Finite Group Databasebeta

← #27428552

Subgroups — #27428552

Order 13568.

796 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#20279017C211yesmaximal, characteristic
2#20277640C211yesmaximal, characteristic
3#20276285C211yesmaximal, characteristic, complemented
4#20276202C211yesmaximal, characteristic, complemented
5#20276153C211yesmaximal, characteristic, complemented
6#20276145C211yesmaximal, characteristic, complemented
7#20275306C211yesmaximal, characteristic, complemented
8#20275298C211yesmaximal, characteristic, complemented
9#20275007C211yesmaximal, characteristic
10#20274917C211yesmaximal, characteristic, complemented
11#20274908C211yesmaximal, characteristic, complemented
12#20274808C211yesmaximal, characteristic, complemented
13#20274801C211yesmaximal, characteristic, complemented
14#20274744C211yesmaximal, characteristic, complemented
15C2×C53×#340C211yesmaximal, characteristic, complemented
16C2×#13133200—21no—
17C2×C4×C22⋊D53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26#16810554—21no—
27#16810549—21no—
28#16810519C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810513—21no—
31#16810504—21no—
32#16810502—21no—
33#16810488C2211yescharacteristic
34#16810486—21no—
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39#16810480C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C22⋊Dic53—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C22⋊D53D411yescharacteristic
60C22×C4×D53C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133200—44no—
64C4×C22⋊D53—22no—
65C4×C22⋊D53—22no—
66#13133175—22no—
67#13133175—41no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919D411yescharacteristic
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53D411yescharacteristic
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921D411yescharacteristic
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86#13133151—22no—
87#13133151—41no—
88#13133148—22no—
89#13133148—41no—
90C22⋊#88322—22no—
91#13133129—22no—
92#13133129—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C23×C4×C53—21no—
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53—22no—
107C22×C8×C53—22no—
108C22×C8×C53—41no—
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—22no—
116C2×C4⋊C4×C53—41no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C24×C53C2×D411yescharacteristic
122C23×D53C2×D411yescharacteristic
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×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—81no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×C4×C53C2411yescharacteristic
141C2×C22⋊D53—22no—
142C2×C22⋊D53—22no—
143C2×C22⋊D53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—44no—
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—42no—
173C4⋊C4×C53#4112yes—
174C22⋊Dic53—22no—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177#11719921—22no—
178#11719921—42no—
179#11719921—42no—
180#11719921—42no—
181#11719921—81no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190#11719919—22no—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C4×Dic53—81no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53#12312yes—
208C23×C53#13512yes—
209C23×C53AGL(1,ℤ/8)12yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—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—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—84no—
237C2×C4×C53—84no—
238C2×C4×C53#12312yes—
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—44no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
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—
267C4×D53—41no—
268C4×D53—41no—
269C4×D53—41no—
270C4×D53—41no—
271C8×C53—22no—
272C8×C53—41no—
273#88322—41no—
274#88322—41no—
275#24941—531nomaximal, Sylow
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53#48214yes—
283C22×C53C2×#12311yescharacteristic
284C22×C53#54611yescharacteristic
285C22×C53#54711yescharacteristic
286C22×C53C2×#13511yescharacteristic
287C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C53#57711yescharacteristic
289C22×C53#57811yescharacteristic
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C2×D53—41no—
297C4×C53—22no—
298C4×C53—42no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53#52911yescharacteristic
303C4×C53#53911yescharacteristic
304C4×C53#55111yescharacteristic
305C4×C53#55211yescharacteristic
306Dic53—41no—
307Dic53—41no—
308Dic53—41no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313#2486—531no—
314#2352—531no—
315#2101—531no—
316#2085—531no—
317#2077—531no—
318#2076—531no—
319#1921—531no—
320#1920—531no—
321#1869—531no—
322#1854—531no—
323#1853—531no—
324#1825—531no—
325#1823—531no—
326#1808—531no—
327C2×#340D5311yescharacteristic, complemented
328C2×C53—24no—
329C2×C53#241911yescharacteristic
330C2×C53C2×#48211yescharacteristic
331C2×C53#309711yescharacteristic
332C2×C53#329611yescharacteristic
333C2×C53#330811yescharacteristic
334C2×C53#333611yescharacteristic
335C2×C53#334511yescharacteristic
