Finite Group Databasebeta

← #27403519

Subgroups — #27403519

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810735C211yesmaximal, characteristic
2#20277663C211yesmaximal, characteristic, complemented
3#20277620C211yesmaximal, characteristic, complemented
4#20275409C211yesmaximal, characteristic, complemented
5#20275401C211yesmaximal, characteristic, complemented
6#20275301C211yesmaximal, characteristic, complemented
7#20275292C211yesmaximal, characteristic, complemented
8#20275065C211yesmaximal, characteristic
9#20274847C211yesmaximal, characteristic, complemented
10#20274723C211yesmaximal, characteristic, complemented
11#20274713C211yesmaximal, characteristic, complemented
12#20274586C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#1879C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133173—21no—
20#16810735C418yescomplemented
21C2×#13133149C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30#16810526C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810517—21no—
34#16810511—21no—
35#16810496—21no—
36#16810495—21no—
37#16810492C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810481—21no—
40#16810481—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719919C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C2×#13133124—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×D8×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#390C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65D4×Dic53—22no—
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53—41no—
69D4×Dic53C2×C414yes—
70D4×Dic53C2×C414yescomplemented
71#13133173—44no—
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×#11719919D411yescharacteristic
81C2×#11719919D411yescharacteristic
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133149C2×C414yescomplemented
92#13133146—22no—
93#13133138—22no—
94#13133138—41no—
95#13133138C2×C414yes—
96#13133138C2×C414yescomplemented
97#13133127—41no—
98#13133127C2×C414yescomplemented
99C4×#88322—41no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×D4×C53C2311yescharacteristic
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C2×D8×C53—22no—
112C2×D8×C53—22no—
113C2×D8×C53C2×C414yes—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C4×D4×C53—22no—
117C4×D4×C53—41no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C42×C53C2311yescharacteristic
124#102×C53—22no—
125#102×C53—41no—
126C24×C53—21no—
127C24×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53C22×C412yes—
137C2×D4×C53C22×C412yes—
138C2×D4×C53#4112yes—
139C2×D4×C53#4112yes—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—24no—
151C22×Dic53—24no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C22×Dic53—81no—
160C22×Dic53—81no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165D8×C53—28no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53C22×C412yes—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—81no—
179C42×C53—21no—
180C42×C53—21no—
181C22⋊Dic53—24no—
182C22⋊Dic53—24no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C22⋊Dic53—81no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53C22×C412yes—
193C4⋊Dic53C22×C412yes—
194#11719919—22no—
195#11719919—41no—
196#11719919—42no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53C22×C412yes—
205C4×Dic53#4112yes—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322C22×C412yes—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—22no—
215C23×C53—22no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53C22×D411yescharacteristic
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—22no—
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—41no—
233D4×C53—41no—
234D4×C53#11714yescomplemented
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
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—84no—
247C2×C4×C53AGL(1,ℤ/8)12yes—
248C2×C4×C53C23×C411yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53H(5,𝔽2)11yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—48no—
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—84no—
279C2×Dic53C4×D412yes—
280C2×Dic53C4×D412yes—
281C8×C53—22no—
282C8×C53—41no—
283#88322—22no—
284#88322—41no—
285#22780—531nomaximal, Sylow
286C22×C53—24no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53C2×C4×D411yescharacteristic
301C22×C53#54611yescharacteristic
302C22×C53#54811yescharacteristic
303C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C53#57511yescharacteristic
305C22×C53#57611yescharacteristic
306C22×C53#57711yescharacteristic
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53C2×#11711yescharacteristic
312C4×C53#52011yescharacteristic
313C4×C53#55211yescharacteristic
314C4×C53#55311yescharacteristic
315Dic53—22no—
316Dic53—22no—
317Dic53—42no—
318Dic53—42no—
319Dic53—41no—
320Dic53—41no—
321Dic53—81no—
322Dic53—81no—
323C2×#442—531no—
324#2354—531no—
325#2349—531no—
326#1936—531no—
327#1934—531no—
328#1920—531no—
329#1919—531no—
330#1879—531no—
331#1879D5311yescharacteristic
332#1836—531no—
333#1804—531no—
334#1802—531no—
335#1771—531no—
336#1762—531no—
337C4×#102—531no—
338C2×C53—44no—
339C2×C53—82no—
340C2×C53#235211yescharacteristic
341C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
342C2×C53#294611yescharacteristic
343C2×C53#330611yescharacteristic
344C2×C53#331911yescharacteristic
