Finite Group Databasebeta

← #27442163

Subgroups — #27442163

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278442C211yesmaximal, characteristic
2#20277599C211yesmaximal, characteristic
3#20276219C211yesmaximal, characteristic, complemented
4#20276216C211yesmaximal, characteristic, complemented
5#20276052C211yesmaximal, characteristic, complemented
6#20275671C211yesmaximal, characteristic, complemented
7#20275481C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20275291C211yesmaximal, characteristic, complemented
10#20274909C211yesmaximal, characteristic
11#20274899C211yesmaximal, characteristic
12#20274808C211yesmaximal, characteristic
13#20274747C211yesmaximal, characteristic
14#20274433C211yesmaximal, characteristic
15C53×#2101C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133129C2211yescharacteristic
27#16810564—21no—
28#16810557—21no—
29#16810522C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810515—21no—
33#16810513—21no—
34#16810497—21no—
35#16810488—21no—
36#16810482—21no—
37#16810480C2211yescharacteristic
38#16810480C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×#11719921C2211yescharacteristic
42#16810468—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#382C2211yescharacteristic
54C53×#340C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53C2311yescharacteristic
61C2×#11719912—22no—
62C2×#11719912—22no—
63C2×#11719912—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—41no—
68#13133175—22no—
69#13133175Q814yes—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53—41no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921D411yescharacteristic
86C2×#11719921D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91#13133150—41no—
92#13133149—22no—
93#13133148—22no—
94#13133148—41no—
95C4⋊#88322—22no—
96#13133146—41no—
97#13133129—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
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×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118C23×D53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
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×C4×D53C2×Q812yes—
141C2×C4×D53#4112yes—
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—21no—
155C4⋊C4×C53—81no—
156C4⋊C4×C53C2×Q812yes—
157C42×C53—21no—
158C42×C53—21no—
159#11719921—24no—
160#11719921—41no—
161#11719921—41no—
162#11719921—42no—
163#11719921—81no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—42no—
171C4⋊Dic53—81no—
172C4⋊Dic53C2×Q812yes—
173C4⋊Dic53#4112yes—
174#11719919—42no—
175#11719919—42no—
176#11719919—81no—
177#11719919—81no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—24no—
181C4×Dic53—42no—
182C4×Dic53—42no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322#4112yes—
189#11719912—44no—
190C23×C53C22×D411yescharacteristic
191C22×D53—21no—
192C22×D53—21no—
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53C22⋊Q812yes—
198C2×C4×C53—22no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
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×C53—41no—
216C2×C4×C53C22×Q811yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53H(5,𝔽2)11yescharacteristic
221C2×C4×C532−1+411yescharacteristic
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—21no—
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×Dic53C22⋊Q812yes—
243C4×D53—22no—
244C4×D53—22no—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—84no—
248C8×C53—41no—
249C8×C53—41no—
250#88322—22no—
251#88322—41no—
252#26019—531nomaximal, Sylow
253C22×C53C2×C22⋊Q811yescharacteristic
254C22×C53#54611yescharacteristic
255C22×C53#54711yescharacteristic
256C22×C53#57511yescharacteristic
257C22×C53#57511yescharacteristic
258C22×C53#57711yescharacteristic
259C22×C53#57711yescharacteristic
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—41no—
264C2×D53—41no—
265C2×D53—41no—
266C2×D53—41no—
267C4×C53—42no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—81no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53#53311yescharacteristic
274C4×C53#54411yescharacteristic
275C4×C53#55111yescharacteristic
276C4×C53#55211yescharacteristic
277Dic53—22no—
278Dic53—42no—
279Dic53—41no—
280Dic53—42no—
281Dic53—41no—
282Dic53—81no—
283Dic53—81no—
284#2442—531no—
285#2346—531no—
286#2101D5311yescharacteristic, complemented
287#2088—531no—
288#2088—531no—
289#2059—531no—
290#1989—531no—
291#1949—531no—
292#1919—531no—
293#1919—531no—
294#1853—531no—
295#1851—531no—
296#1825—531no—
297#1809—531no—
298#1736—531no—
299C2×C53#243811yescharacteristic
300C2×C53#308011yescharacteristic
301C2×C53#308411yescharacteristic
302C2×C53#332111yescharacteristic
303C2×C53#332311yescharacteristic
304C2×C53#334811yescharacteristic
305C2×C53#335111yescharacteristic
