Finite Group Databasebeta

← #27395293

Subgroups — #27395293

Order 13568.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810756C211yesmaximal, characteristic
2C2×#16810525C211yesmaximal, characteristic, complemented
3#20275590C212yesmaximal, complemented
4#20275411C212yesmaximal, complemented
5#20275342C212yesmaximal, complemented
6#20275063C212yesmaximal, complemented
7#20275033C211yesmaximal, characteristic, complemented
8#20274907C211yesmaximal, characteristic, complemented
9#20274645C211yesmaximal, characteristic, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C4×#121×C53C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719922C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719921—21no—
16C22×#11719921—21no—
17#16810756C418yescomplemented
18C2×#13133148C2212yescomplemented
19C2×#13133138C2212yescomplemented
20C2×#13133137C2212yes—
21C2×#13133137C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×#13133128C2212yescomplemented
26#16810525—21no—
27#16810525—21no—
28#16810525C2212yes—
29#16810525C2212yes—
30#16810385C2212yescomplemented
31#16810385C2212yescomplemented
32#16810378C2212yescomplemented
33C2×#121×C53C2211yescharacteristic
34C2×C4×D4×C53—21no—
35C2×C4×D4×C53C2211yescharacteristic, complemented
36C22×C42×C53C2211yescharacteristic
37C53×#390C2211yescharacteristic, complemented
38C53×#389C2211yescharacteristic, complemented
39C53×#388—21no—
40C53×#386—21no—
41C53×#380C2211yescharacteristic
42C4×C4⋊C4×C53C2211yescharacteristic, complemented
43C4×C22⋊C4×C53—21no—
44C24×D53—21no—
45C22×D212—21no—
46C22×D212—22no—
47C22×D212—42no—
48C22×D212—42no—
49C22×D212C2×C414yescomplemented
50C22×D212C2312yes—
51C23×Dic53—21no—
52C2×#11719922—42no—
53C2×#11719922—42no—
54C2×#11719922C2×C418yescomplemented
55C22×#88322C2312yes—
56C22×#88322C2312yes—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—42no—
59C2×C4⋊Dic53C2312yes—
60C2×#11719921—22no—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—44no—
65C2×#11719921—44no—
66#13133148—42no—
67#13133138—42no—
68#13133137—42no—
69#13133137—42no—
70#13133137C2×C418yescomplemented
71C22⋊#88322—22no—
72C22⋊#88322C2×C414yescomplemented
73#13133128—42no—
74C22×D4×C53—21no—
75C22×D4×C53C2311yescharacteristic
76C23×C4×C53—21no—
77C23×C4×C53D411yescharacteristic
78C23×C4×C53D411yescharacteristic
79C23×C4×C53C2311yescharacteristic
80#121×C53—22no—
81#121×C53C2×C414yes—
82C4×D4×C53—44no—
83C4×D4×C53—42no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—21no—
92C2×C22⋊C4×C53—21no—
93C2×C42×C53—21no—
94C2×C42×C53—42no—
95C2×C42×C53D411yescharacteristic
96C2×C42×C53D411yescharacteristic
97C2×C42×C53C2311yescharacteristic
98C2×C42×C53C2311yescharacteristic
99#95×C53—21no—
100#95×C53—21no—
101#95×C53—21no—
102C24×C53—21no—
103C24×C53C2×D411yescharacteristic
104C23×D53—24no—
105C23×D53—22no—
106C23×D53—42no—
107C23×D53—42no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C2×D4×C53—24no—
113C2×D4×C53—21no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53C22×C412yes—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53#4112yes—
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—22no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—22no—
148C2×D212—22no—
149C2×D212—24no—
150C2×D212—22no—
151C2×D212—48no—
152C2×D212—42no—
153C2×D212—42no—
154C2×D212—48no—
155C2×D212C22×C414yes—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53C22×C412yes—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—42no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C42×C53—22no—
167C42×C53—22no—
168C42×C53—41no—
169C42×C53—41no—
170C42×C53—42no—
171#11719922—48no—
172#11719921—44no—
173#11719921—44no—
174#11719921—44no—
175#11719921—44no—
176C4⋊Dic53—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53—42no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322C22×C414yes—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—22no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53C4×D412yes—
