Finite Group Databasebeta

← #27442377

Subgroups — #27442377

Order 13568.

744 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279741C211yesmaximal, characteristic, complemented
2#20276103C211yesmaximal, characteristic
3#20276089C211yesmaximal, characteristic
4#20275851C211yesmaximal, characteristic
5#20275851C211yesmaximal, characteristic
6#20275667C211yesmaximal, characteristic, complemented
7#20275663C211yesmaximal, characteristic, complemented
8#20275481C211yesmaximal, characteristic, complemented
9#20275439C211yesmaximal, characteristic, complemented
10#20275307C211yesmaximal, characteristic, complemented
11#20275293C211yesmaximal, characteristic, complemented
12#20274848C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C53×#2357C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133216—21no—
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29#16810544—21no—
30#16810544—21no—
31#16810522—21no—
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810493C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4×D4×C53C2211yescharacteristic
49C53×#390C2211yescharacteristic
50C53×#390C2211yescharacteristic
51C53×#388—21no—
52C53×#388—21no—
53C53×#382C2211yescharacteristic
54C53×#381—21no—
55C4×C4⋊C4×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×Dic106—21no—
60C22×Dic106—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133217—22no—
66#13133217—22no—
67#13133217—41no—
68#13133217—41no—
69#13133216—44no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C4×Dic53—21no—
81#13133150—22no—
82#13133148—22no—
83C4⋊#88322—22no—
84C4⋊#88322—41no—
85C4⋊#88322—41no—
86#13133138—22no—
87#13133138—22no—
88#13133138—41no—
89#13133138—41no—
90#13133128—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C22×D4×C53C2311yescharacteristic
95C23×C4×C53—21no—
96C4×D4×C53—22no—
97C4×D4×C53Q814yescomplemented
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C42×C53—21no—
109C2×C42×C53—21no—
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53D411yescharacteristic
118#95×C53D411yescharacteristic
119C24×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53C2×Q812yes—
126C22×C4×C53—21no—
127C22×C4×C53—24no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—81no—
131C22×C4×C53—81no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C2×Dic106—42no—
144C2×Dic106—42no—
145C2×Dic106—42no—
146C2×Dic106—42no—
147C2×Dic106—81no—
148C2×Dic106—81no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C2×Q812yes—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—81no—
161C22⋊C4×C53—81no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—81no—
167C42×C53—81no—
168C42×C53C2×Q812yes—
169C42×C53#4112yes—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—42no—
177C4⋊Dic53—42no—
178C4⋊Dic53—42no—
179C4⋊Dic53—81no—
180C4⋊Dic53#4112yes—
181C4⋊Dic53#4112yes—
182#11719919—42no—
183#11719919—42no—
184C4×Dic53—42no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22×D411yescharacteristic
205D4×C53—22no—
206D4×C53—22no—
207D4×C53—41no—
208D4×C53—41no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—84no—
233C2×C4×C53C22⋊Q812yes—
234C2×C4×C53C22⋊Q812yes—
235C2×C4×C53AGL(1,ℤ/8)12yes—
236C2×C4×C53#13712yes—
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C532−1+411yescharacteristic
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255Dic106—84no—
256Dic106—84no—
257#88322—41no—
258#88322—41no—
259#88322—41no—
260#88322—41no—
261#26036—531nomaximal, Sylow
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53C2×C22⋊Q811yescharacteristic
270C22×C53#54611yescharacteristic
271C22×C53#54811yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53C2×#13711yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57711yescharacteristic
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—42no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53#53311yescharacteristic
286C4×C53#54411yescharacteristic
287C4×C53#55511yescharacteristic
288C4×C53#55611yescharacteristic
289Dic53—42no—
290Dic53—42no—
291Dic53—81no—
292Dic53—81no—
293#2551—531no—
294#2357D5311yescharacteristic
295#2067—531no—
296#2065—531no—
297#2023—531no—
298#2023—531no—
299#1988—531no—
300#1987—531no—
301#1949—531no—
302#1940—531no—
303#1921—531no—
304#1919—531no—
305#1836—531no—
306#1802—531no—
307#1762—531no—
308C2×C53—82no—
309C2×C53#243911yescharacteristic
