Finite Group Databasebeta

← #27465370

Subgroups — #27465370

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#389×D53C211yesmaximal, characteristic
2#20277111C211yesmaximal, characteristic
3#20276286C211yesmaximal, characteristic, complemented
4#20276285C211yesmaximal, characteristic, complemented
5#20276202C211yesmaximal, characteristic, complemented
6#20276200C211yesmaximal, characteristic, complemented
7#20275293C211yesmaximal, characteristic, complemented
8#20275287C211yesmaximal, characteristic, complemented
9#20275283C211yesmaximal, characteristic, complemented
10#20274906C211yesmaximal, characteristic
11#20274904C211yesmaximal, characteristic
12#20274804C211yesmaximal, characteristic
13#20274804C211yesmaximal, characteristic
14#20274782C211yesmaximal, characteristic
15C53×#1919C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D53—21no—
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C4⋊C4×D53C2211yescharacteristic
19C2×C42×D53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133129C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25#16810522C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810520—21no—
28#16810519C2211yescharacteristic
29#16810518—21no—
30#16810490—21no—
31#16810488C2211yescharacteristic
32#16810488C2211yescharacteristic
33#16810485—21no—
34#16810480C2211yescharacteristic
35#16810480C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37C4×#11719921—21no—
38#16810469—21no—
39#95×D53—21no—
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C4×C4⋊C4×C53C2211yescharacteristic
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
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C4⋊C4×D53—22no—
76C4⋊C4×D53—22no—
77C4⋊C4×D53—22no—
78C4⋊C4×D53—22no—
79C4⋊C4×D53—44no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53—22no—
87C42×D53—22no—
88#13133151—22no—
89#13133150—22no—
90#13133148—22no—
91#13133129—22no—
92#13133129—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—
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×C53—21no—
111C2×C42×C53C2311yescharacteristic
112#106×C53—22no—
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—24no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—41no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×C4×D53—42no—
146C2×C4×D53#4112yes—
147C2×C4×D53#4112yes—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×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—
162C4⋊C4×C53#4112yes—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—42no—
166#11719921—42no—
167#11719921—42no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—42no—
176C4⋊Dic53#4112yes—
177#11719919—24no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181C4×Dic53—21no—
182C4×Dic53—21no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—42no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C23×C53C22×D411yescharacteristic
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53#12312yes—
198C22×D53#13512yes—
199C22×D53#13712yes—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
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×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—22no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
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×Dic53—41no—
244C2×Dic53#12312yes—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—44no—
250C4×D53—48no—
251C4×D53—44no—
252C4×D53—84no—
253C8×C53—41no—
254C8×C53—41no—
255#88322—41no—
256#88322—41no—
257#27617—531nomaximal, Sylow
258C22×C53C2×#12311yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#53611yescharacteristic
261C22×C53C2×#13511yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53C2×#13711yescharacteristic
264C22×C53#57511yescharacteristic
265C2×D53—22no—
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53#48514yes—
272C4×C53—22no—
273C4×C53—41no—
274C4×C53—42no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53#55211yescharacteristic
279C4×C53#55211yescharacteristic
280C4×C53#55211yescharacteristic
281C4×C53#55211yescharacteristic
282Dic53—22no—
283Dic53—41no—
284Dic53—41no—
285Dic53—42no—
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289C2×#389—531no—
290#2283—531no—
291#2101—531no—
292#2101—531no—
293#2085—531no—
294#2085—531no—
295#1919—531no—
296#1919D5311yescharacteristic, complemented
297#1918—531no—
298#1918—531no—
299#1853—531no—
300#1852—531no—
301#1824—531no—
302#1824—531no—
303#1817—531no—
304C2×C53#248411yescharacteristic
305C2×C53C2×#48511yescharacteristic
306C2×C53#309411yescharacteristic
307C2×C53#310511yescharacteristic
308C2×C53#311711yescharacteristic
309C2×C53#323211yescharacteristic
310C2×C53#323911yescharacteristic
311D53—24no—
