121—1062no—
392C4×D4—1062no—
393C4×D4—2121no—
394C4×D4—2124no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—2121no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4—1061no—
416C2×C22⋊C4D21211yescharacteristic
417C2×C22⋊C4D21211yescharacteristic
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#106—1062no—
423#106—1062no—
424#106—1062no—
425#106—2121no—
426C4⋊C8—22no—
427#102—1062no—
428#102—1062no—
429#102—1062no—
430#102—2121no—
431C22⋊C8—22no—
432C22⋊C8—41no—
433C4×C8—41no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95D21211yescharacteristic
442#95D21211yescharacteristic
443C24—1061no—
444C24C2×D21211yescharacteristic
445C2×D4—532no—
446C2×D4—1064no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—2122no—
453C2×D4—2122no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4C2×D21211yescharacteristic
474C22×C4C2×D21211yescharacteristic
475C22×C4C2×D21211yescharacteristic
476C22×C4C2×D21211yescharacteristic
477C22×C4C2×D21211yescharacteristic
478C22×C4C23×D5311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8#1171994712yes—
491C4⋊C4—532no—
492C4⋊C4—532no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—4241no—
506C4⋊C4—4241no—
507C4⋊C4—4241no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C4⋊C4—2122no—
511C22⋊C4—41no—
512C22⋊C4—1064no—
513C22⋊C4—4241no—
514C22⋊C4—4241no—
515C22⋊C4—4241no—
516C22⋊C4—2122no—
517C22⋊C4—2122no—
518C22⋊C4#1171991314yes—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—4241no—
524C42—2122no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—1062no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23C2×#1171991312yes—
539C23C22×D21211yescharacteristic
540D4—1062no—
541D4—1062no—
542D4—2128no—
543D4—2121no—
544D4—2121no—
545C2×C4—21no—
546C2×C4—21no—
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—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—4244no—
565C2×C4—4244no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
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—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4C2×#1171991312yes—
586C2×C4C2×#1171991312yes—
587C2×C4#41×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4C2×#1171994711yescharacteristic
591C2×C4#1313326211yescharacteristic
592C2×C4#1313326411yescharacteristic
593C8—22no—
594C8—41no—
595C8—41no—
596C8—41no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—4241no—
605C22—4241no—
606C22—4241no—
607C22—4241no—
608C22—2122no—
609C22—2122no—
610C22—2122no—
611C22#1681113511yescharacteristic
612C22#1681126811yescharacteristic
613C22#1681127711yescharacteristic
614C22C22×#1171991311yescharacteristic
615C22#1681142311yescharacteristic
616C22#1681142811yescharacteristic
617C22#1681143511yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—4241no—
624C4—4241no—
625C4—4241no—
626C4—4241no—
627C4—4241no—
628C4—2122no—
629C4—2122no—
630C4#1681115211yescharacteristic
631C4#1681121611yescharacteristic
632C4#1681131611yescharacteristic
633C4#1681134011yescharacteristic
634C2—42no—
635C2—4242no—
636C2#2027874911yescharacteristic
637C2#2028472411yescharacteristic
638C2#2028491611yescharacteristic
639C2#2028654011yescharacteristic
640C2#2028690311yescharacteristic
641C2#2028707311yescharacteristic
642C2#2028732011yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1923C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1923D53
hypocenterC53#27032
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
1. #20279148C2semidirect
3. #20276120C2semidirect
4. #20276088C2semidirect
7. #20275479C2semidirect
8. #20275443C2semidirect
9. #20275301C2semidirect
10. #20275294C2semidirect
11. #20274979C2semidirect
12. #20274910C2semidirect
13. #20274802C2semidirect
14. C4×#13133139C2semidirect
15. C53×#1923C2semidirect
321. #1923D53semidirect
380. C53#27032semidirect

Maximal subgroups

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

subgroupindex
#202791482
#202777292
#202761202
#202760882
#202759052
#202754792
#202754432
#202753012
#202752942
#202749792
#202749102
#202748022
C4×#131331392
C53×#19232
#2703253