Finite Group Databasebeta

← #27422804

Subgroups — #27422804

Order 13568.

1250 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810756C211yesmaximal, characteristic
2C2×#16810680C211yesmaximal, characteristic, complemented
3C2×#16810525C211yesmaximal, characteristic, complemented
4#20276086C212yesmaximal, complemented
5#20276025C212yesmaximal, complemented
6#20275590C212yesmaximal, complemented
7#20275411C212yesmaximal, complemented
8#20275033C211yesmaximal, characteristic, complemented
9#20274907C211yesmaximal, characteristic, complemented
10#20274645C211yesmaximal, characteristic, complemented
11C53×#2854C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719922C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719921—21no—
16C22×#11719921—21no—
17#16810756—24no—
18#16810680—24no—
19C2×#13133148C2212yescomplemented
20C2×C4⋊#88322C2212yescomplemented
21C2×#13133138C2212yescomplemented
22C2×#13133137C2212yes—
23C2×#13133137C2212yes—
24C2×#13133137C2212yes—
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133128C2212yescomplemented
28#16810525—21no—
29#16810525—21no—
30#16810525C2212yes—
31#16810525C2212yes—
32#16810385C2212yescomplemented
33#16810385C2212yescomplemented
34C2×#123×C53—21no—
35C2×#121×C53C2211yescharacteristic
36C22×C42×C53C2211yescharacteristic
37C53×#390C2211yescharacteristic, complemented
38C53×#390C2211yescharacteristic, complemented
39C53×#389C2211yescharacteristic, complemented
40C53×#389C2211yescharacteristic, complemented
41C53×#388—21no—
42C53×#388—21no—
43C53×#386—21no—
44C53×#380C2211yescharacteristic
45C24×D53—21no—
46C22×D212—21no—
47C22×D212—24no—
48C22×D212—22no—
49C22×D212—22no—
50C22×D212—42no—
51C22×D212C2312yes—
52C23×Dic53—21no—
53C2×#11719922—24no—
54C2×#11719922—24no—
55C2×#11719922—42no—
56C22×#88322C2312yes—
57C22×#88322C2312yes—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—42no—
60C2×C4⋊Dic53C2312yes—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—22no—
65C2×#11719921—44no—
66C2×#11719921—44no—
67#13133148—42no—
68C4⋊#88322—24no—
69#13133138—42no—
70#13133137—24no—
71#13133137—24no—
72#13133137—42no—
73C22⋊#88322—22no—
74C22⋊#88322D414yes—
75#13133128—24no—
76C22×D4×C53—21no—
77C22×D4×C53C2311yescharacteristic
78C23×C4×C53D411yescharacteristic
79C23×C4×C53D411yescharacteristic
80C23×C4×C53C2311yescharacteristic
81#123×C53—44no—
82#121×C53—22no—
83#121×C53—22no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53—21no—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—21no—
92C2×C22⋊C4×C53—21no—
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C42×C53—22no—
96C2×C42×C53—22no—
97C2×C42×C53D411yescharacteristic
98C2×C42×C53D411yescharacteristic
99C2×C42×C53C2311yescharacteristic
100C2×C42×C53C2311yescharacteristic
101#95×C53—21no—
102#95×C53—21no—
103#95×C53—21no—
104C24×C53—21no—
105C24×C53C2×D411yescharacteristic
106C23×D53—24no—
107C23×D53—22no—
108C23×D53—42no—
109C23×D53—42no—
110C23×D53—42no—
111C23×D53—42no—
112C23×D53—42no—
113C23×D53—42no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—42no—
119C2×D4×C53—42no—
120C2×D4×C53—42no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—41no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C2×D212—22no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212—22no—
148C2×D212—24no—
149C2×D212—22no—
150C2×D212—48no—
151C2×D212—48no—
152C2×D212—42no—
153C2×D212—42no—
154C2×D212#4114yes—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—42no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C42×C53—22no—
168C42×C53—22no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173#11719922—48no—
174#11719921—44no—
175#11719921—44no—
176#11719921—44no—
