Finite Group Databasebeta

← #27434094

Subgroups — #27434094

Order 13568.

824 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810993C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic
3#20277771C211yesmaximal, characteristic, complemented
4#20276134C211yesmaximal, characteristic, complemented
5#20276124C211yesmaximal, characteristic, complemented
6#20276032C211yesmaximal, characteristic
7#20276032C211yesmaximal, characteristic
8#20275673C211yesmaximal, characteristic, complemented
9#20275439C211yesmaximal, characteristic, complemented
10#20275073C211yesmaximal, characteristic
11#20275073C211yesmaximal, characteristic
12#20274727C211yesmaximal, characteristic, complemented
13#20274717C211yesmaximal, characteristic, complemented
14#20274540C211yesmaximal, characteristic
15C53×#2566C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×#13133217C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21#16810993—24no—
22#16810828—24no—
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35#16810529C2211yescharacteristic
36#16810529C2211yescharacteristic
37#16810519—21no—
38#16810506—21no—
39#16810506—21no—
40C4⋊C4×Dic53—21no—
41C4×C4⋊Dic53C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#396—21no—
53C53×#396—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#385—21no—
57C53×#385—21no—
58C53×#384C2211yescharacteristic
59C53×#382C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C22×Dic106C2311yescharacteristic
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923—22no—
66C2×#11719923—22no—
67C2×#11719923—41no—
68C2×#11719923—41no—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133217—22no—
74#13133217D414yes—
75C4×Dic106—22no—
76C4×Dic106—22no—
77C4×Dic106—41no—
78C4×Dic106—41no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C4⋊#88322—22no—
89C4⋊#88322—41no—
90C4⋊#88322—41no—
91#13133140—22no—
92#13133140—22no—
93#13133140—22no—
94#13133140—22no—
95#13133140—22no—
96#13133140—22no—
97#13133138—22no—
98#13133138—22no—
99#13133128—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×D4×C53C2311yescharacteristic
106C23×C4×C53—21no—
107#125×C53—22no—
108#125×C53—22no—
109C4×D4×C53—22no—
110C4×D4×C53—22no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53—21no—
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—42no—
143C22×C4×C53—81no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
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—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106C2×D412yes—
167C2×Dic106C2×D412yes—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—41no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—81no—
179C4⋊C4×C53—81no—
180C22⋊C4×C53—42no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—81no—
184C22⋊C4×C53—81no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53C2×D412yes—
190C42×C53#4112yes—
191#11719923—44no—
192#11719923—44no—
193C4⋊Dic53—24no—
194C4⋊Dic53—24no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53#4112yes—
199#11719919—42no—
200#11719919—42no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C23×C53—22no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53C22×D411yescharacteristic
224D4×C53—22no—
225D4×C53—22no—
226D4×C53—41no—
227D4×C53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
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—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
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—84no—
253C2×C4×C53#12112yes—
254C2×C4×C53#12112yes—
255C2×C4×C53#13712yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274Dic106—22no—
275Dic106—22no—
276Dic106—22no—
277Dic106—22no—
278Dic106—41no—
279Dic106—41no—
280Dic106—41no—
281Dic106—41no—
282#88322—41no—
283#88322—41no—
284#88322—41no—
285#88322—41no—
286#25331—531nomaximal, Sylow
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53C2×#12111yescharacteristic
295C22×C53#54611yescharacteristic
296C22×C53#54811yescharacteristic
297C22×C53C2×#13711yescharacteristic
298C22×C53#57511yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57711yescharacteristic
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—41no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53#52911yescharacteristic
311C4×C53#54211yescharacteristic
312C4×C53#54611yescharacteristic
313C4×C53#54711yescharacteristic
314Dic53—42no—
315Dic53—42no—
316Dic53—42no—
317Dic53—81no—
318C2×#491—531no—
319C2×#461—531no—
320#2566D5311yescharacteristic
321#2365—531no—
322#2074—531no—
323#2072—531no—
324#2055—531no—
325#2055—531no—
326#1989—531no—
327#1940—531no—
328#1880—531no—
329#1880—531no—
330#1805—531no—
331#1803—531no—
332#1762—531no—
333C2×C53—82no—
334C2×C53#242711yescharacteristic
