Finite Group Databasebeta

← #27452776

Subgroups — #27452776

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279251C211yesmaximal, characteristic, complemented
2#20277790C211yesmaximal, characteristic
3#20276136C211yesmaximal, characteristic, complemented
4#20276108C211yesmaximal, characteristic, complemented
5#20275951C211yesmaximal, characteristic
6#20275395C211yesmaximal, characteristic
7#20275337C211yesmaximal, characteristic
8#20275304C211yesmaximal, characteristic, complemented
9#20275287C211yesmaximal, characteristic, complemented
10#20275271C211yesmaximal, characteristic, complemented
11#20274967C211yesmaximal, characteristic, complemented
12#20274960C211yesmaximal, characteristic
13#20274723C211yesmaximal, characteristic
14#103×Dic53C211yesmaximal, characteristic, complemented
15C53×#2042C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26#16810564—21no—
27#16810553—21no—
28#16810530C2211yescharacteristic
29#16810526C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810516—21no—
34#16810516—21no—
35#16810509—21no—
36#16810498—21no—
37#16810488C2211yescharacteristic
38#16810487—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C22⋊C4×Dic53—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#124×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62Q8×Dic53—22no—
63Q8×Dic53—41no—
64D4×Dic53—22no—
65D4×Dic53—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53D411yescharacteristic
68C2×C4⋊Dic53D411yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84#13133151—22no—
85#13133150—22no—
86#13133146—41no—
87#13133141—22no—
88#13133141—22no—
89#13133138—22no—
90#13133138—22no—
91C8×Dic53—41no—
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#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103C22×Q8×C53C2311yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107#124×C53—22no—
108#124×C53—22no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#103×C53—22no—
116#103×C53—41no—
117#102×C53—22no—
118#102×C53—41no—
119#95×C53—21no—
120C24×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53#4112yes—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53#4112yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—81no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—42no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C42×C53—21no—
164C42×C53—21no—
165C22⋊Dic53—42no—
166C22⋊Dic53—42no—
167C22⋊Dic53—81no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—22no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—41no—
176C4⋊Dic53—42no—
177C4⋊Dic53#4112yes—
178C4⋊Dic53#4112yes—
179#11719919—81no—
180#11719919—81no—
181#11719919—81no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53—81no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C23×C53—22no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53C22×D411yescharacteristic
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—41no—
206Q8×C53—41no—
207D4×C53—22no—
208D4×C53—22no—
209D4×C53—41no—
210D4×C53—41no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53H(5,𝔽2)11yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—22no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—22no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—84no—
256C2×Dic53#13512yes—
257C8×C53—41no—
258C8×C53—41no—
259#88322—41no—
260#88322—41no—
261#26764—531nomaximal, Sylow
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53#52511yescharacteristic
270C22×C53#54611yescharacteristic
271C22×C53#54811yescharacteristic
272C22×C53C2×#13511yescharacteristic
273C22×C53#57511yescharacteristic
274C22×C53#57611yescharacteristic
275C22×C53#57811yescharacteristic
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—42no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53#52911yescharacteristic
282C4×C53#53911yescharacteristic
283C4×C53#55211yescharacteristic
284C4×C53#55311yescharacteristic
285Dic53—22no—
286Dic53—42no—
287Dic53—41no—
288Dic53—41no—
289Dic53—41no—
290Dic53—42no—
291Dic53—81no—
292Dic53—81no—
293#2508—531no—
294#2367—531no—
295#2074—531no—
296#2068—531no—
297#2042—531no—
298#2042D5311yescharacteristic
299#1934—531no—
300#1925—531no—
301#1921—531no—
302#1918—531no—
303#1916—531no—
304#1863—531no—
305#1862—531no—
306#1804—531no—
307C4×#103—531no—
308C2×C53—82no—
309C2×C53#246011yescharacteristic
310C2×C53#310211yescharacteristic
311C2×C53#312411yescharacteristic
312C2×C53#330011yescharacteristic