306D53—42no—
307C22×M16—531no—
308C2×#125—531no—
309C2×#117—531no—
310C22×C4⋊C4—531no—
311C2×#107—531no—
312C2×#106—531no—
313C2×#106—531no—
314C2×#106—531no—
315C2×C4⋊C8—531no—
316C2×C4⋊C8—531no—
317C2×C22⋊C8—531no—
318#400—1061no—
319#398—1061no—
320#389—531no—
321#389—531no—
322#389—531no—
323#389C2×D5311yescharacteristic
324#388—1061no—
325#388—1061no—
326#385—1061no—
327#382—1061no—
328#382C2×D5311yescharacteristic
329#380—531no—
330#380—531no—
331#380—1061no—
332C4×C4⋊C4—531no—
333C4×C4⋊C4—531no—
334C4×C22⋊C4—531no—
335#376—1061no—
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#340C2×D5311yescharacteristic
345#340C2×D5311yescharacteristic
346#340C2×D5311yescharacteristic
347#340C2×D5311yescharacteristic
348#336—531no—
349#336C2×D5311yescharacteristic
350C53#2601911yescharacteristic, complemented, Sylow
351C23×C4—531no—
352C23×C4—1061no—
353C2×M16—1062no—
354C2×M16—1062no—
355C2×M16—2121no—
356C22×C8—531no—
357C22×C8—531no—
358C22×C8C22×D5311yescharacteristic
359C22×C8C22×D5311yescharacteristic
360#125—1062no—
361#125—2121no—
362#117—534no—
363#117—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—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—2121no—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—531no—
382C2×C22⋊C4—531no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—531no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42C22×D5311yescharacteristic
391#107—2121no—
392#106—1062no—
393#106—1062no—
394#106—2121no—
395C4⋊C8—1062no—
396C4⋊C8—2121no—
397C22⋊C8—1062no—
398#95—21no—
399#95—21no—
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—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—531no—
416C22×C4—532no—
417C22×C4—532no—
418C22×C4—531no—
419C22×C4—531no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—4241no—
429C22×C4—2122no—
430C22×C4—2122no—
431C22×C4D4×D5311yescharacteristic
432C22×C4D4×D5311yescharacteristic
433C22×C4C23×D5311yescharacteristic
434M16—2124no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—532no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—81no—
455C4⋊C4—532no—
456C4⋊C4—532no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—4241no—
464C4⋊C4—4241no—
465C4⋊C4—4241no—
466C4⋊C4—2122no—
467C4⋊C4—2122no—
468C4⋊C4—2122no—
469C4⋊C4Q8×D5312yes—
470C22⋊C4—1064no—
471C22⋊C4—4241no—
472C22⋊C4—2121no—
473C22⋊C4—2121no—
474C22⋊C4—2122no—
475C42—21no—
476C42—21no—
477C42—1064no—
478C42—1061no—
479C42—1061no—
480C42—2122no—
481C42—2122no—
482C23—532no—
483C23—1061no—
484C23—1061no—
485C23—1061no—
486C23—1061no—
487C23—1061no—
488C23—1061no—
489C23C2×D4×D5311yescharacteristic
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
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—532no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—1062no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—4244no—
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×C4C2×Q8×D5311yescharacteristic
535C2×C4#41×D5311yescharacteristic
536C2×C4#41×D5311yescharacteristic
537C2×C4#41×D5311yescharacteristic
538C2×C4#1313326311yescharacteristic
539C2×C4#1313326411yescharacteristic
540C8—41no—
541C8—41no—
542C8—1062no—
543C8—2121no—
544C22—1062no—
545C22—1062no—
546C22—1062no—
547C22—2121no—
548C22—2121no—
549C22—2121no—
550C22—2121no—
551C22C22⋊Q8×D5311yescharacteristic
552C22#1681126611yescharacteristic
553C22#1681127111yescharacteristic
554C22#1681142011yescharacteristic
555C22#1681142211yescharacteristic
556C22#1681143011yescharacteristic
557C22#1681143311yescharacteristic
558C4—41no—
559C4—42no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—1062no—
565C4—4241no—
566C4—4241no—
567C4—2122no—
568C4—2121no—
569C4—2121no—
570C4—2122no—
571C4#1681117911yescharacteristic
572C4#1681125311yescharacteristic
573C4#1681129811yescharacteristic
574C4#1681130911yescharacteristic
575C2—2122no—
576C2#2027839411yescharacteristic
577C2#2028453411yescharacteristic
578C2#2028457811yescharacteristic
579C2#2028690011yescharacteristic
580C2#2028693011yescharacteristic
581C2#2028722811yescharacteristic
582C2#2028727311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#26019
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. #20276219C2semidirect
4. #20276216C2semidirect
5. #20276052C2semidirect
6. #20275671C2semidirect
7. #20275481C2semidirect
8. #20275294C2semidirect
9. #20275291C2semidirect
15. C53×#2101C2semidirect
286. #2101D53semidirect
350. C53#26019semidirect

Maximal subgroups

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

subgroupindex
#202784422
#202775992
#202762192
#202762162
#202760522
#202756712
#202754812
#202752942
#202752912
#202749092
#202748992
#202748082
#202747472
#202744332
C53×#21012
#2601953