Finite Group Databasebeta

← #27429440

Subgroups — #27429440

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279635C211yesmaximal, characteristic
2#20277674C211yesmaximal, characteristic, complemented
3#20276288C211yesmaximal, characteristic, complemented
4#20276272C211yesmaximal, characteristic, complemented
5#20276131C211yesmaximal, characteristic
6#20275653C211yesmaximal, characteristic, complemented
7#20275477C211yesmaximal, characteristic, complemented
8#20275344C211yesmaximal, characteristic
9#20275339C211yesmaximal, characteristic
10#20275303C211yesmaximal, characteristic, complemented
11#20275294C211yesmaximal, characteristic, complemented
12#20274847C211yesmaximal, characteristic, complemented
13#20274723C211yesmaximal, characteristic
14#20274542C211yesmaximal, characteristic
15C53×#2071C211yesmaximal, characteristic
16C2×#13133194—21no—
17C2×C4×Dic106C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133127C2211yescharacteristic
25#16810546—21no—
26#16810546—21no—
27#16810528—21no—
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810504—21no—
32#16810504—21no—
33#16810492C2211yescharacteristic
34#16810488C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810481—21no—
37C4⋊C4×Dic53—21no—
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#102×C53C2211yescharacteristic
53C2×#102×C53C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#384C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106D411yescharacteristic
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133194—44no—
64C4×Dic106—22no—
65C4×Dic106—22no—
66D4×Dic53—22no—
67D4×Dic53—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919D411yescharacteristic
78C2×#11719919D411yescharacteristic
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133149—22no—
88#13133148—22no—
89#13133146—22no—
90#13133138—22no—
91#13133138—22no—
92#13133127—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C22×D4×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C42×C53C2311yescharacteristic
113#102×C53—22no—
114#102×C53—22no—
115C24×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53#4112yes—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—42no—
133C22×Dic53—42no—
134C22×Dic53—42no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C2×Dic106—22no—
140C2×Dic106—22no—
141C2×Dic106—41no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—42no—
155C4⋊C4×C53—81no—
156C22⋊C4×C53—42no—
157C22⋊C4×C53—81no—
158C42×C53—21no—
159C42×C53—21no—
160C22⋊Dic53—42no—
161C22⋊Dic53—42no—
162C22⋊Dic53—81no—
163C4⋊Dic53—21no—
164C4⋊Dic53—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—42no—
172C4⋊Dic53#4112yes—
173#11719919—22no—
174#11719919—22no—
175#11719919—42no—
176#11719919—42no—
177#11719919—81no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—42no—
181C4×Dic53—81no—
182C4×Dic53—81no—
183C4×Dic53#4112yes—
184C4×Dic53#4112yes—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C23×C53—22no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53C22×D411yescharacteristic
199D4×C53—22no—
200D4×C53—22no—
201D4×C53—41no—
202D4×C53—41no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—41no—
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×C53AGL(1,ℤ/8)12yes—
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×C4×C532−1+411yescharacteristic
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—22no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—41no—
236C2×Dic53—41no—
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—84no—
247C2×Dic53#12312yes—
248C2×Dic53#12312yes—
249Dic106—44no—
250C8×C53—41no—
251C8×C53—41no—
252#88322—41no—
253#88322—41no—
254#25006—531nomaximal, Sylow
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53C2×#12311yescharacteristic
263C22×C53#54811yescharacteristic
264C22×C53#54811yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57711yescharacteristic
268C22×C53#57711yescharacteristic
269C4×C53—22no—
270C4×C53—41no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53#52911yescharacteristic
275C4×C53#54111yescharacteristic
276C4×C53#55211yescharacteristic
277C4×C53#55611yescharacteristic
278Dic53—22no—
279Dic53—22no—
280Dic53—41no—
281Dic53—41no—
282Dic53—42no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286#2538—531no—
287#2355—531no—
288#2101—531no—
289#2099—531no—
290#2073—531no—
291#2071D5311yescharacteristic
292#1985—531no—
293#1948—531no—
294#1926—531no—
295#1926—531no—
296#1920—531no—
297#1919—531no—
298#1836—531no—
299#1804—531no—
300#1762—531no—
301C2×C53—82no—
