Finite Group Databasebeta

← #27446246

Subgroups — #27446246

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277644C211yesmaximal, characteristic
2#20276151C211yesmaximal, characteristic, complemented
3#20276146C211yesmaximal, characteristic, complemented
4#20276146C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275689C211yesmaximal, characteristic, complemented
7#20275668C211yesmaximal, characteristic, complemented
8#20275514C211yesmaximal, characteristic
9#20275483C211yesmaximal, characteristic, complemented
10#20274912C211yesmaximal, characteristic, complemented
11#20274907C211yesmaximal, characteristic, complemented
12#20274807C211yesmaximal, characteristic, complemented
13#20274799C211yesmaximal, characteristic, complemented
14#20274613C211yesmaximal, characteristic, complemented
15C53×#2563C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133128C2211yescharacteristic
28#16810516—21no—
29#16810509—21no—
30#16810503—21no—
31#16810489C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C22⋊C4×Dic53—21no—
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48C2×C4⋊Q8×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C53×#393—21no—
51C53×#393—21no—
52C53×#390—21no—
53C53×#382—21no—
54C53×#380C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—22no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53—41no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78#13133150—22no—
79#13133150—41no—
80#13133148—22no—
81#13133148—41no—
82C4⋊#88322—22no—
83C22⋊#88322—22no—
84C22⋊#88322—22no—
85C22⋊#88322—41no—
86C22⋊#88322—41no—
87#13133128—22no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93C22×Q8×C53—21no—
94C22×D4×C53D411yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C23×C4×C53C2311yescharacteristic
97C4⋊Q8×C53—22no—
98C4⋊Q8×C53—22no—
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C22⋊C4×C53D411yescharacteristic
113C2×C42×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C2×Q8×C53—42no—
123C2×Q8×C53—81no—
124C2×Q8×C53—81no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—41no—
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×C53—21no—
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×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—42no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—81no—
169C22⋊C4×C53—81no—
170C22⋊C4×C53—81no—
171C22⋊C4×C53—81no—
172C42×C53—42no—
173C42×C53#4112yes—
174C42×C53#4112yes—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—81no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53#4112yes—
187C4×Dic53—42no—
188C4×Dic53—81no—
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#4112yes—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53#13712yes—
215C23×C53C22×D411yescharacteristic
216Q8×C53—84no—
217D4×C53—44no—
218C2×C4×C53—21no—
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×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53#12312yes—
249C2×C4×C53#12312yes—
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×Dic53—22no—
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—
267C2×Dic53—41no—
268C2×Dic53—84no—
269#88322—22no—
270#88322—41no—
271#88322—41no—
272#88322—41no—
273#26313—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53C2×#12311yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53#54711yescharacteristic
291C22×C53C2×#13711yescharacteristic
292C22×C53#57511yescharacteristic
293C22×C53#57611yescharacteristic
294C22×C53#57711yescharacteristic
295C4×C53—22no—
296C4×C53—22no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—81no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53#52911yescharacteristic
304C4×C53#54211yescharacteristic
305C4×C53#55611yescharacteristic
306C4×C53#55911yescharacteristic
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311#2563D5311yescharacteristic
312#2352—531no—
313#2077—531no—
314#2076—531no—
315#2076—531no—
316#2075—531no—
317#1993—531no—
318#1988—531no—
319#1955—531no—
320#1949—531no—
321#1854—531no—
322#1853—531no—
323#1825—531no—
324#1823—531no—
325#1776—531no—
326C2×C53—42no—
327C2×C53—81no—
328C2×C53#244711yescharacteristic
329C2×C53#309511yescharacteristic
330C2×C53#309611yescharacteristic
331C2×C53#331411yescharacteristic
332C2×C53#332111yescharacteristic
333C2×C53#334411yescharacteristic
334C2×C53#334711yescharacteristic
335C2×C4⋊Q8C2×D5311yescharacteristic
336C2×C4×D4C2×D5311yescharacteristic
