Finite Group Databasebeta

← #27469680

Subgroups — #27469680

Order 13568.

750 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#20283066C211yesmaximal, characteristic
2#20277121C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276283C211yesmaximal, characteristic, complemented
5#20276282C211yesmaximal, characteristic, complemented
6#20275674C211yesmaximal, characteristic, complemented
7#20275674C211yesmaximal, characteristic, complemented
8#20274841C211yesmaximal, characteristic, complemented
9#20274841C211yesmaximal, characteristic, complemented
10#20274809C211yesmaximal, characteristic
11#20274809C211yesmaximal, characteristic
12#20274808C211yesmaximal, characteristic
13#20274808C211yesmaximal, characteristic
14#20274472C211yesmaximal, characteristic
15C53×#2101C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×#13133195—21no—
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C42×D53C2211yescharacteristic
21C2×#13133129C2211yescharacteristic
22C2×#13133129C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25#16810527—21no—
26#16810521C2211yescharacteristic
27#16810518—21no—
28#16810492C2211yescharacteristic
29#16810492C2211yescharacteristic
30#16810492C2211yescharacteristic
31#16810489C2211yescharacteristic
32#16810482—21no—
33#16810482—21no—
34#16810482—21no—
35#16810480C2211yescharacteristic
36#16810480C2211yescharacteristic
37#16810474—21no—
38#16810474—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×Dic106—21no—
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216—22no—
67#13133195—44no—
68#13133175—22no—
69#13133175—22no—
70#13133175—22no—
71#13133175—22no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C42×D53—22no—
94C42×D53—22no—
95#13133129—22no—
96#13133129—22no—
97C8×Dic53—22no—
98C8×Dic53—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115C23×D53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—21no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53#4112yes—
141C2×C4×D53#4112yes—
142C2×Dic106—42no—
143C2×Dic106—42no—
144C2×Dic106—42no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×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—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—42no—
160C42×C53—21no—
161C42×C53—21no—
162#11719921—42no—
163#11719921—42no—
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—
174#11719919—42no—
175#11719919—42no—
176#11719919—42no—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—42no—
185C4×Dic53—42no—
186C4×Dic53#4112yes—
187C4×Dic53#4112yes—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C23×C53C22×D411yescharacteristic
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53#12412yes—
202C2×C4×C53—21no—
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×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
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—41no—
240C2×Dic53#12312yes—
241C2×Dic53#12312yes—
242C2×Dic53#12412yes—
243C2×Dic53#13512yes—
244C2×Dic53#13512yes—
245C2×Dic53AGL(1,ℤ/8)12yes—
246C2×Dic53#13712yes—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—22no—
251C4×D53—44no—
252C4×D53—44no—
253Dic106—84no—
254C8×C53—41no—
255C8×C53—41no—
256#88322—41no—
257#88322—41no—
258#27962—531nomaximal, Sylow
259C22×C53C2×#12311yescharacteristic
260C22×C53C2×#12411yescharacteristic
261C22×C53#53611yescharacteristic
262C22×C53C2×#13511yescharacteristic
263C22×C53C2×#13511yescharacteristic
264C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C53C2×#13711yescharacteristic
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—41no—
270C2×D53—41no—
271C2×D53—41no—
272C2×D53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53#55211yescharacteristic
280C4×C53#55211yescharacteristic
281C4×C53#55211yescharacteristic
282C4×C53#55211yescharacteristic
283Dic53—22no—
284Dic53—81no—
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288Dic53#48214yes—
289Dic53#48514yes—
290#2857—531no—
291#2285—531no—
292#2101—531no—
293#2101—531no—
294#2101—531no—
295#2101D5311yescharacteristic, complemented
296#1990—531no—
297#1990—531no—
298#1834—531no—
299#1834—531no—
300#1825—531no—
301#1825—531no—
302#1825—531no—
303#1825—531no—
304C4×C22⋊C8—531no—
305C2×C53#250311yescharacteristic
306C2×C53C2×#48211yescharacteristic
307C2×C53C2×#48511yescharacteristic