302C2×C53#242111yescharacteristic
303C2×C53#309411yescharacteristic
304C2×C53#309911yescharacteristic
305C2×C53#336811yescharacteristic
306C2×C53#337311yescharacteristic
307C2×C53#337611yescharacteristic
308C2×C53#337611yescharacteristic
309C2×C22⋊Q8—1061no—
310C2×C4×Q8—531no—
311C2×C4×D4—531no—
312C2×#106—531no—
313C2×#106—531no—
314C2×C4⋊C8—531no—
315C2×#102—531no—
316C2×#102—531no—
317C2×#102C2×D5311yescharacteristic
318C2×#102C2×D5311yescharacteristic
319C2×#97—531no—
320#396—1061no—
321#396—1061no—
322#391—1061no—
323#390—531no—
324#390C2×D5311yescharacteristic
325#389—531no—
326#389—531no—
327#387—1061no—
328#387—1061no—
329#384—531no—
330#384C2×D5311yescharacteristic
331#382—531no—
332#382—531no—
333#380—1061no—
334C4×C4⋊C4—531no—
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4—1061no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340C2×D5311yescharacteristic
347#340C2×D5311yescharacteristic
348#336—531no—
349#336—531no—
350#336—531no—
351#336C2×D5311yescharacteristic
352C53#2500611yescharacteristic, complemented, Sylow
353C22×Q8—531no—
354C22×D4C22×D5311yescharacteristic
355C23×C4—1061no—
356C22×C8—531no—
357C22×C8—531no—
358C22×C8C22×D5311yescharacteristic
359C22×C8C22×D5311yescharacteristic
360C22⋊Q8—2124no—
361C4×Q8—1062no—
362C4×Q8—1062no—
363C4×D4—1062no—
364C4×D4—1062no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—531no—
387C2×C42—531no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42C22×D5311yescharacteristic
392#106—1062no—
393#106—1062no—
394C4⋊C8—1062no—
395#102—22no—
396#102—22no—
397#102—1062no—
398#102—1062no—
399#97—1062no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409C24—21no—
410C2×Q8—1062no—
411C2×Q8—1062no—
412C2×Q8—2121no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4#1171994712yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—2122no—
431C22×C4—2122no—
432C22×C4—2122no—
433C22×C4D4×D5311yescharacteristic
434C22×C4D4×D5311yescharacteristic
435C22×C4C23×D5311yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—532no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4—1062no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—4241no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—4241no—
474C22⋊C4—2122no—
475C22⋊C4—2122no—
476C42—21no—
477C42—21no—
478C42—532no—
479C42—532no—
480C42—1061no—
481C42—1061no—
482C42—4241no—
483C42—4241no—
484C42—2122no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D5311yescharacteristic
493Q8—2124no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—532no—
513C2×C4—532no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1062no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—4244no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4#1313324112yes—
538C2×C4#41×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#41×D5311yescharacteristic
541C2×C4C2×#1171994711yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326411yescharacteristic
544C8—41no—
545C8—41no—
546C8—2121no—
547C8—2121no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#123×D5311yescharacteristic
556C22#1681127911yescharacteristic
557C22#1681128111yescharacteristic
558C22C2×#1313324111yescharacteristic
559C22#1681142211yescharacteristic
560C22#1681142911yescharacteristic
561C22#1681143311yescharacteristic
562C4—22no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—1062no—
568C4—1062no—
569C4—4241no—
570C4—4241no—
571C4—4241no—
572C4—2121no—
573C4—2122no—
574C4—2121no—
575C4#1681115011yescharacteristic
576C4#1681123011yescharacteristic
577C4#1681131011yescharacteristic
578C4#1681134011yescharacteristic
579C2—82no—
580C2#2027821111yescharacteristic
581C2#2028465011yescharacteristic
582C2#2028470511yescharacteristic
583C2#2028744911yescharacteristic
584C2#2028751111yescharacteristic
585C2#2028753511yescharacteristic
586C2#2028754111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2071C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2071D53
hypocenterC53#25006
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #20277674C2semidirect
3. #20276288C2semidirect
4. #20276272C2semidirect
6. #20275653C2semidirect
7. #20275477C2semidirect
10. #20275303C2semidirect
11. #20275294C2semidirect
12. #20274847C2semidirect
352. C53#25006semidirect

Maximal subgroups

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

subgroupindex
#202796352
#202776742
#202762882
#202762722
#202761312
#202756532
#202754772
#202753442
#202753392
#202753032
#202752942
#202748472
#202747232
#202745422
C53×#20712
#2500653