Finite Group Databasebeta

← #27442378

Subgroups — #27442378

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#20279743C211yesmaximal, characteristic, complemented
2#20277687C211yesmaximal, characteristic
3#20276085C211yesmaximal, characteristic
4#20275851C211yesmaximal, characteristic
5#20275849C211yesmaximal, characteristic
6#20275666C211yesmaximal, characteristic, complemented
7#20275661C211yesmaximal, characteristic, complemented
8#20275480C211yesmaximal, characteristic, complemented
9#20275435C211yesmaximal, characteristic, complemented
10#20275309C211yesmaximal, characteristic, complemented
11#20275290C211yesmaximal, characteristic, complemented
12#20274847C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#20274542C211yesmaximal, characteristic
15C53×#2067C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133216—21no—
18C2×#13133216C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133127C2211yescharacteristic
26#16810544—21no—
27#16810543—21no—
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810520—21no—
31#16810517—21no—
32#16810511—21no—
33#16810492C2211yescharacteristic
34#16810488C2211yescharacteristic
35#16810484—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4×#11719919C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#107×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106—21no—
61C23×Dic53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133217—22no—
65#13133217—41no—
66#13133216—22no—
67#13133216—41no—
68#13133216—44no—
69D4×Dic53—22no—
70D4×Dic53Q814yescomplemented
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919D411yescharacteristic
81C2×#11719919D411yescharacteristic
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133148—22no—
91#13133146—22no—
92#13133146—41no—
93#13133138—22no—
94#13133138—41no—
95#13133127—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53C2311yescharacteristic
114#107×C53—22no—
115C4⋊C8×C53—41no—
116#102×C53—22no—
117#102×C53—41no—
118#95×C53—21no—
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—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—81no—
138C22×Dic53—81no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×Dic106—42no—
144C2×Dic106—42no—
145C2×Dic106—42no—
146C2×Dic106—42no—
147C2×Dic106—81no—
148C2×Dic106—81no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53#4112yes—
162C22⋊C4×C53—42no—
163C22⋊C4×C53—81no—
164C42×C53—21no—
165C42×C53—21no—
166C22⋊Dic53—24no—
167C22⋊Dic53—42no—
168C22⋊Dic53—81no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C4⋊Dic53C2×Q812yes—
178C4⋊Dic53#4112yes—
179#11719919—22no—
180#11719919—22no—
181#11719919—42no—
182#11719919—42no—
183#11719919—81no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—42no—
187C4×Dic53—81no—
188C4×Dic53—81no—
189C4×Dic53C2×Q812yes—
190C4×Dic53#4112yes—
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—22no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53AGL(1,ℤ/8)12yes—
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—84no—
252C2×Dic53C22⋊Q812yes—
253C2×Dic53C22⋊Q812yes—
254C2×Dic53#13712yes—
255Dic106—84no—
256Dic106—84no—
257C8×C53—41no—
258C8×C53—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—42no—
278C4×C53—41no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53#53311yescharacteristic
282C4×C53#54411yescharacteristic
283C4×C53#55511yescharacteristic
284C4×C53#55611yescharacteristic
285Dic53—22no—
286Dic53—22no—
287Dic53—22no—
288Dic53—42no—
289Dic53—41no—
290Dic53—42no—
291Dic53—81no—
292Dic53—81no—
293#2551—531no—
294#2357—531no—
295#2067D5311yescharacteristic
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×D4—531no—
320C2×#107C2×D5311yescharacteristic
321C2×#106—531no—
322C2×C4⋊C8—531no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8C2×D5311yescharacteristic
325C2×#102—531no—
326C2×#102—531no—
327C2×#102C2×D5311yescharacteristic
328C2×#102C2×D5311yescharacteristic
329C2×#97—531no—
330#395—1061no—
331#395—1061no—
332#390—531no—
333#390C2×D5311yescharacteristic
334#389—531no—
335#389—1061no—
336#389C2×D5311yescharacteristic
337#388—1061no—
338#388—1061no—
339#384—531no—
340#382—531no—
341#381—1061no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C22⋊C4—1061no—
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#340C2×D5311yescharacteristic
357#336—531no—
358#336—531no—
359C53#2603611yescharacteristic, complemented, Sylow
360C22×Q8—1061no—
361C22×Q8—1061no—
362C22×D4C22×D5311yescharacteristic
363C23×C4—1061no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8C22×D5311yescharacteristic
367C22×C8C22×D5311yescharacteristic
368C4⋊Q8—1062no—
369C4⋊Q8—1062no—
370C4⋊Q8—2121no—
371C4⋊Q8—2121no—
372C4⋊Q8—2124no—
373C4×D4—534no—
374C4×D4—1062no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42C22×D5311yescharacteristic
401#107—22no—
402#106—1062no—
403C4⋊C8—41no—
404C4⋊C8—1062no—
405C4⋊C8—2121no—
406#102—22no—
407#102—41no—
408#102—1062no—
409#102—2121no—
410#97—1062no—
411#95—21no—
412#95—531no—
413#95—531no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
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×D4Q8×D5312yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—531no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1064no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—4241no—
446C22×C4—4241no—
447C22×C4D4×D5311yescharacteristic
448C22×C4D4×D5311yescharacteristic
449C22×C4C23×D5311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—532no—
469C4⋊C4—532no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—4241no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4#1171994712yes—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—1064no—
487C22⋊C4—4241no—
488C22⋊C4—2122no—
489C42—21no—
490C42—21no—
491C42—532no—
492C42—532no—
493C42—1061no—
494C42—1061no—
495C42—4241no—
496C42—4241no—
497C42—2122no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C2×D4×D5311yescharacteristic
506Q8—4244no—
507Q8—4244no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—532no—
527C2×C4—532no—
528C2×C4—532no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1062no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1062no—
540C2×C4—4244no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4#1313324112yes—
552C2×C4C2×Q8×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4C2×#1171994711yescharacteristic
556C2×C4#1313326211yescharacteristic
557C2×C4#1313326411yescharacteristic
558C8—41no—
559C8—41no—
560C8—2121no—
561C8—2121no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22C22⋊Q8×D5311yescharacteristic
570C22#1681126211yescharacteristic
571C22#1681127911yescharacteristic
572C22C2×#1313324111yescharacteristic
573C22#137×D5311yescharacteristic
574C22#1681142511yescharacteristic
575C22#1681143311yescharacteristic
576C4—22no—
577C4—42no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—1062no—
582C4—1062no—
583C4—1062no—
584C4—4241no—
585C4—4241no—
586C4—2122no—
587C4—2121no—
588C4—2122no—
589C4#1681117511yescharacteristic
590C4#1681125211yescharacteristic
591C4#1681133711yescharacteristic
592C4#1681134011yescharacteristic
593C2—82no—
594C2#2027840111yescharacteristic
595C2#2028452311yescharacteristic
596C2#2028454311yescharacteristic
597C2#2028686211yescharacteristic
598C2#2028688111yescharacteristic
599C2#2028751111yescharacteristic
600C2#2028752411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2067C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2067D53
hypocenterC53#26036
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. #20279743C2semidirect
6. #20275666C2semidirect
7. #20275661C2semidirect
8. #20275480C2semidirect
9. #20275435C2semidirect
10. #20275309C2semidirect
11. #20275290C2semidirect
12. #20274847C2semidirect
70. D4×Dic53Q8semidirect
359. C53#26036semidirect

Maximal subgroups

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

subgroupindex
#202797432
#202776872
#202760852
#202758512
#202758492
#202756662
#202756612
#202754802
#202754352
#202753092
#202752902
#202748472
#202747132
#202745422
C53×#20672
#2603653