Finite Group Databasebeta

← #27403246

Subgroups — #27403246

Order 13568.

978 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810703C211yesmaximal, characteristic
2#20277663C211yesmaximal, characteristic, complemented
3#20277616C211yesmaximal, characteristic, complemented
4#20275414C211yesmaximal, characteristic, complemented
5#20275310C211yesmaximal, characteristic, complemented
6#20275304C211yesmaximal, characteristic, complemented
7#20275066C211yesmaximal, characteristic
8#20275060C211yesmaximal, characteristic
9#20274841C211yesmaximal, characteristic, complemented
10#20274714C211yesmaximal, characteristic, complemented
11#20274713C211yesmaximal, characteristic, complemented
12#20274583C211yesmaximal, characteristic, complemented
13#20274541C211yesmaximal, characteristic, complemented
14#20274539C211yesmaximal, characteristic, complemented
15C53×#1936C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×#13133173—21no—
22C2×C4⋊C4×D53—21no—
23#16810703C418yescomplemented
24C2×#13133150C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810517—21no—
35#16810511—21no—
36#16810497—21no—
37#16810494—21no—
38#16810492C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810482—21no—
41#16810481—21no—
42C4⋊C4×Dic53C2211yescharacteristic
43C4×#11719919C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C23×Dic53—21no—
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719922—22no—
66C2×#11719922—22no—
67C2×#11719922C2×C414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C4×D212—22no—
71C4×D212—41no—
72D4×Dic53—22no—
73D4×Dic53C2×C414yescomplemented
74#13133179—22no—
75#13133179—22no—
76#13133179—41no—
77#13133179C2×C414yes—
78#13133173—44no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919D411yescharacteristic
88C2×#11719919D411yescharacteristic
89C4⋊C4×D53—44no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53C2311yescharacteristic
99C2×C4×Dic53C2311yescharacteristic
100#13133150C2×C414yescomplemented
101#13133146—22no—
102#13133139—22no—
103#13133139C2×C414yes—
104#13133138—41no—
105#13133137—22no—
106#13133127—41no—
107C8×Dic53C2×C414yescomplemented
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124D411yescharacteristic
112#13133124D411yescharacteristic
113C22×D4×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C42×C53C2311yescharacteristic
123#102×C53—41no—
124#102×C53C2×C414yescomplemented
125#97×C53—41no—
126#95×C53—21no—
127C24×C53—21no—
128C23×D53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53C22×C412yes—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—24no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—42no—
146C22×Dic53—42no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212C22×C412yes—
155C2×D212#4112yes—
156C2×D212#4112yes—
157C2×C4×D53—24no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—81no—
165C2×C4×D53—81no—
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—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53C22×C412yes—
179C22⋊C4×C53—42no—
180C22⋊C4×C53—81no—
181C42×C53—21no—
182C42×C53—21no—
183C22⋊Dic53—24no—
184C22⋊Dic53—42no—
185C22⋊Dic53—81no—
186#11719922—28no—
187#11719921—24no—
188#11719921—42no—
189#11719921—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53C22×C412yes—
197#11719919—22no—
198#11719919—22no—
199#11719919—42no—
200#11719919—42no—
201#11719919—81no—
202#11719919—81no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53C22×C412yes—
208C4×Dic53#4112yes—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322C22×C412yes—
215C23×C53—22no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—41no—
233D4×C53—41no—
234C2×C4×C53—22no—
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—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—41no—
249C2×C4×C53C23×C411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C532−1+411yescharacteristic
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—22no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—48no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53C4×D412yes—
274C2×Dic53C4×D412yes—
275C2×Dic53AGL(1,ℤ/8)12yes—
276C2×Dic53AGL(1,ℤ/8)12yes—
277D212—22no—
278D212—22no—
279D212—22no—
280D212#11714yescomplemented
281C4×D53—48no—
282C4×D53—84no—
283C4×D53—84no—
284C8×C53—22no—
285C8×C53—41no—
286#88322—22no—
287#88322—41no—
288#22758—531nomaximal, Sylow
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53C2×C4×D411yescharacteristic
297C22×C53#54611yescharacteristic
298C22×C53#54811yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C53#57611yescharacteristic
302C22×C53#57811yescharacteristic
303C2×D53—24no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—42no—
310C4×C53—41no—
311C4×C53—42no—
312C4×C53—41no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53C2×#11711yescharacteristic
316C4×C53#52011yescharacteristic
317C4×C53#55211yescharacteristic
318C4×C53#55311yescharacteristic
319Dic53—22no—
320Dic53—41no—
321Dic53—42no—
322Dic53—41no—
323Dic53—81no—
324Dic53—81no—
325Dic53#44214yes—
326C2×C4×D8—531no—
327#2354—531no—
328D4×C4⋊C4—531no—
329#1936—531no—
330#1936D5311yescharacteristic, complemented
331#1921—531no—
332#1921—531no—
333#1879—531no—
334#1879—531no—
335#1834—531no—
336#1802—531no—
337#1802—531no—
338#1770—531no—
339#1762—531no—
340#1762—531no—
341C2×C53—82no—
342C2×C53#235211yescharacteristic
343C2×C53C2×#44211yescharacteristic
344C2×C53#295111yescharacteristic
345C2×C53#332511yescharacteristic
346C2×C53#333011yescharacteristic
347C2×C53#336611yescharacteristic
348C2×C53#336711yescharacteristic
349D53—44no—
350C22×D8—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—531no—