310C2×C53#307911yescharacteristic
311C2×C53#308111yescharacteristic
312C2×C53#331811yescharacteristic
313C2×C53#332011yescharacteristic
314C2×C53#337311yescharacteristic
315C2×C53#337511yescharacteristic
316C2×C4⋊Q8—531no—
317C2×C4⋊Q8—531no—
318C2×C4⋊Q8—1061no—
319C2×C4×D4C2×D5311yescharacteristic
320C2×#107—531no—
321C2×#106—531no—
322C2×C4⋊C8—531no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8—531no—
325C2×#102—531no—
326C2×#102—531no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#97—531no—
330#395—1061no—
331#395—1061no—
332#390C2×D5311yescharacteristic
333#390C2×D5311yescharacteristic
334#389—531no—
335#389—531no—
336#389—1061no—
337#388—21no—
338#388—21no—
339#384—531no—
340#382C2×D5311yescharacteristic
341#381—21no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4C2×D5311yescharacteristic
344C4×C4⋊C4C2×D5311yescharacteristic
345C4×C4⋊C4C2×D5311yescharacteristic
346C4×C22⋊C4—21no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#336—531no—
358#336—531no—
359C53#2603611yescharacteristic, complemented, Sylow
360C22×Q8—1061no—
361C22×Q8—1061no—
362C22×D4C22×D5311yescharacteristic
363C23×C4—21no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8—531no—
368C4⋊Q8—1062no—
369C4⋊Q8—1062no—
370C4⋊Q8—2121no—
371C4⋊Q8—2121no—
372C4⋊Q8—2124no—
373C4×D4—22no—
374C4×D4Dic10614yescomplemented
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1061no—
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400C2×C42C22×D5311yescharacteristic
401#107—1062no—
402#106—1062no—
403C4⋊C8—1062no—
404C4⋊C8—2121no—
405C4⋊C8—2121no—
406#102—1062no—
407#102—1062no—
408#102—2121no—
409#102—2121no—
410#97—1062no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95C22⋊D5311yescharacteristic
419#95C22⋊D5311yescharacteristic
420C24—21no—
421C2×Q8—4241no—
422C2×Q8—4241no—
423C2×Q8—2122no—
424C2×Q8—2122no—
425C2×Q8—2122no—
426C2×Q8—2122no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×Dic10612yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C23×D5311yescharacteristic
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—532no—
470C4⋊C4—532no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—4241no—
478C4⋊C4—2122no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4C2×Dic10612yes—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—81no—
494C42—81no—
495C42—2122no—
496C42C2×Dic10612yes—
497C42#1171994512yes—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C22×C22⋊D5311yescharacteristic
506Q8—4244no—
507Q8—4244no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4#1313319612yes—
549C2×C4#1313319612yes—
550C2×C4#1313324212yes—
551C2×C4#1313324712yes—
552C2×C4C22×Dic10611yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4C2×#1171994511yescharacteristic
556C2×C4#1313326311yescharacteristic
557C2×C4#1313326411yescharacteristic
558C8—2121no—
559C8—2121no—
560C8—2121no—
561C8—2121no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C2×#1313319611yescharacteristic
570C22#1681126111yescharacteristic
571C22#1681128011yescharacteristic
572C22C2×#1313324211yescharacteristic
573C22C2×#1313324711yescharacteristic
574C22#1681142611yescharacteristic
575C22#1681143411yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—4241no—
586C4—4241no—
587C4—2122no—
588C4—2122no—
589C4#1681117611yescharacteristic
590C4#1681125111yescharacteristic
591C4#1681133611yescharacteristic
592C4#1681134111yescharacteristic
593C2—82no—
594C2#2027840011yescharacteristic
595C2#2028453011yescharacteristic
596C2#2028454611yescharacteristic
597C2#2028685011yescharacteristic
598C2#2028688011yescharacteristic
599C2#2028750311yescharacteristic
600C2#2028752511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2357C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2357D53
hypocenterC53#26036
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #20279741C2semidirect
6. #20275667C2semidirect
7. #20275663C2semidirect
8. #20275481C2semidirect
9. #20275439C2semidirect
10. #20275307C2semidirect
11. #20275293C2semidirect
12. #20274848C2semidirect
97. C4×D4×C53Q8semidirect
359. C53#26036semidirect
374. C4×D4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202797412
#202761032
#202760892
#202758512
#202756672
#202756632
#202754812
#202754392
#202753072
#202752932
#202748482
#202747152
#202745402
C53×#23572
#2603653