Finite Group Databasebeta

← #27465740

Subgroups — #27465740

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810944C211yesmaximal, characteristic
2#121×Dic53C211yesmaximal, characteristic
3#20277118C211yesmaximal, characteristic, complemented
4#20276200C211yesmaximal, characteristic, complemented
5#20276199C211yesmaximal, characteristic, complemented
6#20276131C211yesmaximal, characteristic, complemented
7#20276131C211yesmaximal, characteristic, complemented
8#20275475C211yesmaximal, characteristic, complemented
9#20275470C211yesmaximal, characteristic, complemented
10#20274811C211yesmaximal, characteristic, complemented
11#20274810C211yesmaximal, characteristic, complemented
12#20274713C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic, complemented
14#20274693C211yesmaximal, characteristic, complemented
15C2×C53×#482C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20#16810944—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810516—21no—
31#16810503—21no—
32#16810492C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810488C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810481—21no—
38#16810481—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4×#11719919—21no—
41C4×#11719919—21no—
42C22⋊C4×Dic53—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#121×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#482—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×C22⋊C8×C53C2211yescharacteristic
61C23×Dic53—21no—
62C23×Dic53D411yescharacteristic
63C23×Dic53D411yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53—44no—
69#13133176—22no—
70#13133176—22no—
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×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53—22no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133150—22no—
94#13133148—22no—
95#13133146—22no—
96#13133146—22no—
97#13133138—22no—
98#13133138—22no—
99C22⋊#88322—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C22×D4×C53—21no—
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#121×C53—22no—
113#121×C53D414yes—
114#117×C53—22no—
115#117×C53—22no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C42×C53—21no—
122#107×C53—22no—
123#106×C53—22no—
124#102×C53—22no—
125#102×C53—22no—
126C22⋊C8×C53—22no—
127C24×C53—21no—
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—24no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53C2×D412yes—
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×C53—21no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D412yes—
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—41no—
158C22×Dic53—41no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53C2×D412yes—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—42no—
180C42×C53—42no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—22no—
202C4×Dic53—22no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C4×Dic53#4112yes—
206C4×Dic53#4112yes—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—22no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53#12112yes—
227C23×C53C22×D411yescharacteristic
228D4×C53—22no—
229D4×C53—22no—
230D4×C53—48no—
231D4×C53—41no—
232D4×C53—41no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
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#12112yes—
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—44no—
264C2×Dic53—44no—
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—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—84no—
278C2×Dic53#12312yes—
279C2×Dic53#12312yes—
280C2×Dic53#13512yes—
281C2×Dic53AGL(1,ℤ/8)12yes—
282C8×C53—41no—
283C8×C53—41no—
284#88322—41no—
285#88322—41no—
286#27646—531nomaximal, Sylow
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53C2×#12111yescharacteristic
302C22×C53C2×#12311yescharacteristic
303C22×C53#53611yescharacteristic
304C22×C53C2×#13511yescharacteristic
305C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C53#57511yescharacteristic
307C22×C53#57511yescharacteristic
308C4×C53—22no—
309C4×C53—42no—
310C4×C53—81no—
311C4×C53—81no—
312C4×C53#54711yescharacteristic
313C4×C53#54711yescharacteristic
314C4×C53#55211yescharacteristic
315C4×C53#55211yescharacteristic
316Dic53—22no—
317Dic53—41no—
318Dic53—41no—
319Dic53—81no—
320Dic53—81no—
321Dic53—81no—
322Dic53—81no—
323Dic53#48214yescomplemented
324C2×#482—531no—
325C2×#482D5311yescharacteristic
326C4×#121—531no—
327#2285—531no—
328#2085—531no—
329#2085—531no—
330#2073—531no—
331#2073—531no—
332#1948—531no—
333#1947—531no—
334#1825—531no—
335#1825—531no—
336#1802—531no—
337#1802—531no—
338#1797—531no—
339C2×C53—42no—
340C2×C53—82no—
341C2×C53#248611yescharacteristic
342C2×C53#303511yescharacteristic
343C2×C53#303611yescharacteristic
344C2×C53C2×#48211yescharacteristic
345C2×C53#309311yescharacteristic
346C2×C53#323311yescharacteristic
347C2×C53#323511yescharacteristic