177#11719921—44no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—42no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322C2×D414yes—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53#12112yes—
199C23×C53AGL(1,ℤ/8)12yes—
200C23×C53AGL(1,ℤ/8)12yes—
201C23×C53C22×D411yescharacteristic
202C22×D53—24no—
203C22×D53—48no—
204C22×D53—48no—
205C22×D53—48no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C22×D53—42no—
212C22×D53—84no—
213C22×D53—84no—
214C22×D53—84no—
215C22×D53—84no—
216D4×C53—42no—
217D4×C53—42no—
218D4×C53—84no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
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—21no—
234C2×C4×C53—44no—
235C2×C4×C53—44no—
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—41no—
244C2×C4×C53—41no—
245C2×C4×C53#12112yes—
246C2×C4×C53AGL(1,ℤ/8)12yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C2×#4112yes—
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53H(5,𝔽2)12yes—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—84no—
255D212—24no—
256D212—24no—
257D212—48no—
258D212—42no—
259D212—42no—
260#88322—24no—
261#88322—42no—
262#24506—531nomaximal, Sylow
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53#46914yes—
277C22×C53C2×#12111yescharacteristic
278C22×C53#52511yescharacteristic
279C22×C53#53511yescharacteristic
280C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C53#57511yescharacteristic
284C2×D53—44no—
285C2×D53—44no—
286C2×D53—44no—
287C2×D53—816no—
288C2×D53—82no—
289C2×D53—82no—
290C2×D53—82no—
291C2×D53—82no—
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#54612yes—
301C4×C53#55112yes—
302Dic53—82no—
303C2×#447—531no—
304C2×#432—531no—
305#2854D5311yescharacteristic, complemented
306C2×#390—531no—
307#2065—532no—
308#2054—532no—
309#1972—532no—
310#1936—532no—
311#1874—531no—
312#1853—531no—
313#1786—531no—
314C2×C53—24no—
315C2×C53—161no—
316C2×C53#241111yescharacteristic
317C2×C53C2×#46911yescharacteristic
318C2×C53#305511yescharacteristic
319C2×C53#310911yescharacteristic
320C2×C53#311411yescharacteristic
321C2×C53#321311yescharacteristic
322C2×C53#321711yescharacteristic
323D53—84no—
324C23×D4—531no—
325C22×D8—532no—
326C2×#123—21no—
327C2×#121C2×D5311yescharacteristic
328C22×C4⋊C4—531no—
329C22×C22⋊C4—1061no—
330C22×C22⋊C4—1061no—
331C22×C42C2×D5311yescharacteristic
332#447—1064no—
333#432—1064no—
334C2×#106—532no—
335C2×C4⋊C8—532no—
336C2×#102—532no—
337C2×#102—532no—
338C2×#102—532no—
339C2×#102—532no—
340C2×C22⋊C8—531no—
341C2×C22⋊C8—531no—
342C2×#97—532no—
343#390—532no—
344#390—532no—
345#390—1061no—
346#390—1061no—
347#390C2×D5311yescharacteristic, complemented
348#390C2×D5311yescharacteristic, complemented
349#389C2×D5311yescharacteristic, complemented
350#389C2×D5311yescharacteristic, complemented
351#388—21no—
352#388—21no—
353#386—21no—
354#380C2×D5311yescharacteristic
355#340—532no—
356#340—532no—
357C53#2450611yescharacteristic, complemented, Sylow
358C25—1061no—
359C22×D4—21no—
360C22×D4—532no—
361C22×D4—1064no—
362C22×D4—1062no—
363C22×D4—1062no—
364C22×D4—1061no—
365C22×D4—2122no—
366C22×D4C22×D5311yescharacteristic
367C23×C4—1061no—
368C23×C4C22⋊D5311yescharacteristic
369C23×C4C22⋊D5311yescharacteristic
370C23×C4C22×D5311yescharacteristic
371C2×D8—1064no—
372C2×D8—1064no—
373C2×D8—2122no—
374C22×C8—532no—
375C22×C8—532no—
376#123—44no—
377#121—22no—
378#121—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—532no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1062no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1062no—
397C2×C22⋊C4—2124no—
398C2×C22⋊C4—2124no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42C22⋊D5311yescharacteristic
402C2×C42C22⋊D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404C2×C42C22×D5311yescharacteristic
