Finite Group Databasebeta

← #27447819

Subgroups — #27447819

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277729C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276211C211yesmaximal, characteristic, complemented
4#20276100C211yesmaximal, characteristic, complemented
5#20275858C211yesmaximal, characteristic, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275304C211yesmaximal, characteristic, complemented
8#20275141C211yesmaximal, characteristic
9#20275032C211yesmaximal, characteristic
10#20274961C211yesmaximal, characteristic, complemented
11#20274902C211yesmaximal, characteristic, complemented
12#20274895C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#2442C211yesmaximal, characteristic, complemented
16C23×#88322C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133177—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810525C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810515—21no—
33#16810500—21no—
34#16810497—21no—
35C4×C4⋊Dic53C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#125×C53C2211yescharacteristic
49C22×C4⋊C4×C53C2211yescharacteristic
50C53×#400—21no—
51C53×#398—21no—
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#385—21no—
55C53×#382C2211yescharacteristic
56C53×#380—21no—
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C22⋊D53—21no—
61C23×Dic53—21no—
62C22×C4×D53—21no—
63C22×#88322—22no—
64C22×#88322—41no—
65C22×#88322—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133191—22no—
71#13133191Q814yes—
72C4×D212—22no—
73C4×D212—41no—
74#13133177—44no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87#13133150—22no—
88#13133150—41no—
89C4⋊#88322—41no—
90C4⋊#88322—41no—
91#13133137—22no—
92#13133137—41no—
93#13133137—41no—
94#13133137—41no—
95C22⋊#88322—22no—
96#13133124—21no—
97#13133124—21no—
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100#125×C53—22no—
101#125×C53—41no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53—41no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—81no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×Q812yes—
144C22×C4×C53#4112yes—
145C22×C4×C53C2411yescharacteristic
146C2×C22⋊D53—24no—
147C2×C22⋊D53—42no—
148C2×C22⋊D53—42no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—81no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212C2×Q812yes—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—81no—
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—42no—
174C4⋊C4×C53—81no—
175C4⋊C4×C53—81no—
176C4⋊C4×C53#4112yes—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—42no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184#11719921—24no—
185#11719921—42no—
186#11719921—42no—
187#11719921—81no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53C2×Q812yes—
194C4⋊Dic53#4112yes—
195#11719919—42no—
196C4×Dic53—42no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—44no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53C22⋊Q812yes—
215C23×C53#13512yes—
216C23×C53AGL(1,ℤ/8)12yes—
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—84no—
256C2×C4×C53C22⋊Q812yes—
257C2×C4×C53C22×Q811yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C22⋊D53—48no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—84no—
274D212—22no—
275D212—22no—
276D212—41no—
277D212—41no—
278C4×D53—84no—
279#88322—41no—
280#88322—41no—
281#88322—41no—
282#88322—41no—
283#26420—531nomaximal, Sylow
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53#47814yes—
291C22×C53C2×C22⋊Q811yescharacteristic
292C22×C53#54611yescharacteristic
293C22×C53#54611yescharacteristic
294C22×C53C2×#13511yescharacteristic
295C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C53#57611yescharacteristic
297C22×C53#57811yescharacteristic
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—81no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C4×C53—22no—
306C4×C53—41no—
307C4×C53—42no—
308C4×C53—41no—
309C4×C53—41no—
310C4×C53—41no—
311C4×C53—81no—
312C4×C53—81no—
313C4×C53—81no—
314C4×C53#52911yescharacteristic
315C4×C53#53811yescharacteristic
316C4×C53#55411yescharacteristic
317C4×C53#55411yescharacteristic
318Dic53—42no—
319Dic53—42no—
320Dic53—81no—
321#2442D5311yescharacteristic, complemented
322#2361—531no—
323#2101—531no—
324#2087—531no—
325#2067—531no—
326#2025—531no—
327#1989—531no—
328#1921—531no—
329#1893—531no—
330#1874—531no—
331#1862—531no—
332#1852—531no—
333#1850—531no—
334#1804—531no—
335C2×#340—531no—
336C2×C53—24no—
337C2×C53#245011yescharacteristic
338C2×C53C2×#47811yescharacteristic
339C2×C53#308311yescharacteristic
340C2×C53#329711yescharacteristic
341C2×C53#330011yescharacteristic
342C2×C53#330811yescharacteristic
343C2×C53#331811yescharacteristic
344D53—82no—
345C23×C8—531no—
346C2×#125C2×D5311yescharacteristic
347C2×#121—531no—
348C2×C4×D4—531no—
349C2×C4×D4—1061no—
350C22×C4⋊C4—531no—
