Finite Group Databasebeta

← #27465620

Subgroups — #27465620

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277324C211yesmaximal, characteristic, complemented
2#20276152C211yesmaximal, characteristic, complemented
3#20276141C211yesmaximal, characteristic, complemented
4#20275901C211yesmaximal, characteristic
5#20275901C211yesmaximal, characteristic
6#20275483C211yesmaximal, characteristic, complemented
7#20275478C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic, complemented
9#20275332C211yesmaximal, characteristic, complemented
10#20275297C211yesmaximal, characteristic, complemented
11#20275290C211yesmaximal, characteristic, complemented
12#20274725C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic, complemented
14#20274699C211yesmaximal, characteristic, complemented
15C53×#2324C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810516—21no—
29#16810516—21no—
30#16810509—21no—
31#16810509—21no—
32#16810493C2211yescharacteristic
33#16810489C2211yescharacteristic
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#121×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#390—21no—
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#386—21no—
56C53×#385—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73#13133150—41no—
74#13133148—22no—
75#13133148—22no—
76#13133148—41no—
77C4⋊#88322—22no—
78C4⋊#88322—22no—
79#13133138—22no—
80#13133138—22no—
81#13133138—41no—
82#13133138—41no—
83C22⋊#88322—41no—
84C22⋊#88322—41no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53—21no—
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94C23×C4×C53C2311yescharacteristic
95#121×C53—22no—
96#121×C53—22no—
97C4×D4×C53—22no—
98C4×D4×C53—22no—
99#117×C53—22no—
100#117×C53—22no—
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×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120C24×C53—21no—
121C24×C53C2×D411yescharacteristic
122C2×D4×C53—24no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—81no—
129C2×D4×C53—81no—
130C2×D4×C53#4112yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—42no—
141C22×C4×C53—81no—
142C22×C4×C53—81no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—81no—
161C4⋊C4×C53#4112yes—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—81no—
167C22⋊C4×C53—81no—
168C22⋊C4×C53—81no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—81no—
172C42×C53#4112yes—
173C42×C53#4112yes—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—81no—
185C4⋊Dic53—81no—
186C4×Dic53—81no—
187C4×Dic53—81no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—22no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53C22×D411yescharacteristic
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—41no—
218D4×C53—41no—
219D4×C53—84no—
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—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
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—84no—
247C2×C4×C53#12312yes—
248C2×C4×C53#12312yes—
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—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—41no—
266C2×Dic53—41no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#88322—41no—
271#27637—531nomaximal, Sylow
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—42no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—42no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—81no—
283C22×C53—81no—
284C22×C53—81no—
285C22×C53—81no—
286C22×C53C2×#12311yescharacteristic
287C22×C53#52511yescharacteristic
288C22×C53#53711yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57711yescharacteristic
291C22×C53#57711yescharacteristic
292C22×C53#57711yescharacteristic
293C4×C53—22no—
294C4×C53—22no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—42no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53#55211yescharacteristic
302C4×C53#55311yescharacteristic
303C4×C53#55511yescharacteristic
304C4×C53#55611yescharacteristic
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53—81no—
309#2324D5311yescharacteristic
310#2314—531no—
311#2077—531no—
312#2075—531no—
313#2034—531no—
314#2034—531no—
315#1949—531no—
316#1948—531no—
317#1926—531no—
318#1925—531no—
319#1920—531no—
320#1919—531no—
321#1804—531no—
322#1802—531no—
323#1798—531no—
324C2×C53—81no—
325C2×C53—82no—
326C2×C53#248511yescharacteristic
327C2×C53#309611yescharacteristic
328C2×C53#309911yescharacteristic
329C2×C53#311911yescharacteristic
330C2×C53#312311yescharacteristic
331C2×C53#325711yescharacteristic
332C2×C53#326311yescharacteristic
333C2×#121C2×D5311yescharacteristic