197C23×C53AGL(1,ℤ/8)12yes—
198C23×C53AGL(1,ℤ/8)12yes—
199C23×C53C22×D411yescharacteristic
200C22×D53—24no—
201C22×D53—48no—
202C22×D53—48no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—48no—
210C22×D53—84no—
211C22×D53—84no—
212C22×D53—84no—
213C22×D53—84no—
214D4×C53—24no—
215D4×C53—48no—
216D4×C53—42no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—44no—
234C2×C4×C53—44no—
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—84no—
244C2×C4×C53C4×D412yes—
245C2×C4×C53AGL(1,ℤ/8)12yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)12yes—
249C2×C4×C53H(5,𝔽2)12yes—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—84no—
254D212—24no—
255D212—24no—
256D212—48no—
257D212—42no—
258D212—42no—
259#88322—24no—
260#88322—42no—
261#22046—531nomaximal, Sylow
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53#44214yes—
276C22×C53C2×C4×D411yescharacteristic
277C22×C53#53411yescharacteristic
278C22×C53#53511yescharacteristic
279C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53#57511yescharacteristic
283C2×D53—44no—
284C2×D53—44no—
285C2×D53—44no—
286C2×D53—816no—
287C2×D53—82no—
288C2×D53—82no—
289C2×D53—82no—
290C2×D53—82no—
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—42no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53#51812yes—
300C4×C53#53812yes—
301Dic53—82no—
302C2×#447—531no—
303C4×#121D5311yescharacteristic, complemented
304C2×#390—531no—
305#1972—532no—
306#1936—532no—
307#1926—532no—
308#1879—532no—
309#1874—531no—
310#1853—531no—
311#1786—531no—
312#1749—531no—
313C2×C53—24no—
314C2×C53—82no—
315C2×C53#232111yescharacteristic
316C2×C53C2×#44211yescharacteristic
317C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C53#319211yescharacteristic
319C2×C53#319411yescharacteristic
320C2×C53#321311yescharacteristic
321C2×C53#321711yescharacteristic
322D53—84no—
323C23×D4—531no—
324C22×D8—532no—
325C2×#121C2×D5311yescharacteristic
326C2×C4×D4—21no—
327C2×C4×D4C2×D5311yescharacteristic, complemented
328C22×C4⋊C4—531no—
329C22×C22⋊C4—1061no—
330C22×C22⋊C4—1061no—
331C22×C42C2×D5311yescharacteristic
332#447—538no—
333C2×#106—532no—
334C2×#102—532no—
335C2×#102—532no—
336C2×#102—532no—
337C2×#102—532no—
338C2×C22⋊C8—531no—
339C2×C22⋊C8—531no—
340C2×#97—532no—
341#390—532no—
342#390—532no—
343#390—1061no—
344#390—1061no—
345#390C2×D5311yescharacteristic, complemented
346#389C2×D5311yescharacteristic, complemented
347#388—21no—
348#386—21no—
349#380C2×D5311yescharacteristic
350C4×C4⋊C4C2×D5311yescharacteristic, complemented
351C4×C22⋊C4—21no—
352#340—532no—
353#340—532no—
354#336—532no—
355C53#2204611yescharacteristic, complemented, Sylow
356C25—1061no—
357C22×D4—21no—
358C22×D4—534no—
359C22×D4—532no—
360C22×D4—1062no—
361C22×D4—1061no—
362C22×D4—2122no—
363C22×D4—2122no—
364C22×D4C22×D5311yescharacteristic
365C23×C4—21no—
366C23×C4—1061no—
367C23×C4C22⋊D5311yescharacteristic
368C23×C4C22⋊D5311yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×D8—538no—
371C2×D8—2122no—
372C2×D8—2122no—
373C22×C8—532no—
374C22×C8—532no—
375#121—22no—
376#121C4×D5314yes—
377C4×D4—42no—
378C4×D4—44no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—532no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—2122no—
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—1062no—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—2124no—
396C2×C22⋊C4—2124no—
397C2×C42—21no—
398C2×C42—42no—
399C2×C42C22⋊D5311yescharacteristic
400C2×C42C22⋊D5311yescharacteristic
401C2×C42C22×D5311yescharacteristic
402C2×C42C22×D5311yescharacteristic
403#106—2122no—
404#102—538no—
405#102—2122no—
406#102—2122no—
407#102—2122no—
408C22⋊C8—534no—
409C22⋊C8—1062no—
410#97—2122no—
411#95—21no—
412#95—21no—
413#95—21no—
414C24—21no—
415C24—1064no—
416C24—1062no—
417C24—2122no—
418C24—2122no—
419C24—2122no—
420C24—2122no—
