Finite Group Databasebeta

← #27512578

Subgroups — #27512578

Order 13568.

698 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#20279140C211yesmaximal, characteristic
2#20278419C211yesmaximal, characteristic
3#20276276C211yesmaximal, characteristic, complemented
4#20276207C211yesmaximal, characteristic, complemented
5#20276142C211yesmaximal, characteristic
6#20276140C211yesmaximal, characteristic
7#20276044C211yesmaximal, characteristic, complemented
8#20275675C211yesmaximal, characteristic, complemented
9#20275472C211yesmaximal, characteristic, complemented
10#20275445C211yesmaximal, characteristic, complemented
11#20275319C211yesmaximal, characteristic
12#20275285C211yesmaximal, characteristic, complemented
13#20274904C211yesmaximal, characteristic
14#20274800C211yesmaximal, characteristic
15C53×#2099C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133212—21no—
18C2×#13133195—21no—
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26C2×#13133129C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810561—21no—
29#16810557—21no—
30#16810549—21no—
31#16810543—21no—
32#16810522C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810509—21no—
36#16810497—21no—
37#16810492C2211yescharacteristic
38#16810480C2211yescharacteristic
39#16810480C2211yescharacteristic
40#16810480C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×#11719921C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#384C2211yescharacteristic
54C53×#340C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133217—22no—
65#13133217—41no—
66#13133212—44no—
67#13133195—44no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53—41no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921D411yescharacteristic
85C2×#11719921D411yescharacteristic
86C2×C4×Dic53—21no—
87C2×C4×Dic53C2311yescharacteristic
88#13133151—22no—
89#13133151—41no—
90#13133150—22no—
91#13133150—41no—
92#13133146—22no—
93#13133129—22no—
94#13133129—41no—
95C4×#88322—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116C23×D53C2×D411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2411yescharacteristic
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—21no—
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—42no—
138C2×C4×D53—42no—
139C2×C4×D53—81no—
140C2×Dic106—42no—
141C2×Dic106—42no—
142C2×Dic106—81no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×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—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—42no—
158C4⋊C4×C53—42no—
159C4⋊C4×C53—42no—
160C42×C53—21no—
161C42×C53—21no—
162#11719921—22no—
163#11719921—42no—
164#11719921—42no—
165#11719921—42no—
166#11719921D814yes—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53—42no—
176C4⋊Dic53#4112yes—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—81no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—42no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322#4112yes—
191C23×C53C22×D411yescharacteristic
192C22×D53—21no—
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53C2×D812yes—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
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×C53C2×D812yes—
216C2×C4×C53#13712yes—
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53H(5,𝔽2)11yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×C4×C532−1+411yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—22no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
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×Dic53C2×D812yes—
244C4×D53—41no—
245C4×D53—41no—
246C4×D53—41no—
247C4×D53—41no—
248C4×D53—84no—
249Dic106—84no—
250C8×C53—41no—
251C8×C53—41no—
252#88322—22no—
253#88322—41no—
254#31422—531nomaximal, Sylow
255C22×C53#53611yescharacteristic
256C22×C53#54611yescharacteristic
257C22×C53#54711yescharacteristic
258C22×C53C22×D811yescharacteristic
259C22×C53C2×#13711yescharacteristic
260C22×C53#57711yescharacteristic
261C22×C53#57811yescharacteristic
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—41no—
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#53911yescharacteristic
276C4×C53#54111yescharacteristic
277C4×C53#55611yescharacteristic
278C4×C53#56311yescharacteristic
279Dic53—22no—
280Dic53—41no—
281Dic53—41no—
282Dic53—42no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286#2496—531no—
287#2440—531no—
288#2099—531no—
289#2099D5311yescharacteristic, complemented
290#2086—531no—
291#2075—531no—
292#2075—531no—
293#2057—531no—
294#1990—531no—
295#1947—531no—
296#1942—531no—
297#1923—531no—
298#1918—531no—
299#1852—531no—
300#1823—531no—
301C2×C53#268911yescharacteristic
302C2×C53#322611yescharacteristic
303C2×C53#324211yescharacteristic
304C2×C53#329411yescharacteristic
305C2×C53#331211yescharacteristic
