Finite Group Databasebeta

← #27447714

Subgroups — #27447714

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277740C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276211C211yesmaximal, characteristic, complemented
4#20276107C211yesmaximal, characteristic, complemented
5#20275853C211yesmaximal, characteristic, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275297C211yesmaximal, characteristic, complemented
8#20275136C211yesmaximal, characteristic
9#20275027C211yesmaximal, characteristic
10#20274982C211yesmaximal, characteristic, complemented
11#20274912C211yesmaximal, characteristic
12#20274895C211yesmaximal, characteristic
13#20274727C211yesmaximal, characteristic, complemented
14#20274431C211yesmaximal, characteristic
15C53×#2439C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29#16810529C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810517—21no—
32#16810511—21no—
33#16810505—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35#16810474—21no—
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×C4⋊Q8×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C53×#396—21no—
51C53×#393—21no—
52C53×#391—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#382C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322—22no—
61C22×#88322—41no—
62C22×#88322—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133196—22no—
68#13133196Q814yescomplemented
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#13133150—41no—
85#13133148—22no—
86C4⋊#88322—41no—
87C4⋊#88322—41no—
88#13133140—22no—
89#13133140—41no—
90#13133140—41no—
91#13133140—41no—
92C22⋊#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53—21no—
98C23×C4×C53C2311yescharacteristic
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×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C22⋊C4×C53D411yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#95×C53—21no—
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—42no—
126C2×Q8×C53—42no—
127C2×Q8×C53—81no—
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—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106C2×Q812yes—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—81no—
164C4⋊C4×C53#4112yes—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—81no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—42no—
174C22⋊Dic53—24no—
175C22⋊Dic53—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C2×Q812yes—
181C4⋊Dic53#4112yes—
182#11719919—24no—
183#11719919—42no—
184#11719919—42no—
185C4×Dic53—42no—
186C4×Dic53—42no—
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—84no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
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—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53C22⋊Q812yes—
240C2×C4×C53#13512yes—
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C53H(5,𝔽2)11yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
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—
259Dic106—22no—
260Dic106—22no—
261Dic106—41no—
262Dic106—41no—
263#88322—41no—
264#88322—41no—
265#88322—41no—
266#88322—41no—
267#26413—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53C2×C22⋊Q811yescharacteristic
276C22×C53#54611yescharacteristic
277C22×C53#54611yescharacteristic
278C22×C53C2×#13511yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—42no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—41no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53#52911yescharacteristic
292C4×C53#53811yescharacteristic
293C4×C53#55411yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—42no—
296Dic53—42no—
297Dic53—42no—
298Dic53—81no—
299#2439D5311yescharacteristic
300#2362—531no—
301#2101—531no—
302#2087—531no—
303#2068—531no—
304#2024—531no—
305#1988—531no—
306#1920—531no—
307#1892—531no—
308#1873—531no—
309#1865—531no—
310#1854—531no—
311#1850—531no—
312#1805—531no—
313#1735—531no—
314C2×C53—42no—
315C2×C53#245011yescharacteristic
316C2×C53#307611yescharacteristic
317C2×C53#308211yescharacteristic
318C2×C53#329611yescharacteristic
319C2×C53#330211yescharacteristic
320C2×C53#331311yescharacteristic
321C2×C53#332111yescharacteristic
322C23×C8—531no—
323C2×C4⋊Q8C2×D5311yescharacteristic
324C2×C22⋊Q8—531no—
325C2×C4×Q8—531no—
326C2×#117—531no—
327C2×#117C2×D5311yescharacteristic
328C2×#107—531no—
329C2×#106—531no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336C2×C22⋊C8—531no—