312C22×C4⋊C4—531no—
313C22×C4⋊C4—531no—
314C22×C4⋊C4—1061no—
315C22×C42—531no—
316C2×#107—531no—
317C2×#107—531no—
318C2×#106—531no—
319C2×#106C2×D5311yescharacteristic
320C2×C22⋊C8—531no—
321C2×C22⋊C8—531no—
322#389—531no—
323#389—531no—
324#389—531no—
325#389—1061no—
326#389—1061no—
327#389C2×D5311yescharacteristic
328#383—1061no—
329#382—531no—
330#382—531no—
331#381—1061no—
332#380—531no—
333#380—531no—
334C4×C4⋊C4—531no—
335C4×C4⋊C4C2×D5311yescharacteristic
336C4×C22⋊C4—1061no—
337#376—1061no—
338C2×#95—1061no—
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#340C2×D5311yescharacteristic
349#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#336—531no—
352#336—531no—
353#336—531no—
354#336C2×D5311yescharacteristic
355C53#2761711yescharacteristic, complemented, Sylow
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—531no—
359C23×C4—1061no—
360C23×C4—1061no—
361C22×C8—531no—
362C22×C8—531no—
363C22×C8C22×D5311yescharacteristic
364C22×C8C22×D5311yescharacteristic
365C2×C4⋊C4—21no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—2124no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—21no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—1062no—
391C2×C42—1062no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394#107—1062no—
395#107—1062no—
396#106—22no—
397#106—1062no—
398C22⋊C8—1062no—
399C22⋊C8—1062no—
400#95—21no—
401#95—21no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412C24—531no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—531no—
418C22×C4—532no—
419C22×C4—532no—
420C22×C4—531no—
421C22×C4—531no—
422C22×C4—1062no—
423C22×C4—1062no—
424C22×C4—1062no—
425C22×C4—1062no—
426C22×C4—1064no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1062no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—2121no—
437C22×C4—2122no—
438C22×C4—2122no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4—2122no—
442C22×C4D4×D5311yescharacteristic
443C22×C4D4×D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—42no—
465C4⋊C4—532no—
466C4⋊C4—1064no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—2122no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4#1171994512yes—
479C22⋊C4—2122no—
480C22⋊C4—2122no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—1062no—
485C42—1062no—
486C42—1061no—
487C42—1061no—
488C42—2122no—
489C23—532no—
490C23—532no—
491C23—532no—
492C23—1061no—
493C23—1061no—
494C23—1061no—
495C23—1061no—
496C23C2×D4×D5311yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
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—41no—
513C2×C4—41no—
514C2×C4—532no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1062no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—4244no—
529C2×C4—2124no—
530C2×C4—2124no—
531C2×C4—2128no—
532C2×C4—2121no—
533C2×C4—2121no—
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#1313324412yes—
544C2×C4#41×D5311yescharacteristic
545C2×C4#41×D5311yescharacteristic
546C2×C4#41×D5311yescharacteristic
547C2×C4C2×#1171994511yescharacteristic
548C2×C4#1313326411yescharacteristic
549C2×C4#1313326511yescharacteristic
550C8—41no—
551C8—41no—
552C8—2121no—
553C8—2121no—
554C22—534no—
555C22—1062no—
556C22—1062no—
557C22—1062no—
558C22—1062no—
559C22—1062no—
560C22—1062no—
561C22#123×D5311yescharacteristic
562C22#1681113211yescharacteristic
563C22#1681119611yescharacteristic
564C22#135×D5311yescharacteristic
565C22C2×#1313324411yescharacteristic
566C22#137×D5311yescharacteristic
567C22#1681141911yescharacteristic
568C4—22no—
569C4—42no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—1062no—
575C4—4241no—
576C4—4241no—
577C4—4241no—
578C4—2122no—
579C4—2121no—
580C4—2121no—
581C4#1681130511yescharacteristic
582C4#1681130911yescharacteristic
583C4#1681131211yescharacteristic
584C4#1681131411yescharacteristic
585C2—1064no—
586C2#2027900011yescharacteristic
587C2#485×D5311yescharacteristic
588C2#2028465211yescharacteristic
589C2#2028476611yescharacteristic
590C2#2028488411yescharacteristic
591C2#2028588111yescharacteristic
592C2#2028594911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1919C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1919D53
hypocenterC53#27617
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276285C2semidirect
5. #20276202C2semidirect
6. #20276200C2semidirect
7. #20275293C2semidirect
8. #20275287C2semidirect
9. #20275283C2semidirect
15. C53×#1919C2semidirect
296. #1919D53semidirect
355. C53#27617semidirect

Maximal subgroups

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

subgroupindex
#389×D532
#202771112
#202762862
#202762852
#202762022
#202762002
#202752932
#202752872
#202752832
#202749062
#202749042
#202748042
#202747822
C53×#19192
#2761753