Finite Group Databasebeta

← #27466187

Subgroups — #27466187

Order 13568.

798 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133193C211yesmaximal, characteristic
2#382×D53C211yesmaximal, characteristic
3#20276287C211yesmaximal, characteristic, complemented
4#20276285C211yesmaximal, characteristic, complemented
5#20276265C211yesmaximal, characteristic, complemented
6#20276261C211yesmaximal, characteristic, complemented
7#20275485C211yesmaximal, characteristic, complemented
8#20275481C211yesmaximal, characteristic, complemented
9#20275019C211yesmaximal, characteristic
10#20275019C211yesmaximal, characteristic
11#20274909C211yesmaximal, characteristic
12#20274904C211yesmaximal, characteristic
13#20274730C211yesmaximal, characteristic, complemented
14#20274692C211yesmaximal, characteristic
15C53×#1805C211yesmaximal, characteristic, complemented
16C2×#13133193C2211yescharacteristic
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C42×D53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29#16810530C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810504—21no—
32#16810494—21no—
33#16810489C2211yescharacteristic
34#16810488C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810487—21no—
37#16810487—21no—
38#16810480C2211yescharacteristic
39C4×#11719919—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41C4×#11719921—21no—
42#95×D53—21no—
43#95×D53—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×Q8×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D411yescharacteristic
63C22×C4×D53C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133193—22no—
67#13133193Q814yes—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C4⋊C4×D53—41no—
77C4⋊C4×D53—41no—
78C4⋊C4×D53—41no—
79C4⋊C4×D53—41no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53—22no—
87C42×D53—22no—
88#13133151—41no—
89#13133150—41no—
90#13133149—41no—
91#13133148—41no—
92C4⋊#88322—22no—
93#13133141—41no—
94#13133141—41no—
95#13133129—22no—
96#13133129—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×Q8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C4×Q8×C53—22no—
112C4×Q8×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C42×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C4⋊C8×C53—22no—
120#103×C53—41no—
121#103×C53—41no—
122#95×C53—21no—
123#95×C53—21no—
124C23×D53C2×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×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×C4×D53—42no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53C2×Q812yes—
162C2×C4×D53#4112yes—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C42×C53—42no—
177C42×C53#4112yes—
178C42×C53#4112yes—
179#11719921—24no—
180#11719921—42no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53C2×Q812yes—
189#11719919—24no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—42no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C23×C53C22×D411yescharacteristic
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—21no—
209C22×D53C22⋊Q812yes—
210C22×D53#13512yes—
211C22×D53#13712yes—
212Q8×C53—22no—
213Q8×C53—22no—
214Q8×C53—41no—
215Q8×C53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
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#12312yes—
233C2×C4×C53#12312yes—
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
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×Dic53C22⋊Q812yes—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—22no—
265C4×D53—44no—
266C4×D53—44no—
267C4×D53—84no—
268C4×D53—84no—
269C8×C53—41no—
270C8×C53—41no—
271#88322—41no—
272#88322—41no—
273#27686—531nomaximal, Sylow
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53C2×#12311yescharacteristic
276C22×C53#53511yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53C2×#13711yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57511yescharacteristic
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—22no—
287C2×D53#47914yes—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—42no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53#55311yescharacteristic
295C4×C53#55311yescharacteristic
296C4×C53#55411yescharacteristic
297C4×C53#55411yescharacteristic
298Dic53—22no—
299Dic53—42no—
300Dic53—41no—
301Dic53—41no—
302Dic53—81no—
303Dic53—81no—
304Dic53—81no—
305C4×C22⋊Q8—531no—
306C2×#382—531no—
307#2101—531no—
308#2101—531no—
309#2097—531no—
310#2097—531no—
311#1949—531no—
312#1949—531no—
313#1872—531no—
314#1872—531no—
315#1853—531no—
316#1852—531no—
317#1805—531no—
318#1805D5311yescharacteristic, complemented
319#1797—531no—
320C2×C53#248711yescharacteristic
