Finite Group Databasebeta

← #27469697

Subgroups — #27469697

Order 13568.

734 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#20283024C211yesmaximal, characteristic
2#382×D53C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276283C211yesmaximal, characteristic, complemented
5#20276282C211yesmaximal, characteristic, complemented
6#20275678C211yesmaximal, characteristic, complemented
7#20275678C211yesmaximal, characteristic, complemented
8#20274844C211yesmaximal, characteristic, complemented
9#20274844C211yesmaximal, characteristic, complemented
10#20274804C211yesmaximal, characteristic
11#20274804C211yesmaximal, characteristic
12#20274800C211yesmaximal, characteristic
13#20274800C211yesmaximal, characteristic
14#20274472C211yesmaximal, characteristic
15C53×#2101C211yesmaximal, characteristic, complemented
16C2×#13133208C2211yescharacteristic
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C42×D53C2211yescharacteristic
20C2×#13133129C2211yescharacteristic
21C2×#13133129C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810509—21no—
26#16810505—21no—
27#16810497—21no—
28#16810494—21no—
29#16810489C2211yescharacteristic
30#16810488C2211yescharacteristic
31#16810488C2211yescharacteristic
32#16810488C2211yescharacteristic
33#16810487—21no—
34#16810487—21no—
35#16810480C2211yescharacteristic
36#16810480C2211yescharacteristic
37#95×D53—21no—
38#95×D53—21no—
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#382C2211yescharacteristic
54C53×#340C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D411yescharacteristic
63C22×C4×D53C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C4⋊C4×D53—22no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53—22no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53—22no—
87C42×D53—22no—
88#13133129—22no—
89#13133129—22no—
90C8×Dic53—22no—
91C8×Dic53—22no—
92#13133124—21no—
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—
102#13133124—21no—
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114C23×D53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2411yescharacteristic
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×C4×D53#4112yes—
146C2×C4×D53#4112yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×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—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—42no—
162C42×C53—21no—
163C42×C53—21no—
164#11719921—42no—
165#11719921—42no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175#11719919—42no—
176#11719919—42no—
177#11719919—42no—
178#11719919—42no—
179C4×Dic53—22no—
180C4×Dic53—22no—
181C4×Dic53#4112yes—
182C4×Dic53#4112yes—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C23×C53C22×D411yescharacteristic
190C22×D53—21no—
191C22×D53—21no—
192C22×D53—21no—
193C22×D53—21no—
194C22×D53#12412yes—
195C22×D53#13512yes—
196C22×D53#13512yes—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
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×#4111yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53#12312yes—
240C2×Dic53#12312yes—
241C2×Dic53#12412yes—
242C4×D53—22no—
243C4×D53—22no—
244C4×D53—22no—
245C4×D53—22no—
246C4×D53—44no—
247C4×D53—44no—
248C4×D53—84no—
249C4×D53—84no—
250C8×C53—41no—
251C8×C53—41no—
252#88322—41no—
253#88322—41no—
254#27964—531nomaximal, Sylow
255C22×C53C2×#12311yescharacteristic
256C22×C53C2×#12411yescharacteristic
257C22×C53#53611yescharacteristic
258C22×C53C2×#13511yescharacteristic
259C22×C53C2×#13511yescharacteristic
260C22×C53#57511yescharacteristic
261C22×C53#57511yescharacteristic
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53#48814yes—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#55211yescharacteristic
276C4×C53#55211yescharacteristic
277C4×C53#55211yescharacteristic
278C4×C53#55211yescharacteristic
279Dic53—22no—
280Dic53—22no—
281Dic53—22no—
282Dic53—81no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286#2851—531no—
287C2×#382—531no—
288#2101—531no—
289#2101—531no—
290#2101—531no—
291#2101D5311yescharacteristic, complemented
292#1991—531no—
293#1991—531no—
294#1835—531no—
295#1835—531no—
296#1824—531no—
297#1824—531no—
298#1823—531no—
299#1823—531no—
300C4×C22⋊C8—531no—
301C2×C53#250311yescharacteristic
302C2×C53#309111yescharacteristic
