Finite Group Databasebeta

← #27422803

Subgroups — #27422803

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×#16810757C211yesmaximal, characteristic
2C2×#16810680C211yesmaximal, characteristic, complemented
3#20283046C211yesmaximal, characteristic, complemented
4#20276089C212yesmaximal, complemented
5#20276025C212yesmaximal, complemented
6#20275590C212yesmaximal, complemented
7#20275415C212yesmaximal, complemented
8#20275032C211yesmaximal, characteristic, complemented
9#20274908C211yesmaximal, characteristic, complemented
10#20274646C211yesmaximal, characteristic, complemented
11C2×C53×#390C211yesmaximal, characteristic, complemented
12C22×#11719922C2212yes—
13C2×#13133200—21no—
14C2×#13133191C2211yescharacteristic
15#16810757—24no—
16#16810680—24no—
17C2×#13133148C2212yescomplemented
18C2×C4⋊#88322C2212yescomplemented
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×#13133137C2212yescomplemented
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25C2×#13133128C2212yescomplemented
26#16810525C2211yescharacteristic, complemented
27#16810525C2211yescharacteristic, complemented
28#16810521C2211yescharacteristic, complemented
29#16810521C2211yescharacteristic, complemented
30#16810514—21no—
31#16810514—21no—
32#16810502—21no—
33#16810483C2211yescharacteristic
34#16810385C2212yescomplemented
35#16810385C2212yescomplemented
36C23×D4×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C22⋊C4×C53—21no—
39C22×C22⋊C4×C53—21no—
40C22×C42×C53C2211yescharacteristic
41C53×#390—21no—
42C53×#390—21no—
43C53×#390C2212yes—
44C53×#390C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C2×#11719922—24no—
48C2×#11719922—24no—
49C2×#11719922—42no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133200—44no—
53#13133191—22no—
54#13133191—22no—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—21no—
59C2×#11719919—21no—
60C2×C22⋊Dic53—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66#13133148—42no—
67C4⋊#88322—24no—
68#13133138—24no—
69#13133138—24no—
70#13133138—42no—
71#13133137—42no—
72C22⋊#88322—22no—
73C22⋊#88322D414yes—
74#13133128—24no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78C25×C53—21no—
79C22×D4×C53—24no—
80C22×D4×C53—21no—
81C22×D4×C53—22no—
82C22×D4×C53—22no—
83C22×D4×C53—42no—
84C22×D4×C53C2312yes—
85C23×C4×C53—21no—
86C23×C4×C53D411yescharacteristic
87C23×C4×C53D411yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—42no—
91C2×C4⋊C4×C53C2312yes—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—22no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—44no—
97C2×C22⋊C4×C53—44no—
98C2×C42×C53—22no—
99C2×C42×C53—22no—
100C2×C42×C53D411yescharacteristic
101C2×C42×C53D411yescharacteristic
102C2×C42×C53C2311yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C24×C53—24no—
105C24×C53—22no—
106C24×C53—42no—
107C24×C53—42no—
108C24×C53—42no—
109C24×C53—42no—
110C24×C53—42no—
111C24×C53—42no—
112C24×C53C2×D411yescharacteristic
113C23×D53—21no—
114C2×D4×C53—24no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—48no—
121C2×D4×C53—48no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53#4114yes—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—41no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53#4112yes—
144C22×C4×C53C2411yescharacteristic
145C2×C22⋊D53—42no—
146C2×C22⋊D53—42no—
147C2×C22⋊D53—42no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—22no—
154C2×D212—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—42no—
158C22⋊C4×C53—44no—
159C22⋊C4×C53—44no—
160C22⋊C4×C53—44no—
161C22⋊C4×C53—44no—
162C42×C53—22no—
163C42×C53—22no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C22⋊Dic53—42no—
169#11719922—48no—
170#11719921—42no—
171#11719921—42no—
172#11719921—42no—
173#11719921—42no—
174#11719921—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—22no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179#11719919—42no—
180#11719919—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—24no—
192C23×C53—48no—
193C23×C53—48no—
194C23×C53—48no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—84no—
202C23×C53—84no—
203C23×C53—84no—
204C23×C53—84no—
205C23×C53#12112yes—
206C23×C53AGL(1,ℤ/8)12yes—
207C23×C53AGL(1,ℤ/8)12yes—
208C23×C53C22×D411yescharacteristic
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216D4×C53—24no—
217D4×C53—24no—
218D4×C53—48no—
219D4×C53—42no—
220D4×C53—42no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
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—21no—
235C2×C4×C53—21no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—84no—
242C2×C4×C53#12112yes—
243C2×C4×C53AGL(1,ℤ/8)12yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4112yes—
247C2×C4×C53H(5,𝔽2)12yes—
248C22⋊D53—84no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
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—44no—
270C22×C53—44no—
271C22×C53—44no—
272C22×C53—816no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53#46914yes—
278C22×C53C2×#12111yescharacteristic
279C22×C53#52511yescharacteristic
280C22×C53#53511yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53#57511yescharacteristic