334C2×C4×D4C2×D5311yescharacteristic
335C2×#117C2×D5311yescharacteristic
336C2×#107—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×C4⋊C8—531no—
341C2×C4⋊C8—531no—
342C2×#102—531no—
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×C22⋊C8—531no—
347C2×C22⋊C8—531no—
348#390—21no—
349#390C2×D5311yescharacteristic
350#389C2×D5311yescharacteristic
351#388—21no—
352#388—1061no—
353#388—1061no—
354#387—1061no—
355#387—1061no—
356#386—21no—
357#385—21no—
358#384—531no—
359#382—531no—
360#380—531no—
361#380—531no—
362#380C2×D5311yescharacteristic
363C4×C4⋊C4—531no—
364C4×C4⋊C4—531no—
365C4×C4⋊C4C2×D5311yescharacteristic
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#336—531no—
375#336—531no—
376C53#2763711yescharacteristic, complemented, Sylow
377C22×D4—21no—
378C22×D4C22×D5311yescharacteristic
379C23×C4—21no—
380C23×C4C22×D5311yescharacteristic
381C22×C8—531no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—531no—
385#121—22no—
386#121—22no—
387C4×D4—22no—
388C4×D4—22no—
389#117—22no—
390#117—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22⋊D5311yescharacteristic
399C2×C4⋊C4C22⋊D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4C22⋊D5311yescharacteristic
411C2×C22⋊C4C22⋊D5311yescharacteristic
412C2×C42—21no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42C22×D5311yescharacteristic
416C2×C42C22×D5311yescharacteristic
417#107—2121no—
418#106—1062no—
419#106—1062no—
420#106—2121no—
421C4⋊C8—1062no—
422C4⋊C8—1062no—
423#102—1062no—
424#102—1062no—
425#102—2121no—
426#102—2121no—
427C22⋊C8—2121no—
428C22⋊C8—2121no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24—21no—
439C24C2×C22⋊D5311yescharacteristic
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4#1171994512yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468C22×C4C2×C22⋊D5311yescharacteristic
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C23×D5311yescharacteristic
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—81no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—4241no—
500C4⋊C4—4241no—
501C4⋊C4#1171994512yes—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4—4241no—
510C22⋊C4—4241no—
511C42—21no—
512C42—21no—
513C42—81no—
514C42—4241no—
515C42—4241no—
516C42#1171994712yes—
517C42#1171994712yes—
518C23—21no—
519C23—22no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×C22⋊D5311yescharacteristic
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537D4—84no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
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—84no—
565C2×C4—2121no—
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#1313319812yes—
578C2×C4#1313319812yes—
579C2×C4C2×#1171994711yescharacteristic
580C2×C4C2×#1171994711yescharacteristic
581C2×C4C2×#1171994511yescharacteristic
582C2×C4C2×#1171994511yescharacteristic
583C2×C4#1313326311yescharacteristic
584C2×C4#1313326411yescharacteristic
585C8—2121no—
586C8—2121no—
587C8—2121no—
588C8—2121no—
589C22—41no—
590C22—41no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×#1313319811yescharacteristic
604C22#1681113411yescharacteristic
605C22#1681120111yescharacteristic
606C22#1681142611yescharacteristic
607C22#1681143111yescharacteristic
608C22#1681143411yescharacteristic
609C22#1681143411yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—4241no—
619C4—4241no—
620C4—4241no—
621C4—4241no—
622C4#1681131311yescharacteristic
623C4#1681131911yescharacteristic
624C4#1681133811yescharacteristic
625C4#1681134311yescharacteristic
626C2—81no—
627C2—82no—
628C2#2027901211yescharacteristic
629C2#2028466711yescharacteristic
630C2#2028470011yescharacteristic
631C2#2028491411yescharacteristic
632C2#2028496711yescharacteristic
633C2#2028612011yescharacteristic
634C2#2028618011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2324C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2324D53
hypocenterC53#27637
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. #20277324C2semidirect
2. #20276152C2semidirect
3. #20276141C2semidirect
6. #20275483C2semidirect
7. #20275478C2semidirect
8. #20275339C2semidirect
9. #20275332C2semidirect
10. #20275297C2semidirect
11. #20275290C2semidirect
12. #20274725C2semidirect
13. #20274715C2semidirect
14. #20274699C2semidirect
376. C53#27637semidirect

Maximal subgroups

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

subgroupindex
#202773242
#202761522
#202761412
#202759012
#202754832
#202754782
#202753392
#202753322
#202752972
#202752902
#202747252
#202747152
#202746992
C53×#23242
#2763753