337C2×#117—531no—
338C22×C4⋊C4—531no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×C4⋊C8—531no—
344C2×C22⋊C8—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347C2×C22⋊C8—531no—
348C2×#97—531no—
349#393—21no—
350#393—21no—
351#390—21no—
352#388—1061no—
353#387—1061no—
354#386—1061no—
355#382—21no—
356#382—531no—
357#380—531no—
358#380—531no—
359#380C2×D5311yescharacteristic
360#380C2×D5311yescharacteristic
361#380C2×D5311yescharacteristic
362#380C2×D5311yescharacteristic
363C4×C4⋊C4—531no—
364C4×C4⋊C4C2×D5311yescharacteristic
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#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378C53#2631311yescharacteristic, complemented, Sylow
379C22×Q8—21no—
380C22×D4C22⋊D5311yescharacteristic
381C23×C4—1061no—
382C23×C4C22×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C22×C8—531no—
385C22×C8—531no—
386C22×C8—531no—
387C22×C8—531no—
388C4⋊Q8—22no—
389C4⋊Q8—22no—
390C4×D4—22no—
391C4×D4—22no—
392#117—1062no—
393#117—2121no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—2121no—
402C2×C4⋊C4C22⋊D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4C22⋊D5311yescharacteristic
415C2×C22⋊C4C22⋊D5311yescharacteristic
416C2×C42—21no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42C22×D5311yescharacteristic
420#107—1062no—
421#107—2121no—
422#106—1062no—
423#106—2121no—
424C4⋊C8—1062no—
425C22⋊C8—1062no—
426C22⋊C8—1062no—
427C22⋊C8—2121no—
428C22⋊C8—2121no—
429#97—1062no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24C2×C22⋊D5311yescharacteristic
441C24C2×C22⋊D5311yescharacteristic
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—81no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—41no—
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—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—4241no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C23×D5311yescharacteristic
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—532no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—2122no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—4241no—
514C22⋊C4—2122no—
515C22⋊C4—2122no—
516C42—42no—
517C42—4241no—
518C42—2122no—
519C42#1171994712yes—
520C42#1171994712yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23#1313324812yes—
535C23C22×C22⋊D5311yescharacteristic
536Q8—84no—
537D4—44no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
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—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—1062no—
569C2×C4—4244no—
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#1313319812yes—
582C2×C4#1313319812yes—
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994711yescharacteristic
587C2×C4#1313326311yescharacteristic
588C2×C4#1313326411yescharacteristic
589C8—1062no—
590C8—2121no—
591C8—2121no—
592C8—2121no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1313319811yescharacteristic
608C22#1681126111yescharacteristic
609C22#1681127211yescharacteristic
610C22C2×#1313324811yescharacteristic
611C22#1681142111yescharacteristic
612C22#1681142611yescharacteristic
613C22#1681143411yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—4241no—
623C4—4241no—
624C4—4241no—
625C4—2122no—
626C4#1681115211yescharacteristic
627C4#1681124211yescharacteristic
628C4#1681134211yescharacteristic
629C4#1681135511yescharacteristic
630C2—42no—
631C2—81no—
632C2#2027849811yescharacteristic
633C2#2028465911yescharacteristic
634C2#2028466711yescharacteristic
635C2#2028679011yescharacteristic
636C2#2028689511yescharacteristic
637C2#2028716411yescharacteristic
638C2#2028721511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2563C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2563D53
hypocenterC53#26313
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. #20276151C2semidirect
3. #20276146C2semidirect
4. #20276146C2semidirect
5. #20276141C2semidirect
6. #20275689C2semidirect
7. #20275668C2semidirect
9. #20275483C2semidirect
10. #20274912C2semidirect
11. #20274907C2semidirect
12. #20274807C2semidirect
13. #20274799C2semidirect
14. #20274613C2semidirect
378. C53#26313semidirect

Maximal subgroups

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

subgroupindex
#202776442
#202761512
#202761462
#202761412
#202756892
#202756682
#202755142
#202754832
#202749122
#202749072
#202748072
#202747992
#202746132
C53×#25632
#2631353