345C2×C53#336611yescharacteristic
346C2×C53#337611yescharacteristic
347C22×D8C2×D5311yescharacteristic
348C2×C4×D4—531no—
349C2×C4×D4—531no—
350C2×C4×D4—531no—
351C2×C4×D4C2×D5311yescharacteristic
352C2×#117—1061no—
353#442—538no—
354C2×#106—531no—
355C2×C4⋊C8—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×#102C2×D5311yescharacteristic
361C2×#102C2×D5311yescharacteristic
362C2×#97—531no—
363C2×#97—531no—
364C2×C4×C8—531no—
365#390—531no—
366#390C2×D5311yescharacteristic
367#389—531no—
368#389—531no—
369#388—1061no—
370#388—1061no—
371#385—1061no—
372#385—1061no—
373#384—531no—
374#382—531no—
375#380—1061no—
376#380—1061no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4—531no—
379C4×C4⋊C4—531no—
380C2×#95—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340C2×D5311yescharacteristic
389#336—531no—
390#336C2×D5311yescharacteristic
391C53#2278011yescharacteristic, complemented, Sylow
392C22×D4C22×D5311yescharacteristic
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4—1061no—
396C23×C4—1061no—
397C23×C4—1061no—
398C2×D8—22no—
399C2×D8—22no—
400C2×D8C4×D5314yes—
401C22×C8—531no—
402C22×C8—531no—
403C22×C8C22×D5311yescharacteristic
404C22×C8C22×D5311yescharacteristic
405C4×D4—22no—
406C4×D4—41no—
407C4×D4—534no—
408C4×D4—534no—
409C4×D4—1062no—
410C4×D4—1062no—
411C4×D4—1062no—
412C4×D4—2121no—
413#117—2124no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42C22×D5311yescharacteristic
439#106—534no—
440C4⋊C8—1062no—
441#102—22no—
442#102—41no—
443#102—534no—
444#102—534no—
445#102—1062no—
446#102—2121no—
447#97—534no—
448#97—2121no—
449C4×C8—2121no—
450#95—531no—
451#95—531no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458C24—21no—
459C24—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×C4×D5312yes—
469C2×D4C2×C4×D5312yes—
470C2×D4#1171994712yes—
471C2×D4#1171994712yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—81no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—1064no—
482C22×C4—1064no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—4241no—
488C22×C4—4241no—
489C22×C4—4241no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4D4×D5311yescharacteristic
495C22×C4D4×D5311yescharacteristic
496C22×C4C23×D5311yescharacteristic
497D8—28no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—532no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8C2×C4×D5312yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—532no—
515C4⋊C4—532no—
516C4⋊C4—1062no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—4241no—
524C4⋊C4—4241no—
525C4⋊C4—4241no—
526C4⋊C4—2121no—
527C4⋊C4—2122no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C22⋊C4—1064no—
532C22⋊C4—1064no—
533C22⋊C4—4241no—
534C22⋊C4—2122no—
535C22⋊C4—2122no—
536C42—21no—
537C42—21no—
538C42—532no—
539C42—532no—
540C42—1061no—
541C42—1061no—
542C42—2122no—
543C42—2122no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×D4×D5311yescharacteristic
559D4—22no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4#1313317314yescomplemented
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—532no—
580C2×C4—532no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—4244no—
594C2×C4—4244no—
595C2×C4—2128no—
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#1313324112yes—
607C2×C4C22×C4×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4#1313326211yescharacteristic
612C2×C4#1313326411yescharacteristic
613C8—22no—
614C8—41no—
615C8—1062no—
616C8—2121no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C4×D4×D5311yescharacteristic
632C22#1681126011yescharacteristic
633C22#1681127911yescharacteristic
634C22C2×#1313324111yescharacteristic
635C22#1681142011yescharacteristic
636C22#1681142711yescharacteristic
637C22#1681143311yescharacteristic
638C4—22no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—1062no—
643C4—1062no—
644C4—4241no—
645C4—4241no—
646C4—2122no—
647C4—2122no—
648C4—2121no—
649C4—2121no—
650C4C2×#1313317311yescharacteristic
651C4#1681110211yescharacteristic
652C4#1681131011yescharacteristic
653C4#1681131611yescharacteristic
654C2—44no—
655C2—82no—
656C2#2027764211yescharacteristic
657C2C4×#1313324111yescharacteristic
658C2#2028354411yescharacteristic
659C2#2028668011yescharacteristic
660C2#2028687511yescharacteristic
661C2#2028742611yescharacteristic
662C2#2028754611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1879C2
socleC23×C53C22×D4
prime radicalC22×D8×C53C22
cosocleC22×C4×C53C24
hypercenter#1879D53
hypocenterC53#22780
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #20277663C2semidirect
3. #20277620C2semidirect
4. #20275409C2semidirect
5. #20275401C2semidirect
6. #20275301C2semidirect
7. #20275292C2semidirect
9. #20274847C2semidirect
10. #20274723C2semidirect
11. #20274713C2semidirect
12. #20274586C2semidirect
13. #20274542C2semidirect
14. C4×#13133138C2semidirect
20. #16810735C4semidirect
70. D4×Dic53C2×C4semidirect
91. #13133149C2×C4semidirect
96. #13133138C2×C4semidirect
98. #13133127C2×C4semidirect
234. D4×C53#117semidirect
391. C53#22780semidirect
566. D4#13133173semidirect

Maximal subgroups

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

subgroupindex
C2×#168107352
#202776632
#202776202
#202754092
#202754012
#202753012
#202752922
#202750652
#202748472
#202747232
#202747132
#202745862
#202745422
C4×#131331382
C53×#18792
#2278053