348C2×#121C2×D5311yescharacteristic
349C2×C4×D4—531no—
350C2×C4×D4—1061no—
351C2×#117—531no—
352C2×#117C2×D5311yescharacteristic
353C22×C42—531no—
354#482—24no—
355#482—1064no—
356C2×#107—531no—
357C2×#107C2×D5311yescharacteristic
358C2×#106—531no—
359C2×#106C2×D5311yescharacteristic
360C2×C4⋊C8—531no—
361C2×C4⋊C8—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102C2×D5311yescharacteristic
365C2×#102C2×D5311yescharacteristic
366C2×C22⋊C8—531no—
367C2×C22⋊C8C2×D5311yescharacteristic
368#390—531no—
369#389—531no—
370#388—1061no—
371#386—1061no—
372#384—531no—
373#384—531no—
374#382—531no—
375#382—531no—
376#380—531no—
377#380—1061no—
378#380—1061no—
379C4×C4⋊C4—531no—
380C4×C4⋊C4—1061no—
381C4×C4⋊C4—1061no—
382C4×C22⋊C4—1061no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#336—531no—
392#336—531no—
393C53#2764611yescharacteristic, complemented, Sylow
394C22×D4—21no—
395C22×D4C22×D5311yescharacteristic
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—1061no—
399C23×C4C22×D5311yescharacteristic
400C22×C8—531no—
401C22×C8—531no—
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
404#121—22no—
405#121C22⋊D5314yes—
406C4×D4—1062no—
407C4×D4—1062no—
408C4×D4—2124no—
409#117—22no—
410#117—22no—
411#117—1062no—
412#117—1062no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
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⋊C4—1061no—
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C4⋊C4C22×D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C42—21no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—1062no—
438C2×C42—1062no—
439C2×C42—1061no—
440C2×C42—1061no—
441#107—22no—
442#107—1062no—
443#106—22no—
444#106—1062no—
445C4⋊C8—1062no—
446C4⋊C8—1062no—
447#102—22no—
448#102—22no—
449#102—1062no—
450#102—1062no—
451C22⋊C8—22no—
452C22⋊C8—1062no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460C24—21no—
461C24D4×D5311yescharacteristic
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4C2×C22⋊D5312yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2121no—
494C22×C4—2121no—
495C22×C4D4×D5311yescharacteristic
496C22×C4#1171994712yes—
497C22×C4C2×C22⋊D5312yes—
498C22×C4C23×D5311yescharacteristic
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C4⋊C4—2122no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4C2×C22⋊D5312yes—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—2122no—
533C22⋊C4—2122no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C22⋊C4—2122no—
537C42—42no—
538C42—532no—
539C42—532no—
540C42—1062no—
541C42—1062no—
542C42—1061no—
543C42—1061no—
544C42—2122no—
545C42—2122no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#1313318912yes—
560C23C2×D4×D5311yescharacteristic
561D4—22no—
562D4—22no—
563D4—48no—
564D4—41no—
565D4—41no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—4244no—
594C2×C4—2124no—
595C2×C4—2124no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4#1313318912yes—
609C2×C4C22×C22⋊D5311yescharacteristic
610C2×C4#41×D5311yescharacteristic
611C2×C4#41×D5311yescharacteristic
612C2×C4C2×#1171994711yescharacteristic
613C2×C4#1313326411yescharacteristic
614C2×C4#1313326411yescharacteristic
615C8—41no—
616C8—41no—
617C8—2121no—
618C8—2121no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×#1313318911yescharacteristic
634C22#123×D5311yescharacteristic
635C22#1681119511yescharacteristic
636C22#135×D5311yescharacteristic
637C22AGL(1,ℤ/8)×D5311yescharacteristic
638C22#1681141911yescharacteristic
639C22#1681142011yescharacteristic
640C4—22no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—534no—
645C4—1062no—
646C4—4241no—
647C4—4241no—
648C4—4241no—
649C4—4241no—
650C4—2121no—
651C4—2121no—
652C4#1681127211yescharacteristic
653C4#1681127211yescharacteristic
654C4#1681131011yescharacteristic
655C4#1681131011yescharacteristic
656C2—42no—
657C2—82no—
658C2#2027902011yescharacteristic
659C2#2028416611yescharacteristic
660C2#2028417611yescharacteristic
661C2#482×D5311yescharacteristic
662C2#2028464311yescharacteristic
663C2#2028589111yescharacteristic
664C2#2028591011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#482C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#482D53
hypocenterC53#27646
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
3. #20277118C2semidirect
4. #20276200C2semidirect
5. #20276199C2semidirect
6. #20276131C2semidirect
7. #20276131C2semidirect
8. #20275475C2semidirect
9. #20275470C2semidirect
10. #20274811C2semidirect
11. #20274810C2semidirect
12. #20274713C2semidirect
13. #20274713C2semidirect
14. #20274693C2semidirect
323. Dic53#482semidirect
393. C53#27646semidirect

Maximal subgroups

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

subgroupindex
C2×#168109442
#121×Dic532
#202771182
#202762002
#202761992
#202761312
#202754752
#202754702
#202748112
#202748102
#202747132
#202746932
C2×C53×#4822
#2764653