Finite Group Databasebeta

← #27466247

Subgroups — #27466247

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277338C211yesmaximal, characteristic
2#20276283C211yesmaximal, characteristic, complemented
3#20276273C211yesmaximal, characteristic, complemented
4#20276263C211yesmaximal, characteristic, complemented
5#20276253C211yesmaximal, characteristic, complemented
6#20275483C211yesmaximal, characteristic, complemented
7#20275473C211yesmaximal, characteristic, complemented
8#20275028C211yesmaximal, characteristic
9#20275023C211yesmaximal, characteristic
10#20274912C211yesmaximal, characteristic
11#20274902C211yesmaximal, characteristic
12#20274730C211yesmaximal, characteristic, complemented
13#20274720C211yesmaximal, characteristic, complemented
14#20274699C211yesmaximal, characteristic
15C53×#2323C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810522—21no—
31#16810522—21no—
32#16810498—21no—
33#16810498—21no—
34#16810493C2211yescharacteristic
35#16810493C2211yescharacteristic
36#16810489C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#388—21no—
56C53×#387—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—41no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—41no—
72C2×C4⋊Dic53—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—41no—
80#13133150—41no—
81#13133150—41no—
82#13133148—41no—
83C4⋊#88322—22no—
84C4⋊#88322—22no—
85#13133141—41no—
86#13133141—41no—
87#13133141—41no—
88#13133141—41no—
89C22⋊#88322—22no—
90C22⋊#88322—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×Q8×C53C2311yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53Q814yes—
99C4×Q8×C53—22no—
100C4×Q8×C53—22no—
101#117×C53—22no—
102#117×C53—22no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C2×Q812yes—
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×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×Q812yes—
145C22×C4×C53#4112yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C22×Dic53—81no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—81no—
164C4⋊C4×C53C2×Q812yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—81no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—42no—
172C42×C53—81no—
173C42×C53#4112yes—
174C42×C53#4112yes—
175C22⋊Dic53—42no—
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—42no—
185C4⋊Dic53—81no—
186C4⋊Dic53—81no—
187C4⋊Dic53—81no—
188C4⋊Dic53—81no—
189C4×Dic53—42no—
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—
200C2×#88322—21no—
201C2×#88322—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C22⋊Q812yes—
209C23×C53C22×D411yescharacteristic
210Q8×C53—22no—
211Q8×C53—22no—
212Q8×C53—41no—
213Q8×C53—41no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
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×C53C22⋊Q812yes—
243C2×C4×C53#12312yes—
244C2×C4×C53#12312yes—
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
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—84no—
264#88322—41no—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#27690—531nomaximal, Sylow
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53C2×C22⋊Q811yescharacteristic
277C22×C53C2×#12311yescharacteristic
278C22×C53#53511yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C22×C53#57811yescharacteristic
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—22no—
286C4×C53—42no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53#55311yescharacteristic
293C4×C53#55311yescharacteristic
294C4×C53#55411yescharacteristic
295C4×C53#55411yescharacteristic
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300#2323D5311yescharacteristic
301#2317—531no—
302#2101—531no—
303#2099—531no—
304#2097—531no—
305#2095—531no—
306#1949—531no—
307#1947—531no—
308#1873—531no—
309#1872—531no—
310#1854—531no—
311#1852—531no—
312#1805—531no—
313#1803—531no—
314#1798—531no—
315C2×C53—42no—
316C2×C53#248711yescharacteristic
317C2×C53#308211yescharacteristic
318C2×C53#308311yescharacteristic
319C2×C53#309711yescharacteristic
320C2×C53#309811yescharacteristic
321C2×C53#322311yescharacteristic
322C2×C53#322411yescharacteristic
323C2×C22⋊Q8C2×D5311yescharacteristic
324C2×C4×Q8C2×D5311yescharacteristic
325C2×#117—531no—
