Finite Group Databasebeta

← #27452584

Subgroups — #27452584

Order 13568.

762 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277740C211yesmaximal, characteristic, complemented
2#20276032C211yesmaximal, characteristic, complemented
3#20275911C211yesmaximal, characteristic, complemented
4#20275863C211yesmaximal, characteristic, complemented
5#20275718C211yesmaximal, characteristic
6#20275673C211yesmaximal, characteristic, complemented
7#20275415C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic, complemented
9#20275330C211yesmaximal, characteristic, complemented
10#20275300C211yesmaximal, characteristic, complemented
11#20275073C211yesmaximal, characteristic, complemented
12#20275038C211yesmaximal, characteristic
13#20274807C211yesmaximal, characteristic, complemented
14#20274548C211yesmaximal, characteristic, complemented
15C53×#2490C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810517—21no—
34#16810511—21no—
35#16810505—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37#16810474—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#125×C53C2211yescharacteristic
49C2×#121×C53C2211yescharacteristic
50C53×#397—21no—
51C53×#394—21no—
52C53×#390—21no—
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#388—21no—
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719923—22no—
61C2×#11719923—41no—
62C2×#11719923—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133196—22no—
68#13133196—22no—
69C4×Dic106—22no—
70C4×Dic106—41no—
71#13133176—22no—
72#13133176—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84#13133148—22no—
85C4⋊#88322—41no—
86#13133140—22no—
87#13133140—22no—
88#13133140—41no—
89#13133138—22no—
90#13133138—22no—
91#13133138—41no—
92C22⋊#88322—41no—
93C22⋊#88322—41no—
94#13133128—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×D4×C53—21no—
100C22×D4×C53C2311yescharacteristic
101C23×C4×C53C2311yescharacteristic
102#125×C53—22no—
103#125×C53—41no—
104#121×C53—22no—
105#121×C53—41no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
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×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—42no—
135C2×D4×C53—42no—
136C2×D4×C53—81no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106#4112yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53#4112yes—
170C22⋊C4×C53—41no—
171C22⋊C4×C53—41no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—81no—
175C22⋊C4×C53—81no—
176C22⋊C4×C53—81no—
177C22⋊C4×C53—81no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—81no—
183C22⋊Dic53—24no—
184C22⋊Dic53—42no—
185#11719923—44no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53#4112yes—
191C4⋊Dic53#4112yes—
192#11719919—24no—
193#11719919—42no—
194#11719919—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
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—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53C22×D411yescharacteristic
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—84no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
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—21no—
245C2×C4×C53—21no—
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—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C53H(5,𝔽2)11yescharacteristic
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276Dic106—22no—
277Dic106—22no—
278Dic106—41no—
279Dic106—41no—
280#88322—41no—
281#88322—41no—
282#88322—41no—
283#88322—41no—
284#26751—531nomaximal, Sylow
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53#52511yescharacteristic
300C22×C53#54611yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53#57511yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57711yescharacteristic
305C22×C53#57711yescharacteristic
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53—42no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53—81no—
313C4×C53—81no—
314C4×C53#52911yescharacteristic
315C4×C53#53811yescharacteristic
316C4×C53#55511yescharacteristic
317C4×C53#55911yescharacteristic
318Dic53—42no—
319Dic53—42no—
320Dic53—42no—
321Dic53—81no—
322#2490D5311yescharacteristic
323#2362—531no—
324#2055—531no—
325#2035—531no—
326#2026—531no—
327#2000—531no—
328#1989—531no—
329#1936—531no—
330#1926—531no—
331#1924—531no—
332#1920—531no—
333#1880—531no—
334#1875—531no—
335#1825—531no—
336#1763—531no—
337C2×C53—81no—
338C2×C53—161no—
339C2×C53#245911yescharacteristic
340C2×C53#311111yescharacteristic
341C2×C53#312611yescharacteristic
342C2×C53#330711yescharacteristic
343C2×C53#331111yescharacteristic
344C2×C53#331711yescharacteristic
345C2×C53#332611yescharacteristic
346C22×SD16—531no—
347C2×#125C2×D5311yescharacteristic
