Finite Group Databasebeta

← #27446184

Subgroups — #27446184

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279881C211yesmaximal, characteristic
2#20276152C211yesmaximal, characteristic, complemented
3#20276152C211yesmaximal, characteristic, complemented
4#20276151C211yesmaximal, characteristic, complemented
5#20276151C211yesmaximal, characteristic, complemented
6#20275683C211yesmaximal, characteristic, complemented
7#20275672C211yesmaximal, characteristic, complemented
8#20275514C211yesmaximal, characteristic
9#20275476C211yesmaximal, characteristic, complemented
10#20274914C211yesmaximal, characteristic, complemented
11#20274914C211yesmaximal, characteristic, complemented
12#20274810C211yesmaximal, characteristic, complemented
13#20274810C211yesmaximal, characteristic, complemented
14#20274613C211yesmaximal, characteristic, complemented
15C53×#2354C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×#13133128C2211yescharacteristic
27#16810561—21no—
28#16810537—21no—
29#16810517—21no—
30#16810517—21no—
31#16810483C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34#16810483C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
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×D4×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#390—21no—
52C53×#388—21no—
53C53×#388—21no—
54C53×#384C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133209—22no—
64#13133209—22no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C4×Dic53—21no—
76#13133150—22no—
77#13133148—22no—
78#13133148—41no—
79#13133148—41no—
80C4⋊#88322—22no—
81C22⋊#88322—22no—
82C22⋊#88322—22no—
83C22⋊#88322—41no—
84C22⋊#88322—41no—
85#13133128—22no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×D4×C53D411yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96#117×C53—22no—
97#117×C53—22no—
98#117×C53—41no—
99#117×C53—41no—
100C2×C4⋊C4×C53—21no—
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×C53—21no—
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C2×D4×C53—22no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C22×C4×C53—21no—
126C22×C4×C53—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—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—81no—
160C4⋊C4×C53#4112yes—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—81no—
166C22⋊C4×C53—81no—
167C42×C53—42no—
168C42×C53—42no—
169C42×C53—42no—
170C42×C53#4112yes—
171C42×C53#4112yes—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184#11719919—42no—
185#11719919—42no—
186C4×Dic53—81no—
187C2×#88322—21no—
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#4112yes—
199C23×C53—21no—
200C23×C53—21no—
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×C53C22×D411yescharacteristic
214D4×C53—44no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
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#12312yes—
246C2×C4×C53#12312yes—
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53H(5,𝔽2)11yescharacteristic
252C2×C4×C532−1+411yescharacteristic
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—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265#88322—22no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#26309—531nomaximal, Sylow
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53C2×#12311yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54711yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57511yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57711yescharacteristic
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—42no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#52911yescharacteristic
300C4×C53#54211yescharacteristic
301C4×C53#55611yescharacteristic
302C4×C53#55911yescharacteristic
303Dic53—81no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307#2562—531no—
308#2354D5311yescharacteristic
309#2077—531no—
310#2077—531no—
311#2077—531no—
312#2077—531no—
313#1992—531no—
314#1989—531no—
315#1955—531no—
316#1948—531no—
317#1854—531no—
318#1854—531no—
319#1825—531no—
320#1825—531no—
321#1776—531no—
322C2×C53—81no—
323C2×C53—81no—
324C2×C53#244711yescharacteristic
325C2×C53#309311yescharacteristic
326C2×C53#309911yescharacteristic
327C2×C53#330711yescharacteristic
328C2×C53#331711yescharacteristic
329C2×C53#334811yescharacteristic
330C2×C53#335011yescharacteristic
331C2×#125—531no—
332C2×C4×D4C2×D5311yescharacteristic
333C2×#117C2×D5311yescharacteristic
334C2×#117C2×D5311yescharacteristic
335C2×#107—531no—
336C2×#106—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×C4⋊C8—531no—
340C2×C22⋊C8—531no—
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—531no—
344C2×#97—531no—
345#400—1061no—
346#394—1061no—
347#390—21no—
348#388—21no—
349#388—21no—
350#388—1061no—
351#388—1061no—
352#384C2×D5311yescharacteristic
353#380—531no—
354#380—531no—
355#380—531no—
356#380—531no—
357#380C2×D5311yescharacteristic
358#380C2×D5311yescharacteristic
359C4×C4⋊C4—21no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4C2×D5311yescharacteristic
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374C53#2630911yescharacteristic, complemented, Sylow
375C22×D4C22⋊D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C22×C8—531no—
379C22×C8—531no—
380C22×C8—531no—
381C22×C8—531no—
382#125—1062no—
383#125—1062no—
384C4×D4—22no—
385C4×D4—22no—
386#117—22no—
387#117—22no—
388#117—41no—
389#117—41no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4C22⋊D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4C22⋊D5311yescharacteristic
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418#107—1062no—
419#106—1062no—
420#106—2121no—
421#106—2121no—
422C4⋊C8—1062no—
423C22⋊C8—1062no—
424C22⋊C8—1062no—
425C22⋊C8—2121no—
426C22⋊C8—2121no—
427#97—1062no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24C2×C22⋊D5311yescharacteristic
437C24C2×C22⋊D5311yescharacteristic
438C2×D4—22no—
439C2×D4—41no—
440C2×D4—41no—
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—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466C2×C8—532no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4#1171994712yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—4241no—
506C22⋊C4—4241no—
507C22⋊C4—2122no—
508C22⋊C4—2122no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—4241no—
513C42#1171994512yes—
514C42#1171994512yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23C22×C22⋊D5311yescharacteristic
530D4—44no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
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—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—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4#1313320012yes—
574C2×C4#1313320012yes—
575C2×C4#41×D5311yescharacteristic
576C2×C4C2×#1171994711yescharacteristic
577C2×C4C2×#1171994511yescharacteristic
578C2×C4C2×#1171994511yescharacteristic
579C2×C4#1313326211yescharacteristic
580C2×C4#1313326411yescharacteristic
581C8—1062no—
582C8—2121no—
583C8—2121no—
584C8—2121no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22C2×#1313320011yescharacteristic
600C22#1681126711yescharacteristic
601C22#1681127211yescharacteristic
602C22#1681142111yescharacteristic
603C22#1681142111yescharacteristic
604C22#1681143111yescharacteristic
605C22#1681143411yescharacteristic
606C4—22no—
607C4—22no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—4241no—
615C4—4241no—
616C4—4241no—
617C4—4241no—
618C4#1681115511yescharacteristic
619C4#1681123811yescharacteristic
620C4#1681134311yescharacteristic
621C4#1681135611yescharacteristic
622C2—81no—
623C2—81no—
624C2#2027849611yescharacteristic
625C2#2028464611yescharacteristic
626C2#2028470411yescharacteristic
627C2#2028669111yescharacteristic
628C2#2028684111yescharacteristic
629C2#2028723011yescharacteristic
630C2#2028726011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2354C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2354D53
hypocenterC53#26309
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #20276152C2semidirect
3. #20276152C2semidirect
4. #20276151C2semidirect
5. #20276151C2semidirect
6. #20275683C2semidirect
7. #20275672C2semidirect
9. #20275476C2semidirect
10. #20274914C2semidirect
11. #20274914C2semidirect
12. #20274810C2semidirect
13. #20274810C2semidirect
14. #20274613C2semidirect
374. C53#26309semidirect

Maximal subgroups

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

subgroupindex
#202798812
#202761522
#202761512
#202756832
#202756722
#202755142
#202754762
#202749142
#202748102
#202746132
C53×#23542
#2630953