Finite Group Databasebeta

← #27430324

Subgroups — #27430324

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279636C211yesmaximal, characteristic
2#20277670C211yesmaximal, characteristic, complemented
3#20276242C211yesmaximal, characteristic
4#20276237C211yesmaximal, characteristic
5#20276202C211yesmaximal, characteristic, complemented
6#20276186C211yesmaximal, characteristic, complemented
7#20275653C211yesmaximal, characteristic, complemented
8#20275472C211yesmaximal, characteristic, complemented
9#20275342C211yesmaximal, characteristic
10#20275333C211yesmaximal, characteristic
11#20275306C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274724C211yesmaximal, characteristic
14#20274539C211yesmaximal, characteristic
15C53×#1919C211yesmaximal, characteristic, complemented
16C2×#13133195—21no—
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30#16810547—21no—
31#16810545—21no—
32#16810529—21no—
33#16810525C2211yescharacteristic
34#16810524C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810510—21no—
37#16810510—21no—
38#16810493C2211yescharacteristic
39#16810492C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810488C2211yescharacteristic
42#16810482—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45C4×C4⋊Dic53—21no—
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C4×C4⋊C4×C53C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106D411yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133195—44no—
65C4×Dic106—22no—
66C4×Dic106—22no—
67#13133179—22no—
68#13133179—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919D411yescharacteristic
79C2×#11719919D411yescharacteristic
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133151—22no—
90#13133151—41no—
91#13133150—41no—
92#13133150—41no—
93#13133148—41no—
94#13133146—22no—
95#13133139—22no—
96#13133139—41no—
97#13133137—41no—
98#13133137Q814yescomplemented
99#13133127—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53—21no—
115C2×C42×C53C2311yescharacteristic
116#106×C53—22no—
117#95×C53—21no—
118#95×C53—21no—
119C23×D53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212C2×Q812yes—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×Dic106—22no—
144C2×Dic106—22no—
145C2×Dic106—41no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53#4112yes—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—42no—
160C4⋊C4×C53C2×Q812yes—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—42no—
164#11719921—42no—
165#11719921—42no—
166#11719921—42no—
167#11719921—81no—
168#11719921—81no—
169C4⋊Dic53—22no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—42no—
177C4⋊Dic53—81no—
178#11719919—22no—
179#11719919—22no—
180#11719919—42no—
181#11719919—42no—
182#11719919—81no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—81no—
186C4×Dic53—81no—
187C4×Dic53#4112yes—
188C4×Dic53#4112yes—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322C2×Q812yes—
195C23×C53C22×D411yescharacteristic
196C22×D53—22no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53AGL(1,ℤ/8)12yes—
221C2×C4×C53C22×Q811yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53#12312yes—
247C2×Dic53#12312yes—
248D212—22no—
249D212—22no—
250D212—41no—
251D212—41no—
252C4×D53—84no—
253Dic106—44no—
254C8×C53—22no—
255C8×C53—41no—
256#88322—22no—
257#88322—41no—
258#25072—531nomaximal, Sylow
259C22×C53C2×#12311yescharacteristic
260C22×C53D4×Q811yescharacteristic
261C22×C53#55011yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57611yescharacteristic
265C22×C53#57811yescharacteristic
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C2×D53—81no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53#53311yescharacteristic
280C4×C53#54211yescharacteristic
281C4×C53#55211yescharacteristic
282C4×C53#55411yescharacteristic
283Dic53—22no—
284Dic53—22no—
285Dic53—41no—
286Dic53—41no—
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290#2538—531no—
291#2355—531no—
292#2093—531no—
293#2092—531no—
294#2085—531no—
295#2083—531no—
296#1985—531no—
297#1947—531no—
298#1926—531no—
299#1925—531no—
300#1921—531no—
301#1919D5311yescharacteristic, complemented
302#1847—531no—
303#1804—531no—
304#1762—531no—
305C2×C53#242311yescharacteristic
306C2×C53#309411yescharacteristic
307C2×C53#309811yescharacteristic
308C2×C53#338611yescharacteristic
309C2×C53#338911yescharacteristic
310C2×C53#339911yescharacteristic
311C2×C53#340211yescharacteristic
312D53—82no—
313C2×C22⋊Q8—1061no—
314C2×C4×Q8—531no—
315C2×C4×D4—531no—
316C2×#107—531no—
317C2×#107—531no—
318C2×#107—531no—