351C22×C4⋊C4C2×D5311yescharacteristic
352C2×#107—531no—
353C2×#107—531no—
354C2×C4⋊C8—531no—
355C2×C4⋊C8—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×C22⋊C8—531no—
361#400—21no—
362#398—21no—
363#390—531no—
364#389—531no—
365#389C2×D5311yescharacteristic
366#389C2×D5311yescharacteristic
367#388—1061no—
368#386—1061no—
369#385—21no—
370#385—1061no—
371#382C2×D5311yescharacteristic
372#380—21no—
373#380C2×D5311yescharacteristic
374C4×C4⋊C4—531no—
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#336—531no—
387#336—531no—
388C53#2642011yescharacteristic, complemented, Sylow
389C22×D4—531no—
390C22×D4—1061no—
391C23×C4—21no—
392C23×C4—1061no—
393C23×C4—1061no—
394C23×C4C22×D5311yescharacteristic
395C22×C8—531no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—1062no—
400C22×C8—2121no—
401C22×C8—2121no—
402#125—22no—
403#125—41no—
404#121—534no—
405#121—1062no—
406C4×D4—1062no—
407C4×D4—2121no—
408C4×D4—2124no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—41no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—2121no—
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4C22⋊D5311yescharacteristic
433C2×C22⋊C4C22⋊D5311yescharacteristic
434C2×C42—1061no—
435C2×C42C22×D5311yescharacteristic
436C2×C42C22×D5311yescharacteristic
437#107—1062no—
438#107—2121no—
439C4⋊C8—2121no—
440C4⋊C8—2121no—
441#102—1062no—
442#102—2121no—
443#102—2121no—
444#102—2121no—
445C22⋊C8—1062no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95C22⋊D5311yescharacteristic
454#95C22⋊D5311yescharacteristic
455C24—1061no—
456C24C2×C22⋊D5311yescharacteristic
457C2×D4—532no—
458C2×D4—1064no—
459C2×D4—1061no—
460C2×D4—1061no—
461C2×D4—1061no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—2122no—
465C2×D4—2122no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—81no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—4241no—
481C22×C4—4241no—
482C22×C4—2122no—
483C22×C4—2122no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4Q8×D5312yes—
487C22×C4#1171994912yes—
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C2×C22⋊D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—2124no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—532no—
520C4⋊C4—532no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4#1171994712yes—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—42no—
531C22⋊C4—1064no—
532C22⋊C4—4241no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—2122no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—1062no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23#1313319312yes—
552C23#1313323712yes—
553C23#1313324212yes—
554C23C22×C22⋊D5311yescharacteristic
555D4—1062no—
556D4—1062no—
557D4—2128no—
558D4—2121no—
559D4—2121no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—4244no—
594C2×C4—4244no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4#1313319312yes—
603C2×C4C2×Q8×D5311yescharacteristic
604C2×C4#41×D5311yescharacteristic
605C2×C4C2×#1171994711yescharacteristic
606C2×C4C2×#1171994911yescharacteristic
607C2×C4#1313326211yescharacteristic
608C2×C4#1313326311yescharacteristic
609C8—2121no—
610C8—2121no—
611C8—2121no—
612C8—2121no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—4241no—
620C22—4241no—
621C22—4241no—
622C22—4241no—
623C22—2122no—
624C22—2122no—
625C22—2122no—
626C22#1681092014yes—
627C22C2×#1313319311yescharacteristic
628C22#1681126111yescharacteristic
629C22#1681126711yescharacteristic
630C22C2×#1313323711yescharacteristic
631C22C2×#1313324211yescharacteristic
632C22#1681142611yescharacteristic
633C22#1681143811yescharacteristic
634C4—22no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—4241no—
644C4—2122no—
645C4—2122no—
646C4#1681115911yescharacteristic
647C4#1681121311yescharacteristic
648C4#1681132711yescharacteristic
649C4#1681133111yescharacteristic
650C2—24no—
651C2—4242no—
652C2#2027853511yescharacteristic
653C2C2×#1681092011yescharacteristic
654C2#2028456211yescharacteristic
655C2#2028656511yescharacteristic
656C2#2028659611yescharacteristic
657C2#2028670611yescharacteristic
658C2#2028685011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2442C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2442D53
hypocenterC53#26420
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276211C2semidirect
4. #20276100C2semidirect
5. #20275858C2semidirect
6. #20275673C2semidirect
7. #20275304C2semidirect
10. #20274961C2semidirect
11. #20274902C2semidirect
12. #20274895C2semidirect
13. #20274722C2semidirect
14. C2×#16810385C2semidirect
15. C53×#2442C2semidirect
321. #2442D53semidirect
388. C53#26420semidirect

Maximal subgroups

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

subgroupindex
#202777292
#202762822
#202762112
#202761002
#202758582
#202756732
#202753042
#202751412
#202750322
#202749612
#202749022
#202748952
#202747222
C2×#168103852
C53×#24422
#2642053