313C2×C53#331911yescharacteristic
314C2×C53#336111yescharacteristic
315C2×C53#337711yescharacteristic
316C2×#124C2×D5311yescharacteristic
317C2×C4×Q8—531no—
318C2×C4×D4—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×C4⋊C8—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×#103C2×D5311yescharacteristic
325C2×#103C2×D5311yescharacteristic
326C2×#102—531no—
327C2×#102—531no—
328C2×#102C2×D5311yescharacteristic
329C2×#102C2×D5311yescharacteristic
330C2×C4×C8—531no—
331#400—1061no—
332#398—1061no—
333#391—531no—
334#390—531no—
335#389—531no—
336#389—531no—
337#389—531no—
338#389C2×D5311yescharacteristic
339#388—1061no—
340#388—1061no—
341#387—1061no—
342#385—1061no—
343#382—531no—
344#382—1061no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348C4×C22⋊C4—1061no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#336—531no—
356#336—531no—
357#336—531no—
358#336C2×D5311yescharacteristic
359C53#2676411yescharacteristic, complemented, Sylow
360C22×Q8C22×D5311yescharacteristic
361C22×D4C22×D5311yescharacteristic
362C23×C4—1061no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8C22×D5311yescharacteristic
366C22×C8C22×D5311yescharacteristic
367#124—22no—
368#124—22no—
369C4×Q8—1062no—
370C4×Q8—2121no—
371C4×D4—1062no—
372C4×D4—2121no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42C22×D5311yescharacteristic
397#107—1062no—
398#107—1062no—
399C4⋊C8—2121no—
400#103—22no—
401#103—41no—
402#103—1062no—
403#103—1062no—
404#102—22no—
405#102—41no—
406#102—1062no—
407#102—1062no—
408C4×C8—2121no—
409#95—21no—
410#95—531no—
411#95—531no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1171994712yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#1171994712yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—4241no—
446C22×C4—2122no—
447C22×C4—2122no—
448C22×C4D4×D5311yescharacteristic
449C22×C4D4×D5311yescharacteristic
450C22×C4C23×D5311yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—532no—
469C4⋊C4—532no—
470C4⋊C4—1062no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—4241no—
478C4⋊C4—4241no—
479C4⋊C4—4241no—
480C4⋊C4—2121no—
481C4⋊C4—2122no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—4241no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C42—21no—
488C42—21no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42—4241no—
494C42—2122no—
495C42—2122no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C2×D4×D5311yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—532no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—4244no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4#41×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4C2×#1171994711yescharacteristic
555C2×C4C2×#1171994711yescharacteristic
556C2×C4#1313326211yescharacteristic
557C2×C4#1313326411yescharacteristic
558C8—41no—
559C8—41no—
560C8—2121no—
561C8—2121no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#1681112911yescharacteristic
570C22#1681127011yescharacteristic
571C22#1681128111yescharacteristic
572C22#135×D5311yescharacteristic
573C22#1681141911yescharacteristic
574C22#1681142511yescharacteristic
575C22#1681143911yescharacteristic
576C4—41no—
577C4—41no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—1062no—
582C4—4241no—
583C4—4241no—
584C4—2122no—
585C4—2121no—
586C4—2121no—
587C4—2121no—
588C4—2122no—
589C4#1681115011yescharacteristic
590C4#1681121811yescharacteristic
591C4#1681131011yescharacteristic
592C4#1681131611yescharacteristic
593C2—82no—
594C2#2027866011yescharacteristic
595C2#2028472611yescharacteristic
596C2#2028497811yescharacteristic
597C2#2028660911yescharacteristic
598C2#2028687811yescharacteristic
599C2#2028737211yescharacteristic
600C2#2028755611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2042C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2042D53
hypocenterC53#26764
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
1. #20279251C2semidirect
3. #20276136C2semidirect
4. #20276108C2semidirect
8. #20275304C2semidirect
9. #20275287C2semidirect
10. #20275271C2semidirect
11. #20274967C2semidirect
14. #103×Dic53C2semidirect
359. C53#26764semidirect

Maximal subgroups

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

subgroupindex
#202792512
#202777902
#202761362
#202761082
#202759512
#202753952
#202753372
#202753042
#202752872
#202752712
#202749672
#202749602
#202747232
#103×Dic532
C53×#20422
#2676453