321C2×C53C2×#47911yescharacteristic
322C2×C53#308011yescharacteristic
323C2×C53#309111yescharacteristic
324C2×C53#309511yescharacteristic
325C2×C53#321111yescharacteristic
326C2×C53#321511yescharacteristic
327D53—24no—
328C2×C22⋊Q8—531no—
329C2×C4×Q8C2×D5311yescharacteristic
330C22×C4⋊C4—531no—
331C22×C4⋊C4—531no—
332C22×C42—531no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#106—531no—
336C2×#106—531no—
337C2×C4⋊C8—531no—
338C2×C4⋊C8C2×D5311yescharacteristic
339C2×#103—531no—
340C2×#103—531no—
341C2×#103C2×D5311yescharacteristic
342C2×#103C2×D5311yescharacteristic
343C2×C22⋊C8—531no—
344C2×C22⋊C8—531no—
345#391—531no—
346#389—531no—
347#387—1061no—
348#385—1061no—
349#382—531no—
350#382—531no—
351#382—531no—
352#382—1061no—
353#382—1061no—
354#382C2×D5311yescharacteristic
355#380—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C22⋊C4—1061no—
359C2×#95—1061no—
360C2×#95—1061no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371C53#2768611yescharacteristic, complemented, Sylow
372C22×Q8C22×D5311yescharacteristic
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—1061no—
377C23×C4—1061no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22⋊Q8—534no—
383C22⋊Q8—1062no—
384C4×Q8—22no—
385C4×Q8—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—2121no—
396C2×C4⋊C4—2121no—
397C2×C4⋊C4—2121no—
398C2×C4⋊C4—2121no—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C42—21no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—1062no—
409C2×C42—1062no—
410C2×C42—1061no—
411C2×C42C22×D5311yescharacteristic
412#107—2121no—
413#107—2121no—
414#106—2121no—
415#106—2121no—
416C4⋊C8—22no—
417C4⋊C8—1062no—
418#103—41no—
419#103—41no—
420#103—2121no—
421#103—2121no—
422C22⋊C8—1062no—
423C22⋊C8—1062no—
424#95—21no—
425#95—21no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24—531no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8Q8×D5312yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2121no—
466C22×C4—2121no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4D4×D5311yescharacteristic
474C22×C4D4×D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—532no—
496C4⋊C4—1064no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C22⋊C4—1064no—
508C22⋊C4—2122no—
509C42—42no—
510C42—1062no—
511C42—1062no—
512C42—1061no—
513C42—1061no—
514C42—2122no—
515C42#1171994512yes—
516C42#1171994712yes—
517C23—532no—
518C23—532no—
519C23—532no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23C2×D4×D5311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—532no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—4244no—
560C2×C4—4244no—
561C2×C4—2124no—
562C2×C4—2124no—
563C2×C4—2121no—
564C2×C4—2121no—
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#1313319912yes—
575C2×C4#1313319912yes—
576C2×C4C2×Q8×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994511yescharacteristic
580C2×C4#1313326211yescharacteristic
581C2×C4#1313326311yescharacteristic
582C8—41no—
583C8—41no—
584C8—2121no—
585C8—2121no—
586C22—534no—
587C22—1062no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22—1062no—
592C22—1062no—
593C22C22⋊Q8×D5311yescharacteristic
594C22C2×#1313319911yescharacteristic
595C22#1681118911yescharacteristic
596C22#135×D5311yescharacteristic
597C22#137×D5311yescharacteristic
598C22#1681142011yescharacteristic
599C22#1681142211yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—1062no—
607C4—4241no—
608C4—4241no—
609C4—4241no—
610C4—2122no—
611C4—2121no—
612C4—2121no—
613C4#1681131711yescharacteristic
614C4#1681132011yescharacteristic
615C4#1681132811yescharacteristic
616C4#1681133111yescharacteristic
617C2—1064no—
618C2#2027902811yescharacteristic
619C2#479×D5311yescharacteristic
620C2#2028453411yescharacteristic
621C2#2028462611yescharacteristic
622C2#2028466111yescharacteristic
623C2#2028568411yescharacteristic
624C2#2028571611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1805C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1805D53
hypocenterC53#27686
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
3. #20276287C2semidirect
4. #20276285C2semidirect
5. #20276265C2semidirect
6. #20276261C2semidirect
7. #20275485C2semidirect
8. #20275481C2semidirect
13. #20274730C2semidirect
15. C53×#1805C2semidirect
318. #1805D53semidirect
371. C53#27686semidirect

Maximal subgroups

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

subgroupindex
C4×#131331932
#382×D532
#202762872
#202762852
#202762652
#202762612
#202754852
#202754812
#202750192
#202749092
#202749042
#202747302
#202746922
C53×#18052
#2768653