337#396—21no—
338#393—21no—
339#391—21no—
340#391—531no—
341#389—531no—
342#389C2×D5311yescharacteristic
343#389C2×D5311yescharacteristic
344#388—1061no—
345#388—1061no—
346#387—21no—
347#387—1061no—
348#382C2×D5311yescharacteristic
349#380C2×D5311yescharacteristic
350C4×C4⋊C4—531no—
351C4×C4⋊C4—1061no—
352C4×C22⋊C4C2×D5311yescharacteristic
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#340—531no—
363#336—531no—
364#336—531no—
365C53#2641311yescharacteristic, complemented, Sylow
366C22×Q8—21no—
367C22×Q8—531no—
368C23×C4C22×D5311yescharacteristic
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—1062no—
374C22×C8—2121no—
375C22×C8—2121no—
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378C22⋊Q8—534no—
379C22⋊Q8—1062no—
380C4×Q8—1062no—
381C4×Q8—2121no—
382#117—22no—
383#117—41no—
384#117—1062no—
385#117—2121no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4C22⋊D5311yescharacteristic
404C2×C22⋊C4C22⋊D5311yescharacteristic
405C2×C42—21no—
406C2×C42—1061no—
407C2×C42—1061no—
408C2×C42—1061no—
409C2×C42C22×D5311yescharacteristic
410C2×C42C22×D5311yescharacteristic
411#107—2121no—
412#106—1062no—
413C4⋊C8—2121no—
414C4⋊C8—2121no—
415#103—1062no—
416#103—2121no—
417#103—2121no—
418#103—2121no—
419C22⋊C8—1062no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95C22⋊D5311yescharacteristic
430#95C22⋊D5311yescharacteristic
431C24C2×C22⋊D5311yescharacteristic
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—81no—
435C2×Q8—532no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×Q8—1061no—
440C2×Q8—1061no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4Q8×D5312yes—
454C22×C4#1171994912yes—
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C2×C22⋊D5311yescharacteristic
459C22×C4C2×C22⋊D5311yescharacteristic
460C22×C4C23×D5311yescharacteristic
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—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—2124no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—532no—
485C4⋊C4—532no—
486C4⋊C4—1064no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4#1171994712yes—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—81no—
498C22⋊C4—1064no—
499C22⋊C4—2122no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—2122no—
506C42—2122no—
507C42—2122no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23#1313319312yes—
515C23C22×C22⋊D5311yescharacteristic
516Q8—84no—
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
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—41no—
549C2×C4—41no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4#1313319312yes—
563C2×C4#1313323712yes—
564C2×C4C2×Q8×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4C2×#1171994711yescharacteristic
567C2×C4C2×#1171994911yescharacteristic
568C2×C4#1313326211yescharacteristic
569C2×C4#1313326311yescharacteristic
570C8—2121no—
571C8—2121no—
572C8—2121no—
573C8—2121no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1313319311yescharacteristic
582C22#1681126111yescharacteristic
583C22#1681126711yescharacteristic
584C22C2×#1313323711yescharacteristic
585C22#1681142111yescharacteristic
586C22#1681143411yescharacteristic
587C22#1681143811yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—4241no—
598C4—2122no—
599C4—2122no—
600C4—2122no—
601C4#1681115911yescharacteristic
602C4#1681121311yescharacteristic
603C4#1681132711yescharacteristic
604C4#1681133111yescharacteristic
605C2—42no—
606C2#2027853511yescharacteristic
607C2#2028449711yescharacteristic
608C2#2028455111yescharacteristic
609C2#2028655011yescharacteristic
610C2#2028662611yescharacteristic
611C2#2028678111yescharacteristic
612C2#2028689511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2439C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2439D53
hypocenterC53#26413
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276211C2semidirect
4. #20276107C2semidirect
5. #20275853C2semidirect
6. #20275668C2semidirect
7. #20275297C2semidirect
10. #20274982C2semidirect
13. #20274727C2semidirect
68. #13133196Q8semidirect
365. C53#26413semidirect

Maximal subgroups

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

subgroupindex
#202777402
#202762822
#202762112
#202761072
#202758532
#202756682
#202752972
#202751362
#202750272
#202749822
#202749122
#202748952
#202747272
#202744312
C53×#24392
#2641353