308C2×C53#313111yescharacteristic
309C2×C53#314111yescharacteristic
310C2×C53#322811yescharacteristic
311C2×C53#323111yescharacteristic
312D53—42no—
313C2×C4⋊Q8—531no—
314C2×C22⋊Q8—1061no—
315C2×#117—531no—
316C2×#117—531no—
317C22×C42—531no—
318C2×C22⋊C8—531no—
319C2×C22⋊C8—531no—
320C2×C4×C8—531no—
321C2×C4×C8—531no—
322#391—1061no—
323#389—531no—
324#389—1061no—
325#389C2×D5311yescharacteristic
326#384—531no—
327#384—531no—
328#384—531no—
329#382—531no—
330#382C2×D5311yescharacteristic
331#380—531no—
332#380—531no—
333#380—1061no—
334#380—1061no—
335#380—1061no—
336C4×C4⋊C4—1061no—
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#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340C2×D5311yescharacteristic
351#340C2×D5311yescharacteristic
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#336—531no—
355#336C2×D5311yescharacteristic
356C53#2796211yescharacteristic, complemented, Sylow
357C22×Q8—1061no—
358C23×C4—531no—
359C23×C4—531no—
360C23×C4—531no—
361C22×C8—531no—
362C22×C8—531no—
363C22×C8C22×D5311yescharacteristic
364C22×C8C22×D5311yescharacteristic
365C4⋊Q8—1062no—
366C4⋊Q8—1062no—
367C22⋊Q8—2124no—
368#117—1062no—
369#117—1062no—
370#117—1062no—
371#117—1062no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C42—531no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—1062no—
397C2×C42—1062no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42C22×D5311yescharacteristic
401C22⋊C8—1062no—
402C22⋊C8—1062no—
403C4×C8—1062no—
404C4×C8—1062no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413C24—531no—
414C2×Q8—2122no—
415C2×Q8—2122no—
416C2×Q8—2122no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—531no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—1062no—
427C22×C4—1062no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4D4×D5311yescharacteristic
441C22×C4D4×D5311yescharacteristic
442C22×C4C23×D5311yescharacteristic
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C4⋊C4—21no—
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—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4—2122no—
480C22⋊C4—2122no—
481C22⋊C4—2122no—
482C22⋊C4—2122no—
483C22⋊C4—2122no—
484C42—21no—
485C42—21no—
486C42—532no—
487C42—532no—
488C42—1062no—
489C42—1062no—
490C42—2122no—
491C42—2122no—
492C23—532no—
493C23—1061no—
494C23—1061no—
495C23—1061no—
496C23—1061no—
497C23—1061no—
498C23—1061no—
499C23C2×D4×D5311yescharacteristic
500Q8—4244no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—532no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—2124no—
533C2×C4—2124no—
534C2×C4—2121no—
535C2×C4—2121no—
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—2121no—
546C2×C4#41×D5311yescharacteristic
547C2×C4#41×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4#1313326411yescharacteristic
551C2×C4#1313326511yescharacteristic
552C8—41no—
553C8—41no—
554C8—2121no—
555C8—2121no—
556C22—1062no—
557C22—1062no—
558C22—1062no—
559C22—2121no—
560C22—2121no—
561C22—2121no—
562C22—2121no—
563C22#123×D5311yescharacteristic
564C22#124×D5311yescharacteristic
565C22#1681119611yescharacteristic
566C22#135×D5311yescharacteristic
567C22#135×D5311yescharacteristic
568C22AGL(1,ℤ/8)×D5311yescharacteristic
569C22#137×D5311yescharacteristic
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—534no—
577C4—534no—
578C4—1062no—
579C4—4241no—
580C4—4241no—
581C4—4241no—
582C4—4241no—
583C4#1681130511yescharacteristic
584C4#1681130511yescharacteristic
585C4#1681130911yescharacteristic
586C4#1681130911yescharacteristic
587C2—2122no—
588C2#2027919511yescharacteristic
589C2#482×D5311yescharacteristic
590C2#485×D5311yescharacteristic
591C2#2028503611yescharacteristic
592C2#2028510711yescharacteristic
593C2#2028585011yescharacteristic
594C2#2028587311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#27962
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. #20275674C2semidirect
7. #20275674C2semidirect
8. #20274841C2semidirect
9. #20274841C2semidirect
15. C53×#2101C2semidirect
295. #2101D53semidirect
356. C53#27962semidirect

Maximal subgroups

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

subgroupindex
#202830662
#202771212
#202762842
#202762832
#202762822
#202756742
#202748412
#202748092
#202748082
#202744722
C53×#21012
#2796253