306C2×C53#333411yescharacteristic
307C2×C53#335611yescharacteristic
308D53—42no—
309C2×C4⋊Q8—531no—
310C2×#126—1061no—
311C2×C22⋊Q8—1061no—
312C22×C4⋊C4—531no—
313C2×#107—531no—
314C2×#107—531no—
315C2×#107—531no—
316C2×#107—531no—
317C2×C4⋊C8—531no—
318C2×C22⋊C8—531no—
319C2×C22⋊C8—531no—
320C2×C4×C8—531no—
321#400—1061no—
322#398—1061no—
323#397—1061no—
324#395—1061no—
325#389—531no—
326#389—531no—
327#389—531no—
328#389C2×D5311yescharacteristic
329#387—1061no—
330#385—1061no—
331#384—531no—
332#384C2×D5311yescharacteristic
333#380—531no—
334#380—531no—
335#380—531no—
336C4×C4⋊C4—531no—
337C4×C22⋊C4—531no—
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#340C2×D5311yescharacteristic
349#340C2×D5311yescharacteristic
350#336—531no—
351#336C2×D5311yescharacteristic
352C53#3142211yescharacteristic, complemented, Sylow
353C22×Q8—1061no—
354C23×C4—531no—
355C23×C4—1061no—
356C22×C8—531no—
357C22×C8—531no—
358C22×C8C22×D5311yescharacteristic
359C22×C8C22×D5311yescharacteristic
360C4⋊Q8—1062no—
361C4⋊Q8—2121no—
362#126—2124no—
363C22⋊Q8—2124no—
364C2×C4⋊C4—21no—
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—1062no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—2121no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—531no—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C42—531no—
390C2×C42—1061no—
391C2×C42C22×D5311yescharacteristic
392#107—1062no—
393#107—1062no—
394#107—2121no—
395#107—2121no—
396C4⋊C8—1062no—
397C22⋊C8—1062no—
398C22⋊C8—2121no—
399C4×C8—2121no—
400#95—21no—
401#95—531no—
402#95—531no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410C24—531no—
411C2×Q8—4241no—
412C2×Q8—2122no—
413C2×Q8—2122no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—531no—
419C22×C4—531no—
420C22×C4—531no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—4241no—
432C22×C4—2122no—
433C22×C4—2122no—
434C22×C4D4×D5311yescharacteristic
435C22×C4D4×D5311yescharacteristic
436C22×C4C23×D5311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—532no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—532no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—4241no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C4⋊C4—2122no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C22⋊C4—534no—
476C22⋊C4—1062no—
477C22⋊C4—2122no—
478C22⋊C4—2122no—
479C22⋊C4—2122no—
480C42—21no—
481C42—21no—
482C42—1061no—
483C42—1061no—
484C42—2122no—
485C23—532no—
486C23—1061no—
487C23—1061no—
488C23—1061no—
489C23—1061no—
490C23—1061no—
491C23—1061no—
492C23C2×D4×D5311yescharacteristic
493Q8—4244no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—532no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1062no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—4244no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
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×C4D8×D5312yes—
537C2×C4#1313324612yes—
538C2×C4#41×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#1313326311yescharacteristic
541C2×C4#1313326411yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—41no—
545C8—41no—
546C8—1062no—
547C8—2121no—
548C22—1062no—
549C22—1062no—
550C22—1062no—
551C22—2121no—
552C22—2121no—
553C22—2121no—
554C22—2121no—
555C22#1681119311yescharacteristic
556C22#1681126511yescharacteristic
557C22#1681127511yescharacteristic
558C22C2×D8×D5311yescharacteristic
559C22C2×#1313324611yescharacteristic
560C22#1681143211yescharacteristic
561C22#1681143911yescharacteristic
562C4—22no—
563C4—22no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—1062no—
569C4—4241no—
570C4—4241no—
571C4—4241no—
572C4—2122no—
573C4—2121no—
574C4—2121no—
575C4#1681121411yescharacteristic
576C4#1681123211yescharacteristic
577C4#1681134011yescharacteristic
578C4#1681137711yescharacteristic
579C2—2122no—
580C2#2028115911yescharacteristic
581C2#2028583211yescharacteristic
582C2#2028597811yescharacteristic
583C2#2028652611yescharacteristic
584C2#2028676411yescharacteristic
585C2#2028706111yescharacteristic
586C2#2028732711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2099C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2099D53
hypocenterC53#31422
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
3. #20276276C2semidirect
4. #20276207C2semidirect
7. #20276044C2semidirect
8. #20275675C2semidirect
9. #20275472C2semidirect
10. #20275445C2semidirect
12. #20275285C2semidirect
15. C53×#2099C2semidirect
289. #2099D53semidirect
352. C53#31422semidirect

Maximal subgroups

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

subgroupindex
#202791402
#202784192
#202762762
#202762072
#202761422
#202761402
#202760442
#202756752
#202754722
#202754452
#202753192
#202752852
#202749042
#202748002
C53×#20992
#3142253