Finite Group Databasebeta

← #27447772

Subgroups — #27447772

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278428C211yesmaximal, characteristic
2#20276278C211yesmaximal, characteristic, complemented
3#20276206C211yesmaximal, characteristic, complemented
4#20276110C211yesmaximal, characteristic, complemented
5#20275866C211yesmaximal, characteristic, complemented
6#20275672C211yesmaximal, characteristic, complemented
7#20275293C211yesmaximal, characteristic, complemented
8#20275139C211yesmaximal, characteristic
9#20275028C211yesmaximal, characteristic
10#20274983C211yesmaximal, characteristic, complemented
11#20274914C211yesmaximal, characteristic
12#20274895C211yesmaximal, characteristic
13#20274720C211yesmaximal, characteristic, complemented
14#20274431C211yesmaximal, characteristic
15C53×#2362C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810559—21no—
29#16810546—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810520—21no—
33#16810506—21no—
34#16810493C2211yescharacteristic
35#16810483C2211yescharacteristic
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×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#387—21no—
56C4×C4⋊C4×C53—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322—22no—
60C22×#88322—41no—
61C22×#88322—41no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133209—22no—
67#13133209—41no—
68#13133176—22no—
69#13133176—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—41no—
84#13133148—22no—
85C4⋊#88322—41no—
86C4⋊#88322—41no—
87#13133141—22no—
88#13133141—41no—
89#13133141—41no—
90#13133141—41no—
91C22⋊#88322—22no—
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—41no—
101#117×C53—22no—
102#117×C53—41no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53—21no—
114C2×C42×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
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—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—42no—
158C4⋊C4×C53C2×Q812yes—
159C4⋊C4×C53#4112yes—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—42no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—42no—
169C42×C53—42no—
170C42×C53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—42no—
181C4⋊Dic53—81no—
182C4⋊Dic53#4112yes—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186#11719919—81no—
187C4×Dic53—42no—
188C2×#88322—21no—
189C2×#88322—21no—
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—44no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53C22⋊Q812yes—
208C23×C53C22×D411yescharacteristic
209Q8×C53—22no—
210Q8×C53—22no—
211Q8×C53—41no—
212Q8×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
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—21no—
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#13512yes—
244C2×C4×C53C22×Q811yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×Dic53—22no—
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—
262#88322—41no—
263#88322—41no—
264#88322—41no—
265#88322—41no—
266#26417—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53#54611yescharacteristic
276C22×C53#54611yescharacteristic
277C22×C53C2×#13511yescharacteristic
278C22×C53#57511yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57811yescharacteristic
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—42no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—81no—
290C4×C53#52911yescharacteristic
291C4×C53#53811yescharacteristic
292C4×C53#55411yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—42no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298#2441—531no—
299#2362D5311yescharacteristic
300#2100—531no—
301#2086—531no—
302#2068—531no—
303#2026—531no—
304#1989—531no—
305#1919—531no—
306#1892—531no—
307#1873—531no—
308#1865—531no—
309#1854—531no—
310#1850—531no—
311#1803—531no—
312#1735—531no—
313C2×C53—42no—
314C2×C53#245011yescharacteristic
315C2×C53#307611yescharacteristic
316C2×C53#308211yescharacteristic
317C2×C53#330111yescharacteristic
318C2×C53#330211yescharacteristic
319C2×C53#331511yescharacteristic
320C2×C53#331711yescharacteristic
321C23×C8—531no—
322C2×#125—531no—
323C2×C22⋊Q8C2×D5311yescharacteristic
324C2×C4×Q8C2×D5311yescharacteristic
325C2×#117—531no—
326C2×#117C2×D5311yescharacteristic
327C2×#107—531no—
328C2×#106—531no—
329C2×C4⋊C8—531no—
330C2×C4⋊C8—531no—
331C2×#103—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×C22⋊C8—531no—
336#399—1061no—
337#396—1061no—
338#391C2×D5311yescharacteristic
339#389—531no—
340#389—531no—
341#389—1061no—
342#389C2×D5311yescharacteristic
343#388—21no—
344#388—21no—
345#387—21no—
346#387—1061no—
347#384—531no—
348#380—531no—
349C4×C4⋊C4—21no—
350C4×C4⋊C4C2×D5311yescharacteristic
351C4×C22⋊C4C2×D5311yescharacteristic
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#336—531no—
363#336—531no—
364C53#2641711yescharacteristic, complemented, Sylow
365C22×Q8C22×D5311yescharacteristic
366C23×C4C22×D5311yescharacteristic
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—1062no—
372C22×C8—2121no—
373C22×C8—2121no—
374#125—1062no—
375#125—2121no—
376C22⋊Q8—22no—
377C22⋊Q8Dic10614yes—
378C4×Q8—22no—
379C4×Q8—41no—
380#117—22no—
381#117—41no—
382#117—1062no—
383#117—2121no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
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—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4C22⋊D5311yescharacteristic
405C2×C22⋊C4C22⋊D5311yescharacteristic
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—1061no—
410C2×C42C22×D5311yescharacteristic
411C2×C42C22×D5311yescharacteristic
412#107—2121no—
413#106—1062no—
414C4⋊C8—2121no—
415C4⋊C8—2121no—
416#103—1062no—
417#103—2121no—
418#103—2121no—
419#103—2121no—
420C22⋊C8—1062no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95C22⋊D5311yescharacteristic
429#95C22⋊D5311yescharacteristic
430C24C2×C22⋊D5311yescharacteristic
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C2×Dic10612yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4C2×Dic10612yes—
450C22×C4#1171994512yes—
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—2124no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—532no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C4⋊C4C2×Dic10612yes—
492C4⋊C4#1171994712yes—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—4241no—
498C22⋊C4—2122no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—2122no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1313319612yes—
514C23C22×C22⋊D5311yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
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—1062no—
549C2×C4—2121no—
550C2×C4—2121no—
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#1313319612yes—
561C2×C4#1313323712yes—
562C2×C4C22×Dic10611yescharacteristic
563C2×C4#41×D5311yescharacteristic
564C2×C4C2×#1171994711yescharacteristic
565C2×C4C2×#1171994511yescharacteristic
566C2×C4#1313326211yescharacteristic
567C2×C4#1313326311yescharacteristic
568C8—2121no—
569C8—2121no—
570C8—2121no—
571C8—2121no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22C2×#1313319611yescharacteristic
580C22#1681126111yescharacteristic
581C22#1681126711yescharacteristic
582C22C2×#1313323711yescharacteristic
583C22#1681142111yescharacteristic
584C22#1681143111yescharacteristic
585C22#1681143811yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—2122no—
599C4#1681115311yescharacteristic
600C4#1681120611yescharacteristic
601C4#1681133011yescharacteristic
602C4#1681133211yescharacteristic
603C2—42no—
604C2#2027853311yescharacteristic
605C2#2028450311yescharacteristic
606C2#2028455711yescharacteristic
607C2#2028661711yescharacteristic
608C2#2028662611yescharacteristic
609C2#2028681111yescharacteristic
610C2#2028683511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2362C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2362D53
hypocenterC53#26417
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #20276278C2semidirect
3. #20276206C2semidirect
4. #20276110C2semidirect
5. #20275866C2semidirect
6. #20275672C2semidirect
7. #20275293C2semidirect
10. #20274983C2semidirect
13. #20274720C2semidirect
364. C53#26417semidirect

Maximal subgroups

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

subgroupindex
#202784282
#202762782
#202762062
#202761102
#202758662
#202756722
#202752932
#202751392
#202750282
#202749832
#202749142
#202748952
#202747202
#202744312
C53×#23622
#2641753