303C2×C53#309111yescharacteristic
304C2×C53C2×#48811yescharacteristic
305C2×C53#314011yescharacteristic
306C2×C53#323211yescharacteristic
307C2×C53#323211yescharacteristic
308D53—24no—
309C2×#125—531no—
310C22×C4⋊C4—531no—
311C22×C4⋊C4—531no—
312C22×C42—531no—
313C2×C22⋊C8—531no—
314C2×C22⋊C8—531no—
315C2×C4×C8—531no—
316C2×C4×C8—531no—
317#389—531no—
318#389C2×D5311yescharacteristic
319#387—1061no—
320#387—1061no—
321#385—1061no—
322#385—1061no—
323#382—531no—
324#382—531no—
325#382—531no—
326#382—531no—
327#382—1061no—
328#382—1061no—
329#382C2×D5311yescharacteristic
330#380—531no—
331#380—531no—
332C2×#95—1061no—
333C2×#95—1061no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—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#2796411yescharacteristic, complemented, Sylow
353C23×C4—531no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—1061no—
357C23×C4—1061no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362#125—1062no—
363#125—1062no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1061no—
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×C4⋊C4C22×D5311yescharacteristic
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C42—531no—
385C2×C42—531no—
386C2×C42—531no—
387C2×C42—1062no—
388C2×C42—1062no—
389C2×C42C22×D5311yescharacteristic
390C22⋊C8—1062no—
391C22⋊C8—1062no—
392C4×C8—1062no—
393C4×C8—1062no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408C24—531no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—532no—
417C22×C4—532no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1062no—
421C22×C4—1062no—
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—2121no—
431C22×C4—2121no—
432C22×C4—2122no—
433C22×C4—2122no—
434C22×C4—2122no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4—2122no—
438C22×C4D4×D5311yescharacteristic
439C22×C4D4×D5311yescharacteristic
440C22×C4C23×D5311yescharacteristic
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—2122no—
471C4⋊C4—2122no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C22⋊C4—2122no—
476C22⋊C4—2122no—
477C42—21no—
478C42—21no—
479C42—532no—
480C42—532no—
481C42—1062no—
482C42—1062no—
483C23—532no—
484C23—532no—
485C23—532no—
486C23—1061no—
487C23—1061no—
488C23—1061no—
489C23—1061no—
490C23C2×D4×D5311yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
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—532no—
510C2×C4—532no—
511C2×C4—532no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1062no—
522C2×C4—4244no—
523C2×C4—4244no—
524C2×C4—2124no—
525C2×C4—2124no—
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—2121no—
538C2×C4#41×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#41×D5311yescharacteristic
541C2×C4#41×D5311yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—41no—
545C8—41no—
546C8—2121no—
547C8—2121no—
548C22—534no—
549C22—1062no—
550C22—1062no—
551C22—1062no—
552C22—1062no—
553C22—1062no—
554C22—1062no—
555C22#123×D5311yescharacteristic
556C22#124×D5311yescharacteristic
557C22#1681119611yescharacteristic
558C22#135×D5311yescharacteristic
559C22#135×D5311yescharacteristic
560C22#1681141911yescharacteristic
561C22#1681142211yescharacteristic
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—1062no—
569C4—1062no—
570C4—1062no—
571C4—4241no—
572C4—4241no—
573C4—4241no—
574C4—4241no—
575C4#1681130511yescharacteristic
576C4#1681130511yescharacteristic
577C4#1681130911yescharacteristic
578C4#1681130911yescharacteristic
579C2—1064no—
580C2#2027919511yescharacteristic
581C2#2028462711yescharacteristic
582C2#2028463311yescharacteristic
583C2#488×D5311yescharacteristic
584C2#2028510011yescharacteristic
585C2#2028588111yescharacteristic
586C2#2028588711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#27964
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #20276284C2semidirect
4. #20276283C2semidirect
5. #20276282C2semidirect
6. #20275678C2semidirect
7. #20275678C2semidirect
8. #20274844C2semidirect
9. #20274844C2semidirect
15. C53×#2101C2semidirect
291. #2101D53semidirect
352. C53#27964semidirect

Maximal subgroups

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

subgroupindex
#202830242
#382×D532
#202762842
#202762832
#202762822
#202756782
#202748442
#202748042
#202748002
#202744722
C53×#21012
#2796453