421C24—2122no—
422C24—2122no—
423C24C2×C22⋊D5311yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—534no—
431C2×D4—1064no—
432C2×D4—1062no—
433C2×D4—1062no—
434C2×D4—1062no—
435C2×D4—1062no—
436C2×D4—1062no—
437C2×D4—2128no—
438C2×D4—2122no—
439C2×D4—2122no—
440C2×D4—2128no—
441C2×D4C2×C4×D5312yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—1062no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4C2×C4×D5312yes—
461C22×C4#1171994512yes—
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C23×D5311yescharacteristic
468D8—2128no—
469C2×C8—534no—
470C2×C8—1062no—
471C2×C8—1062no—
472C2×C8—1062no—
473C2×C8—1062no—
474C2×C8—1062no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—1062no—
480C4⋊C4—1062no—
481C4⋊C4—2122no—
482C4⋊C4C2×C4×D5312yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—2124no—
489C22⋊C4—2124no—
490C22⋊C4—2124no—
491C22⋊C4—2124no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—42no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—1064no—
509C23—4244no—
510C23—4244no—
511C23—4244no—
512C23—4244no—
513C23—2128no—
514C23—2128no—
515C23—2122no—
516C23—2122no—
517C23—2122no—
518C23—2122no—
519C23—2122no—
520C23—2122no—
521C23—2128no—
522C23C4×C22⋊D5312yes—
523C23#1313324212yes—
524C23#1313324312yes—
525C23C22×C22⋊D5311yescharacteristic
526D4—24no—
527D4—48no—
528D4—42no—
529D4—1064no—
530D4—1064no—
531D4—2128no—
532D4—2122no—
533D4—2122no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
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—84no—
561C2×C4—4244no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4C4×C22⋊D5312yes—
566C2×C4#1313324212yes—
567C2×C4C22×C4×D5311yescharacteristic
568C2×C4C2×#1171994511yescharacteristic
569C2×C4#1313326212yes—
570C2×C4#1313326312yes—
571C8—1064no—
572C8—2122no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—42416no—
587C22—4242no—
588C22—4242no—
589C22—4242no—
590C22—4242no—
591C22—2124no—
592C22—2124no—
593C22—2124no—
594C22#1681073314yes—
595C22C2×C4×C22⋊D5311yescharacteristic
596C22#1681118211yescharacteristic
597C22#1681118711yescharacteristic
598C22C2×#1313324211yescharacteristic
599C22C2×#1313324211yescharacteristic
600C22C2×#1313324311yescharacteristic
601C22#1681142111yescharacteristic
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—4242no—
611C4#1681109512yes—
612C4#1681120612yes—
613C2—24no—
614C2—82no—
615C2—4244no—
616C2#2027736911yescharacteristic
617C2C2×#1681073311yescharacteristic
618C2C4×#1313324211yescharacteristic
619C2#2028551211yescharacteristic
620C2#2028552511yescharacteristic
621C2#2028569811yescharacteristic
622C2#2028573011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
prime radicalC2×#16810756C2
cosocleC22×C4×C53C24
hypercenterC4×#121D53
hypocenterC53#22046
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#16810525C2semidirect
3. #20275590C2semidirect
4. #20275411C2semidirect
5. #20275342C2semidirect
6. #20275063C2semidirect
7. #20275033C2semidirect
8. #20274907C2semidirect
9. #20274645C2semidirect
10. #20274477C2semidirect
11. C4×#121×C53C2semidirect
17. #16810756C4semidirect
18. C2×#13133148C22semidirect
19. C2×#13133138C22semidirect
25. C2×#13133128C22semidirect
30. #16810385C22semidirect
31. #16810385C22semidirect
32. #16810378C22semidirect
35. C2×C4×D4×C53C22semidirect
37. C53×#390C22semidirect
38. C53×#389C22semidirect
42. C4×C4⋊C4×C53C22semidirect
49. C22×D212C2×C4semidirect
54. C2×#11719922C2×C4semidirect
70. #13133137C2×C4semidirect
72. C22⋊#88322C2×C4semidirect
303. C4×#121D53semidirect
327. C2×C4×D4C2×D53semidirect
345. #390C2×D53semidirect
346. #389C2×D53semidirect
350. C4×C4⋊C4C2×D53semidirect
355. C53#22046semidirect

Maximal subgroups

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

subgroupindex
C2×#168107562
C2×#168105252
#202755902
#202754112
#202753422
#202750632
#202750332
#202749072
#202746452
#202744772
C4×#121×C532
#2204653