319C2×#107—531no—
320C2×#106—531no—
321C2×#106C2×D5311yescharacteristic
322C2×C4⋊C8—531no—
323C2×#102—531no—
324C2×#102—531no—
325C2×#102—531no—
326C2×#102—531no—
327C2×#97—531no—
328#396—1061no—
329#396—1061no—
330#391—1061no—
331#390—531no—
332#390—531no—
333#389—531no—
334#389C2×D5311yescharacteristic
335#387—1061no—
336#387—1061no—
337#384—531no—
338#384—531no—
339#382—531no—
340#382—531no—
341#380—1061no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—1061no—
345C4×C4⋊C4C2×D5311yescharacteristic
346#340—531no—
347#340—531no—
348#340—531no—
349#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#340C2×D5311yescharacteristic
352#336—531no—
353#336—531no—
354#336—531no—
355#336C2×D5311yescharacteristic
356C53#2507211yescharacteristic, complemented, Sylow
357C22×Q8—531no—
358C22×D4—531no—
359C23×C4—1061no—
360C22×C8—531no—
361C22×C8—531no—
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C22⋊Q8—2124no—
365C4×Q8—1062no—
366C4×Q8—1062no—
367C4×D4—1062no—
368C4×D4—1062no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C42—21no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42C22×D5311yescharacteristic
396#107—1062no—
397#107—2121no—
398#107—2121no—
399#107—2121no—
400#106—22no—
401#106—2121no—
402C4⋊C8—1062no—
403#102—534no—
404#102—1062no—
405#102—2121no—
406#102—2121no—
407#97—1062no—
408#95—21no—
409#95—21no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—1061no—
418C2×Q8—1062no—
419C2×Q8—1062no—
420C2×Q8—2121no—
421C2×D4—532no—
422C2×D4—1061no—
423C2×D4—1061no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1061no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—2122no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4D4×D5311yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—532no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8#1171994512yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—1062no—
465C4⋊C4—1062no—
466C4⋊C4—1062no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—4241no—
474C4⋊C4—4241no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4Q8×D5312yes—
479C22⋊C4—4241no—
480C22⋊C4—4241no—
481C22⋊C4—2122no—
482C22⋊C4—2122no—
483C22⋊C4—2122no—
484C42—21no—
485C42—21no—
486C42—42no—
487C42—532no—
488C42—532no—
489C42—1061no—
490C42—1061no—
491C42—4241no—
492C42—4241no—
493C23—1062no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23—2121no—
498C23—2121no—
499C23—2121no—
500C23C2×D4×D5311yescharacteristic
501Q8—2124no—
502D4—1062no—
503D4—1062no—
504D4—2121no—
505D4—2121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—4244no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4#1313324412yes—
546C2×C4C2×Q8×D5311yescharacteristic
547C2×C4#41×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C2×C4C2×#1171994511yescharacteristic
550C2×C4#1313326311yescharacteristic
551C2×C4#1313326411yescharacteristic
552C8—22no—
553C8—41no—
554C8—1062no—
555C8—2121no—
556C22—4241no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22—2122no—
561C22—2122no—
562C22—2122no—
563C22#123×D5311yescharacteristic
564C22#1681128611yescharacteristic
565C22#1681129411yescharacteristic
566C22C2×#1313324411yescharacteristic
567C22#1681141911yescharacteristic
568C22#1681142511yescharacteristic
569C22#1681143911yescharacteristic
570C4—22no—
571C4—41no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—1062no—
577C4—1062no—
578C4—4241no—
579C4—4241no—
580C4—4241no—
581C4—2121no—
582C4—2121no—
583C4#1681117711yescharacteristic
584C4#1681123911yescharacteristic
585C4#1681130611yescharacteristic
586C4#1681132911yescharacteristic
587C2—4242no—
588C2#2027823911yescharacteristic
589C2#2028465211yescharacteristic
590C2#2028469611yescharacteristic
591C2#2028763511yescharacteristic
592C2#2028767211yescharacteristic
593C2#2028775511yescharacteristic
594C2#2028779511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1919C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1919D53
hypocenterC53#25072
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
2. #20277670C2semidirect
5. #20276202C2semidirect
6. #20276186C2semidirect
7. #20275653C2semidirect
8. #20275472C2semidirect
11. #20275306C2semidirect
12. #20274880C2semidirect
15. C53×#1919C2semidirect
98. #13133137Q8semidirect
301. #1919D53semidirect
356. C53#25072semidirect

Maximal subgroups

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

subgroupindex
#202796362
#202776702
#202762422
#202762372
#202762022
#202761862
#202756532
#202754722
#202753422
#202753332
#202753062
#202748802
#202747242
#202745392
C53×#19192
#2507253