285C2×D53—81no—
286C2×D53—81no—
287C2×D53—81no—
288C2×D53—81no—
289C2×D53—81no—
290C2×D53—81no—
291C2×D53—81no—
292C4×C53—22no—
293C4×C53—22no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—82no—
298C4×C53#54612yes—
299C4×C53#55112yes—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303C2×#447—531no—
304C2×#432—531no—
305#2854—531no—
306C2×#390D5311yescharacteristic, complemented
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—84no—
316C2×C53#241111yescharacteristic
317C2×C53C2×#46911yescharacteristic
318C2×C53#305511yescharacteristic
319C2×C53#310911yescharacteristic
320C2×C53#311411yescharacteristic
321C2×C53#321311yescharacteristic
322C2×C53#321711yescharacteristic
323D53—161no—
324C23×D4C2×D5311yescharacteristic
325C22×D8—532no—
326C2×#123—1061no—
327C2×#121—531no—
328C22×C4⋊C4C2×D5311yescharacteristic
329C22×C22⋊C4—21no—
330C22×C22⋊C4—21no—
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—21no—
344#390—21no—
345#390—531no—
346#390—531no—
347#390C2×D5312yes—
348#390C2×D5312yes—
349#389—531no—
350#389—531no—
351#388—1061no—
352#388—1061no—
353#386—1061no—
354#380—531no—
355#340—532no—
356#340—532no—
357C53#2450611yescharacteristic, complemented, Sylow
358C25—21no—
359C22×D4—24no—
360C22×D4—22no—
361C22×D4—22no—
362C22×D4—21no—
363C22×D4—42no—
364C22×D4—531no—
365C22×D4—1061no—
366C22×D4C22×D5312yes—
367C23×C4—21no—
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—2124no—
377#121—1062no—
378#121—1062no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—42no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4C22×D5312yes—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—44no—
392C2×C22⋊C4—44no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
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—1061no—
415#95—1061no—
416#95—1061no—
417C24—24no—
418C24—22no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—1061no—
426C24C2×C22⋊D5311yescharacteristic
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1062no—
440C2×D4—1062no—
441C2×D4—2122no—
442C2×D4—2122no—
443C2×D4—2122no—
444C2×D4#1171994514yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
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—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1062no—
481C4⋊C4—1062no—
482C4⋊C4—2122no—
483C4⋊C4—2122no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—2122no—
489C22⋊C4—2122no—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C22⋊C4—2122no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—24no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23—2121no—
519C23—2121no—
520C23—2121no—
521C23—2121no—
522C23—2121no—
523C23—2121no—
524C23—2121no—
525C23#1313318912yes—
526C23#1313324212yes—
527C23#1313324312yes—
528C23C22×C22⋊D5311yescharacteristic
529D4—24no—
530D4—24no—
531D4—48no—
532D4—42no—
533D4—42no—
534D4—4244no—
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—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4#1313318912yes—
568C2×C4#1313324312yes—
569C2×C4C2×D4×D5311yescharacteristic
570C2×C4C2×#1171994711yescharacteristic
571C2×C4C2×#1171994512yes—
572C2×C4#1313326212yes—
573C8—1064no—
574C8—2122no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—816no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—4241no—
590C22—4241no—
591C22—4241no—
592C22—4241no—
593C22—4241no—
594C22—4241no—
595C22—4241no—
596C22#1681087414yes—
597C22C2×#1313318911yescharacteristic
598C22#1681113411yescharacteristic
599C22#1681119011yescharacteristic
600C22C2×#1313324211yescharacteristic
601C22C2×#1313324311yescharacteristic
602C22C2×#1313324311yescharacteristic
603C22#1681142111yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—82no—
610C4—4241no—
611C4—4241no—
612C4—4241no—
613C4#1681126312yes—
614C4#1681129912yes—
615C2—24no—
616C2—84no—
617C2—8481no—
618C2#2027810011yescharacteristic
619C2C2×#1681087411yescharacteristic
620C2#2028433011yescharacteristic
621C2#2028480611yescharacteristic
622C2#2028485311yescharacteristic
623C2#2028570211yescharacteristic
624C2#2028573311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#390C2
socleC23×C53C22×D4
prime radicalC2×#16810757C2
hypercenterC2×#390D53
hypocenterC53#24506
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#16810680C2semidirect
3. #20283046C2semidirect
4. #20276089C2semidirect
5. #20276025C2semidirect
6. #20275590C2semidirect
7. #20275415C2semidirect
8. #20275032C2semidirect
9. #20274908C2semidirect
10. #20274646C2semidirect
11. C2×C53×#390C2semidirect
17. C2×#13133148C22semidirect
18. C2×C4⋊#88322C22semidirect
22. C2×#13133137C22semidirect
25. C2×#13133128C22semidirect
26. #16810525C22semidirect
27. #16810525C22semidirect
28. #16810521C22semidirect
29. #16810521C22semidirect
34. #16810385C22semidirect
35. #16810385C22semidirect
306. C2×#390D53semidirect
357. C53#24506semidirect

Maximal subgroups

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

subgroupindex
C2×#168107572
C2×#168106802
#202830462
#202760892
#202760252
#202755902
#202754152
#202750322
#202749082
#202746462
C2×C53×#3902
#2450653