326C2×#117C2×D5311yescharacteristic
327C22×C4⋊C4—531no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#106—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×#103—531no—
337C2×#103—531no—
338C2×C22⋊C8—531no—
339C2×C22⋊C8—531no—
340#391C2×D5311yescharacteristic
341#389—21no—
342#389—1061no—
343#389—1061no—
344#389C2×D5311yescharacteristic
345#388—21no—
346#388—21no—
347#387—21no—
348#385—1061no—
349#385—1061no—
350#384—531no—
351#384—531no—
352#382—531no—
353#382—531no—
354#380C2×D5311yescharacteristic
355C4×C4⋊C4C2×D5311yescharacteristic
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#336—531no—
365#336—531no—
366C53#2769011yescharacteristic, complemented, Sylow
367C22×Q8C22×D5311yescharacteristic
368C23×C4—1061no—
369C23×C4C22×D5311yescharacteristic
370C22×C8—531no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—531no—
374C22⋊Q8—22no—
375C22⋊Q8Dic10614yes—
376C4×Q8—22no—
377C4×Q8—22no—
378#117—22no—
379#117—22no—
380#117—2121no—
381#117—2121no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—2121no—
395C2×C4⋊C4—2121no—
396C2×C4⋊C4C22⋊D5311yescharacteristic
397C2×C4⋊C4C22⋊D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4C22⋊D5311yescharacteristic
405C2×C22⋊C4C22⋊D5311yescharacteristic
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—1061no—
409C2×C42C22×D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411#107—2121no—
412#107—2121no—
413#107—2121no—
414#106—2121no—
415C4⋊C8—1062no—
416C4⋊C8—1062no—
417#103—2121no—
418#103—2121no—
419#103—2121no—
420#103—2121no—
421C22⋊C8—1062no—
422C22⋊C8—1062no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432C24C2×C22⋊D5311yescharacteristic
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×Dic10612yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—4241no—
452C22×C4—4241no—
453C22×C4—2122no—
454C22×C4C2×Dic10612yes—
455C22×C4#1171994512yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C23×D5311yescharacteristic
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—4241no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4C2×Dic10612yes—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—81no—
502C22⋊C4—2122no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—81no—
507C42—2122no—
508C42#1171994712yes—
509C42#1171994712yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#1313319612yes—
517C23C22×C22⋊D5311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—4244no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4#1313319612yes—
564C2×C4#1313319812yes—
565C2×C4#1313319812yes—
566C2×C4C22×Dic10611yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4#1313326311yescharacteristic
571C2×C4#1313326311yescharacteristic
572C8—2121no—
573C8—2121no—
574C8—2121no—
575C8—2121no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1313319611yescharacteristic
584C22C2×#1313319811yescharacteristic
585C22#1681118711yescharacteristic
586C22#1681142611yescharacteristic
587C22#1681143111yescharacteristic
588C22#1681143811yescharacteristic
589C22#1681143811yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—4241no—
600C4—4241no—
601C4—4241no—
602C4—4241no—
603C4#1681131911yescharacteristic
604C4#1681131911yescharacteristic
605C4#1681133011yescharacteristic
606C4#1681133011yescharacteristic
607C2—42no—
608C2#2027903211yescharacteristic
609C2#2028455711yescharacteristic
610C2#2028456611yescharacteristic
611C2#2028467811yescharacteristic
612C2#2028468911yescharacteristic
613C2#2028579211yescharacteristic
614C2#2028580311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2323C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2323D53
hypocenterC53#27690
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. #20276283C2semidirect
3. #20276273C2semidirect
4. #20276263C2semidirect
5. #20276253C2semidirect
6. #20275483C2semidirect
7. #20275473C2semidirect
12. #20274730C2semidirect
13. #20274720C2semidirect
366. C53#27690semidirect

Maximal subgroups

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

subgroupindex
#202773382
#202762832
#202762732
#202762632
#202762532
#202754832
#202754732
#202750282
#202750232
#202749122
#202749022
#202747302
#202747202
#202746992
C53×#23232
#2769053