354C2×C4×D4—531no—
355C2×#117—1061no—
356C22×C4⋊C4—1061no—
357C4×D8—538no—
358C2×#107—531no—
359C2×C4⋊C8—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#102C2×D5311yescharacteristic
365C2×#102C2×D5311yescharacteristic
366C2×#97—531no—
367C2×#97C2×D5311yescharacteristic
368C2×C4×C8—531no—
369#390—531no—
370#390C2×D5311yescharacteristic
371#389—531no—
372#389C2×D5311yescharacteristic
373#388—1061no—
374#388—1061no—
375#385—1061no—
376#385—1061no—
377#384—531no—
378#384—531no—
379#380—1061no—
380#380—1061no—
381C4×C4⋊C4—531no—
382C4×C4⋊C4—531no—
383C4×C4⋊C4—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340C2×D5311yescharacteristic
391#340C2×D5311yescharacteristic
392#336—531no—
393#336—531no—
394C53#2275811yescharacteristic, complemented, Sylow
395C22×D4—531no—
396C22×D4C22×D5311yescharacteristic
397C23×C4—1061no—
398C23×C4—1061no—
399C23×C4—1061no—
400C23×C4—1061no—
401C2×D8—534no—
402C2×D8—1062no—
403C2×D8—1062no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408C4×D4—534no—
409C4×D4—534no—
410C4×D4—1062no—
411C4×D4—1062no—
412C4×D4—1062no—
413C4×D4—1062no—
414C4×D4—2121no—
415C4×D4—2121no—
416#117—2124no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—2124no—
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—1061no—
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445#107—534no—
446C4⋊C8—1062no—
447#102—41no—
448#102—534no—
449#102—1062no—
450#102—1062no—
451#102—2121no—
452#102C4×D5314yescomplemented
453#97—41no—
454#97—2121no—
455C4×C8—534no—
456#95—21no—
457#95—531no—
458#95—531no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462C24—21no—
463C24—1061no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—532no—
470C2×D4—532no—
471C2×D4—532no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4C2×C4×D5312yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1064no—
484C22×C4—1064no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—4241no—
489C22×C4—4241no—
490C22×C4—2122no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4D4×D5311yescharacteristic
499C22×C4D4×D5311yescharacteristic
500C22×C4C23×D5311yescharacteristic
501D8—1068no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—532no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8C2×C4×D5312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—532no—
521C4⋊C4—1062no—
522C4⋊C4—1062no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—4241no—
529C4⋊C4—4241no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4C2×C4×D5312yes—
534C22⋊C4—42no—
535C22⋊C4—81no—
536C22⋊C4—1064no—
537C22⋊C4—1064no—
538C22⋊C4—4241no—
539C22⋊C4—2122no—
540C22⋊C4—2122no—
541C22⋊C4—2122no—
542C42—21no—
543C42—21no—
544C42—532no—
545C42—532no—
546C42—1061no—
547C42—1061no—
548C42—2122no—
549C42—2122no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—1062no—
558C23—1062no—
559C23—1062no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23—2121no—
564C23C2×D4×D5311yescharacteristic
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—534no—
570D4—1062no—
571D4—1062no—
572D4—1062no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—532no—
591C2×C4—532no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1062no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—4244no—
602C2×C4—4244no—
603C2×C4—2128no—
604C2×C4—2128no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4C22×C4×D5311yescharacteristic
614C2×C4#41×D5311yescharacteristic
615C2×C4#41×D5311yescharacteristic
616C2×C4#41×D5311yescharacteristic
617C2×C4#1313326211yescharacteristic
618C2×C4#1313326411yescharacteristic
619C8—22no—
620C8—41no—
621C8—1062no—
622C8—2121no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—1064no—
631C22—2122no—
632C22—2122no—
633C22—2122no—
634C22—2122no—
635C22—2122no—
636C22—2122no—
637C22C4×D4×D5311yescharacteristic
638C22#1681126211yescharacteristic
639C22#1681128111yescharacteristic
640C22AGL(1,ℤ/8)×D5311yescharacteristic
641C22AGL(1,ℤ/8)×D5311yescharacteristic
642C22#1681142711yescharacteristic
643C22#1681143911yescharacteristic
644C4—41no—
645C4—42no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—534no—
650C4—1062no—
651C4—4241no—
652C4—4241no—
653C4—2121no—
654C4—2122no—
655C4—2121no—
656C4#117×D5311yescharacteristic
657C4#1681110211yescharacteristic
658C4#1681130911yescharacteristic
659C4#1681131811yescharacteristic
660C2—82no—
661C2—2124no—
662C2#2027764311yescharacteristic
663C2#442×D5311yescharacteristic
664C2#2028358711yescharacteristic
665C2#2028695211yescharacteristic
666C2#2028701511yescharacteristic
667C2#2028741911yescharacteristic
668C2#2028743011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1936C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
cosocleC22×C4×C53C24
hypercenter#1936D53
hypocenterC53#22758
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #20277663C2semidirect
3. #20277616C2semidirect
4. #20275414C2semidirect
5. #20275310C2semidirect
6. #20275304C2semidirect
9. #20274841C2semidirect
10. #20274714C2semidirect
11. #20274713C2semidirect
12. #20274583C2semidirect
13. #20274541C2semidirect
14. #20274539C2semidirect
15. C53×#1936C2semidirect
23. #16810703C4semidirect
73. D4×Dic53C2×C4semidirect
100. #13133150C2×C4semidirect
107. C8×Dic53C2×C4semidirect
124. #102×C53C2×C4semidirect
280. D212#117semidirect
330. #1936D53semidirect
394. C53#22758semidirect
452. #102C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107032
#202776632
#202776162
#202754142
#202753102
#202753042
#202750662
#202750602
#202748412
#202747142
#202747132
#202745832
#202745412
#202745392
C53×#19362
#2275853