348C2×C22⋊Q8—531no—
349C2×#121C2×D5311yescharacteristic
350C2×C4×Q8—531no—
351C2×#117—531no—
352C2×#106—531no—
353C2×C4⋊C8—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×C22⋊C8—531no—
361C2×C22⋊C8—531no—
362C2×#97—531no—
363#397—21no—
364#394—21no—
365#391—531no—
366#390—21no—
367#390C2×D5311yescharacteristic
368#390C2×D5311yescharacteristic
369#389—531no—
370#388—21no—
371#388—1061no—
372#388—1061no—
373#387—1061no—
374#380C2×D5311yescharacteristic
375C4×C4⋊C4—531no—
376C4×C4⋊C4—1061no—
377C4×C4⋊C4C2×D5311yescharacteristic
378C4×C22⋊C4C2×D5311yescharacteristic
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#336—531no—
388#336—531no—
389C53#2675111yescharacteristic, complemented, Sylow
390C22×Q8—531no—
391C22×D4—21no—
392C22×D4C22×D5311yescharacteristic
393C23×C4C22×D5311yescharacteristic
394C2×SD16—1062no—
395C2×SD16—2121no—
396C2×SD16—2121no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401#125—22no—
402#125—41no—
403C22⋊Q8—1062no—
404C22⋊Q8—1062no—
405#121—22no—
406#121—41no—
407C4×Q8—1062no—
408C4×Q8—2121no—
409#117—1062no—
410#117—2121no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4C22⋊D5311yescharacteristic
431C2×C22⋊C4C22⋊D5311yescharacteristic
432C2×C42—21no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42C22×D5311yescharacteristic
437C2×C42C22×D5311yescharacteristic
438#106—1062no—
439C4⋊C8—2121no—
440#103—1062no—
441#103—1062no—
442#103—2121no—
443#102—1062no—
444#102—1062no—
445#102—2121no—
446C22⋊C8—2121no—
447C22⋊C8—2121no—
448#97—2121no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95C22⋊D5311yescharacteristic
456#95C22⋊D5311yescharacteristic
457C24—21no—
458C24C2×C22⋊D5311yescharacteristic
459C2×Q8—532no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—81no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C2×C22⋊D5311yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491C22×C4C2×C22⋊D5311yescharacteristic
492C22×C4C23×D5311yescharacteristic
493SD16—2124no—
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—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—532no—
514C4⋊C4—532no—
515C4⋊C4—1064no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4#1171994712yes—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—81no—
529C22⋊C4—81no—
530C22⋊C4—81no—
531C22⋊C4—1064no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—81no—
538C42—2122no—
539C42—2122no—
540C42—2122no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C22×C22⋊D5311yescharacteristic
556Q8—1062no—
557Q8—1062no—
558Q8—2121no—
559Q8—2121no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
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—21no—
580C2×C4—21no—
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—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4#41×D5311yescharacteristic
606C2×C4#41×D5311yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4#1313326211yescharacteristic
610C2×C4#1313326311yescharacteristic
611C8—2121no—
612C8—2121no—
613C8—2121no—
614C8—2121no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#1681113011yescharacteristic
630C22#1681126111yescharacteristic
631C22#1681126711yescharacteristic
632C22#1681142111yescharacteristic
633C22#1681142111yescharacteristic
634C22#1681143411yescharacteristic
635C22#1681143411yescharacteristic
636C4—41no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—4241no—
645C4—2122no—
646C4—2122no—
647C4—2122no—
648C4#1681115211yescharacteristic
649C4#1681120711yescharacteristic
650C4#1681133511yescharacteristic
651C4#1681135511yescharacteristic
652C2—81no—
653C2—161no—
654C2#2027864811yescharacteristic
655C2#2028481811yescharacteristic
656C2#2028500111yescharacteristic
657C2#2028668511yescharacteristic
658C2#2028675111yescharacteristic
659C2#2028684111yescharacteristic
660C2#2028696211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2490C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2490D53
hypocenterC53#26751
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2490

Complements

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

NH ≅ G/Ntype
1. #20277740C2semidirect
2. #20276032C2semidirect
3. #20275911C2semidirect
4. #20275863C2semidirect
6. #20275673C2semidirect
7. #20275415C2semidirect
8. #20275339C2semidirect
9. #20275330C2semidirect
10. #20275300C2semidirect
11. #20275073C2semidirect
13. #20274807C2semidirect
14. #20274548C2semidirect
389. C53#26751semidirect

Maximal subgroups

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

subgroupindex
#202777402
#202760322
#202759112
#202758632
#202757182
#202756732
#202754152
#202753392
#202753302
#202753002
#202750732
#202750382
#202748072
#202745482
C53×#24902
#2675153