336D53—81no—
337C23×C8C2×D5311yescharacteristic
338C2×#123—1061no—
339C2×C4×D4—531no—
340C2×#117—531no—
341C22×C4⋊C4C2×D5311yescharacteristic
342C22×C4⋊C4C2×D5311yescharacteristic
343C2×#107—531no—
344C2×#107—531no—
345C2×#106—531no—
346C2×#106—531no—
347C2×C22⋊C8—531no—
348C2×C22⋊C8—531no—
349C2×C22⋊C8—531no—
350#398—1061no—
351#397—1061no—
352#389—531no—
353#389—531no—
354#388—1061no—
355#387—1061no—
356#386—1061no—
357#382—531no—
358#381—1061no—
359#380—531no—
360#380—531no—
361#380—531no—
362#380—531no—
363#380—531no—
364C4×C4⋊C4—531no—
365C4×C22⋊C4—1061no—
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#340C2×D5311yescharacteristic
375#340C2×D5311yescharacteristic
376#340C2×D5311yescharacteristic
377#340C2×D5311yescharacteristic
378#336—531no—
379#336—531no—
380C53#2494111yescharacteristic, complemented, Sylow
381C22×D4—531no—
382C23×C4—21no—
383C23×C4—21no—
384C23×C4—531no—
385C23×C4C22×D5311yescharacteristic
386C22×C8—22no—
387C22×C8—22no—
388C22×C8—41no—
389C22×C8—531no—
390C22×C8—531no—
391C22×C8C22×D5311yescharacteristic
392C22×C8C22×D5311yescharacteristic
393#123—2124no—
394C4×D4—1062no—
395C4×D4—1062no—
396#117—1062no—
397#117—2121no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—41no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C4⋊C4C22×D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C42—531no—
424C2×C42—1061no—
425C2×C42—1061no—
426#107—1062no—
427#107—2121no—
428#106—1062no—
429#106—2121no—
430C22⋊C8—1062no—
431C22⋊C8—1062no—
432C22⋊C8—2121no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443C24—531no—
444C24D4×D5311yescharacteristic
445C2×D4—1062no—
446C2×D4—1062no—
447C2×D4—1062no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4#1171994512yes—
476C22×C4D4×D5311yescharacteristic
477C22×C4#1171994712yes—
478C22×C4C23×D5311yescharacteristic
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—44no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8#1171994512yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—1062no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4#1171994512yes—
513C22⋊C4—1062no—
514C22⋊C4—1062no—
515C22⋊C4—4241no—
516C22⋊C4—2122no—
517C22⋊C4—2122no—
518C22⋊C4—2122no—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—1061no—
522C42—1061no—
523C42—4241no—
524C42—2122no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—1061no—
530C23—1061no—
531C23—1061no—
532C23—1061no—
533C23—1061no—
534C23—1061no—
535C23—1061no—
536C23#1313319912yes—
537C23#1313323612yes—
538C23#1313324412yes—
539C23C2×D4×D5311yescharacteristic
540D4—2124no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
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—84no—
560C2×C4—84no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
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×C4—2121no—
586C2×C4#1313319912yes—
587C2×C4C2×#1171994711yescharacteristic
588C2×C4C2×#1171994511yescharacteristic
589C2×C4C2×#1171994511yescharacteristic
590C2×C4C2×#1171994511yescharacteristic
591C2×C4#1313326211yescharacteristic
592C2×C4#1313326511yescharacteristic
593C8—22no—
594C8—41no—
595C8—2121no—
596C8—2121no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—2121no—
604C22—2121no—
605C22—2121no—
606C22—2121no—
607C22—2121no—
608C22—2121no—
609C22—2121no—
610C22#1681094514yes—
611C22C2×#1313319911yescharacteristic
612C22#1681126311yescharacteristic
613C22#1681127411yescharacteristic
614C22C2×#1313323611yescharacteristic
615C22C2×#1313324411yescharacteristic
616C22#1681142911yescharacteristic
617C22#1681143611yescharacteristic
618C4—22no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—4241no—
624C4—4241no—
625C4—4241no—
626C4—4241no—
627C4—2121no—
628C4—2121no—
629C4—2121no—
630C4#1681115711yescharacteristic
631C4#1681122011yescharacteristic
632C4#1681129911yescharacteristic
633C4#1681130811yescharacteristic
634C2—24no—
635C2—4241no—
636C2#2027819211yescharacteristic
637C2C2×#1681094511yescharacteristic
638C2#2028468011yescharacteristic
639C2#2028654611yescharacteristic
640C2#2028670211yescharacteristic
641C2#2028707911yescharacteristic
642C2#2028718111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC23×C53C22×D4
prime radicalC22×C22⋊D53D4
hypercenterC2×#340D53
hypocenterC53#24941
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #20276285C2semidirect
4. #20276202C2semidirect
5. #20276153C2semidirect
6. #20276145C2semidirect
7. #20275306C2semidirect
8. #20275298C2semidirect
10. #20274917C2semidirect
11. #20274908C2semidirect
12. #20274808C2semidirect
13. #20274801C2semidirect
14. #20274744C2semidirect
15. C2×C53×#340C2semidirect
327. C2×#340D53semidirect
380. C53#24941semidirect

Maximal subgroups

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

subgroupindex
#202790172
#202776402
#202762852
#202762022
#202761532
#202761452
#202753062
#202752982
#202750072
#202749172
#202749082
#202748082
#202748012
#202747442
C2×C53×#3402
#2494153