335C2×C53#303711yescharacteristic
336C2×C53#304311yescharacteristic
337C2×C53#330711yescharacteristic
338C2×C53#332011yescharacteristic
339C2×C53#337211yescharacteristic
340C2×C53#337611yescharacteristic
341C22×SD16—531no—
342C22×SD16—531no—
343C2×C4⋊Q8—531no—
344C2×#125C2×D5311yescharacteristic
345C2×C4×Q8—531no—
346C2×C4×Q8—531no—
347C2×C4×D4C2×D5311yescharacteristic
348#491—1064no—
349#461—1064no—
350C2×C4⋊C8—531no—
351C2×C4⋊C8—531no—
352C2×C4⋊C8—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#97—531no—
362#396—21no—
363#396—21no—
364#391—531no—
365#391—531no—
366#390C2×D5311yescharacteristic
367#390C2×D5311yescharacteristic
368#389—1061no—
369#387—1061no—
370#387—1061no—
371#385—21no—
372#385—21no—
373#384C2×D5311yescharacteristic
374#382C2×D5311yescharacteristic
375C4×C4⋊C4—531no—
376C4×C4⋊C4—1061no—
377C4×C4⋊C4C2×D5311yescharacteristic
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#336—531no—
385#336—531no—
386C53#2533111yescharacteristic, complemented, Sylow
387C22×Q8—531no—
388C22×Q8—531no—
389C22×D4C22×D5311yescharacteristic
390C23×C4—21no—
391C2×SD16—1062no—
392C2×SD16—1062no—
393C2×SD16—1062no—
394C2×SD16—1062no—
395C2×SD16—2121no—
396C2×SD16—2121no—
397C22×C8—531no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C4⋊Q8—534no—
402C4⋊Q8—1062no—
403#125—22no—
404#125—22no—
405C4×Q8—1062no—
406C4×Q8—1062no—
407C4×Q8—2121no—
408C4×Q8—2121no—
409C4×D4—22no—
410C4×D4—22no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4C22⋊D5311yescharacteristic
422C2×C4⋊C4C22⋊D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42C22×D5311yescharacteristic
435C2×C42C22×D5311yescharacteristic
436C2×C42C22×D5311yescharacteristic
437C4⋊C8—1062no—
438C4⋊C8—2121no—
439C4⋊C8—2121no—
440#103—1062no—
441#103—1062no—
442#103—1062no—
443#103—1062no—
444#103—1062no—
445#103—1062no—
446#102—1062no—
447#102—1062no—
448#97—1062no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95C22⋊D5311yescharacteristic
457#95C22⋊D5311yescharacteristic
458C24—21no—
459C2×Q8—532no—
460C2×Q8—532no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4#1171994912yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
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×C4C2×C22⋊D5311yescharacteristic
493C22×C4C23×D5311yescharacteristic
494SD16—2124no—
495SD16—2124no—
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—
506C2×C8—1061no—
507C2×C8—1061no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—532no—
521C4⋊C4—1064no—
522C4⋊C4—1064no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—2122no—
538C42—2122no—
539C42—2122no—
540C42D4×D5312yes—
541C42#1171994712yes—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C22×C22⋊D5311yescharacteristic
550Q8—1062no—
551Q8—1062no—
552Q8—1062no—
553Q8—1062no—
554Q8—2121no—
555Q8—2121no—
556Q8—2121no—
557Q8—2121no—
558D4—22no—
559D4—22no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
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—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4#1313318812yes—
600C2×C4#1313318812yes—
601C2×C4#1313324712yes—
602C2×C4C2×D4×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994911yescharacteristic
606C2×C4#1313326311yescharacteristic
607C2×C4#1313326511yescharacteristic
608C8—2121no—
609C8—2121no—
610C8—2121no—
611C8—2121no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22C2×#1313318811yescharacteristic
620C22#1681126111yescharacteristic
621C22#1681128411yescharacteristic
622C22C2×#1313324711yescharacteristic
623C22#1681142111yescharacteristic
624C22#1681143411yescharacteristic
625C22#1681143411yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—4241no—
636C4—2122no—
637C4—2122no—
638C4—2122no—
639C4#1681115911yescharacteristic
640C4#1681124611yescharacteristic
641C4#1681126411yescharacteristic
642C4#1681127511yescharacteristic
643C2—82no—
644C2#2027828811yescharacteristic
645C2#2028418311yescharacteristic
646C2#2028423211yescharacteristic
647C2#2028668511yescharacteristic
648C2#2028688011yescharacteristic
649C2#2028749611yescharacteristic
650C2#2028754211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2566C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2566D53
hypocenterC53#25331
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#16810993C2semidirect
3. #20277771C2semidirect
4. #20276134C2semidirect
5. #20276124C2semidirect
8. #20275673C2semidirect
9. #20275439C2semidirect
12. #20274727C2semidirect
13. #20274717C2semidirect
386. C53#25331semidirect

Maximal subgroups

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

subgroupindex
C2×#168109932
C2×#168108282
#202777712
#202761342
#202761242
#202760322
#202756732
#202754392
#202750732
#202747272
#202747172
#202745402
C53×#25662
#2533153