405#106—2122no—
406C4⋊C8—1064no—
407#102—1064no—
408#102—1064no—
409#102—2122no—
410#102—2122no—
411C22⋊C8—534no—
412C22⋊C8—1062no—
413#97—1064no—
414#95—21no—
415#95—21no—
416#95—21no—
417C24—21no—
418C24—1064no—
419C24—1062no—
420C24—2122no—
421C24—2122no—
422C24—2122no—
423C24—2122no—
424C24—2122no—
425C24—2122no—
426C24C2×C22⋊D5311yescharacteristic
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—534no—
435C2×D4—1064no—
436C2×D4—1062no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1062no—
440C2×D4—1062no—
441C2×D4—2128no—
442C2×D4—2128no—
443C2×D4—2122no—
444C2×D4—2122no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—1062no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4D4×D5312yes—
461C22×C4#1171994712yes—
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—21no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—2122no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—2124no—
491C22⋊C4—2124no—
492C22⋊C4—2124no—
493C22⋊C4—2124no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—1064no—
512C23—4244no—
513C23—4244no—
514C23—4244no—
515C23—4244no—
516C23—2128no—
517C23—2128no—
518C23—2128no—
519C23—2122no—
520C23—2122no—
521C23—2122no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23#1313318912yes—
526C23#1313324212yes—
527C23#1313324312yes—
528C23C22×C22⋊D5311yescharacteristic
529D4—42no—
530D4—42no—
531D4—84no—
532D4—1064no—
533D4—1064no—
534D4—2128no—
535D4—2122no—
536D4—2122no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—4244no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4#1313318912yes—
568C2×C4#1313324212yes—
569C2×C4C2×D4×D5311yescharacteristic
570C2×C4#41×D5312yes—
571C2×C4C2×#1171994711yescharacteristic
572C2×C4#1313326312yes—
573C8—1064no—
574C8—2122no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—42416no—
589C22—4242no—
590C22—4242no—
591C22—4242no—
592C22—4242no—
593C22—2124no—
594C22—2124no—
595C22—2124no—
596C22#1681087414yes—
597C22C2×#1313318911yescharacteristic
598C22#1681113011yescharacteristic
599C22#1681118711yescharacteristic
600C22C2×#1313324211yescharacteristic
601C22C2×#1313324211yescharacteristic
602C22C2×#1313324311yescharacteristic
603C22#1681142111yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—4242no—
613C4#1681126512yes—
614C4#1681129612yes—
615C2—24no—
616C2—161no—
617C2—4244no—
618C2#2027810111yescharacteristic
619C2C2×#1681087411yescharacteristic
620C2#2028433611yescharacteristic
621C2#2028480211yescharacteristic
622C2#2028484911yescharacteristic
623C2#2028569811yescharacteristic
624C2#2028573011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2854C2
socleC23×C53C22×D4
prime radicalC2×#16810756C2
hypercenter#2854D53
hypocenterC53#24506
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×#16810680C2semidirect
3. C2×#16810525C2semidirect
4. #20276086C2semidirect
5. #20276025C2semidirect
6. #20275590C2semidirect
7. #20275411C2semidirect
8. #20275033C2semidirect
9. #20274907C2semidirect
10. #20274645C2semidirect
11. C53×#2854C2semidirect
19. C2×#13133148C22semidirect
20. C2×C4⋊#88322C22semidirect
21. C2×#13133138C22semidirect
27. C2×#13133128C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#390C22semidirect
38. C53×#390C22semidirect
39. C53×#389C22semidirect
40. C53×#389C22semidirect
305. #2854D53semidirect
347. #390C2×D53semidirect
348. #390C2×D53semidirect
349. #389C2×D53semidirect
350. #389C2×D53semidirect
357. C53#24506semidirect

Maximal subgroups

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

subgroupindex
C2×#168107562
C2×#168106802
C2×#168105252
#202760862
#202760252
#202755902
#202754112
#202750332
#202749072
#202746452
C53×#28542
#2450653