Finite Group Databasebeta

← #27452717

Subgroups — #27452717

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#20279253C211yesmaximal, characteristic, complemented
2#20276128C211yesmaximal, characteristic
3#20276089C211yesmaximal, characteristic
4#20275951C211yesmaximal, characteristic
5#20275951C211yesmaximal, characteristic
6#20275394C211yesmaximal, characteristic, complemented
7#20275346C211yesmaximal, characteristic, complemented
8#20275300C211yesmaximal, characteristic, complemented
9#20275293C211yesmaximal, characteristic, complemented
10#20275274C211yesmaximal, characteristic, complemented
11#20274983C211yesmaximal, characteristic, complemented
12#20274976C211yesmaximal, characteristic
13#20274730C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic
15C53×#2367C211yesmaximal, characteristic
16C2×#13133148C2211yescharacteristic
17C2×#13133148C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133141C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810564—21no—
29#16810554—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810504—21no—
35#16810489C2211yescharacteristic
36#16810487—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#124×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#390C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C4×Dic53—21no—
72#13133148—22no—
73#13133148—22no—
74C4⋊#88322—41no—
75#13133141—22no—
76#13133141—22no—
77#13133141—22no—
78#13133141—41no—
79#13133138—22no—
80#13133138—22no—
81#13133138—22no—
82#13133138—41no—
83C4×#88322—41no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53C2311yescharacteristic
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94#124×C53—22no—
95#124×C53—22no—
96C4×Q8×C53—22no—
97C4×Q8×C53—41no—
98C4×D4×C53—22no—
99C4×D4×C53—41no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
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×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—81no—
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53#4112yes—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—81no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—42no—
169C42×C53—81no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53#4112yes—
178#11719919—42no—
179#11719919—81no—
180#11719919—81no—
181#11719919—81no—
182C4×Dic53—42no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
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—21no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—84no—
238C2×C4×C53#13512yes—
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C2×Dic53—22no—
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—41no—
256C2×Dic53—41no—
257#88322—41no—
258#88322—41no—
259#88322—41no—
260#88322—41no—
261#26760—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#57711yescharacteristic
275C22×C53#57711yescharacteristic
276C4×C53—22no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—42no—
280C4×C53—41no—
281C4×C53—42no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—81no—
285C4×C53#52911yescharacteristic
286C4×C53#53911yescharacteristic
287C4×C53#55211yescharacteristic
288C4×C53#55311yescharacteristic
289Dic53—42no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293#2508—531no—
294#2367D5311yescharacteristic
295#2073—531no—
296#2065—531no—
297#2042—531no—
298#2042—531no—
299#1933—531no—
300#1927—531no—
301#1920—531no—
302#1919—531no—
303#1916—531no—
304#1865—531no—
305#1864—531no—
306#1805—531no—
307C4×#102—531no—
308C2×C53—82no—
309C2×C53#246011yescharacteristic
310C2×C53#310211yescharacteristic
311C2×C53#312511yescharacteristic
312C2×C53#329611yescharacteristic
313C2×C53#332111yescharacteristic
314C2×C53#336011yescharacteristic
315C2×C53#337111yescharacteristic
316C2×#124C2×D5311yescharacteristic
317C2×C4×Q8C2×D5311yescharacteristic
318C2×C4×D4C2×D5311yescharacteristic
319C2×#106—531no—
320C2×#106—531no—
321C2×C4⋊C8—531no—
322C2×#103—531no—
323C2×#103—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#102—531no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#102—531no—
330C2×C4×C8—531no—
331#400—1061no—
332#398—1061no—
333#391C2×D5311yescharacteristic
334#390C2×D5311yescharacteristic
335#389—531no—
336#389—531no—
337#389—531no—
338#389—531no—
339#388—21no—
340#388—21no—
341#387—1061no—
342#385—21no—
343#382—531no—
344#382—1061no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4C2×D5311yescharacteristic
347C4×C4⋊C4C2×D5311yescharacteristic
348C4×C22⋊C4—21no—
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#336—531no—
359C53#2676011yescharacteristic, complemented, Sylow
360C22×Q8C22×D5311yescharacteristic
361C22×D4C22×D5311yescharacteristic
362C23×C4—21no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—531no—
366C22×C8—531no—
367#124—22no—
368#124—22no—
369C4×Q8—22no—
370C4×Q8—41no—
371C4×D4—22no—
372C4×D4—41no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22⋊D5311yescharacteristic
383C2×C4⋊C4C22⋊D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—1061no—
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396C2×C42C22×D5311yescharacteristic
397#106—1062no—
398#106—1062no—
399C4⋊C8—2121no—
400#103—1062no—
401#103—1062no—
402#103—1062no—
403#103—2121no—
404#102—1062no—
405#102—1062no—
406#102—1062no—
407#102—2121no—
408C4×C8—2121no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95C22⋊D5311yescharacteristic
420#95C22⋊D5311yescharacteristic
421C24—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1171994512yes—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#1171994512yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C23×D5311yescharacteristic
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—532no—
470C4⋊C4—1061no—
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—2122no—
481C4⋊C4#1171994712yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—81no—
495C42—2122no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22×C22⋊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—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—84no—
539C2×C4—1062no—
540C2×C4—2121no—
541C2×C4—2121no—
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#1313323712yes—
552C2×C4#41×D5311yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C2×C4C2×#1171994511yescharacteristic
555C2×C4C2×#1171994511yescharacteristic
556C2×C4#1313326211yescharacteristic
557C2×C4#1313326411yescharacteristic
558C8—2121no—
559C8—2121no—
560C8—2121no—
561C8—2121no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#1681113411yescharacteristic
570C22#1681126711yescharacteristic
571C22#1681128011yescharacteristic
572C22C2×#1313323711yescharacteristic
573C22#1681142111yescharacteristic
574C22#1681143111yescharacteristic
575C22#1681143411yescharacteristic
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—4241no—
586C4—4241no—
587C4—4241no—
588C4—2122no—
589C4#1681115311yescharacteristic
590C4#1681121511yescharacteristic
591C4#1681131311yescharacteristic
592C4#1681132211yescharacteristic
593C2—82no—
594C2#2027866111yescharacteristic
595C2#2028473311yescharacteristic
596C2#2028499211yescharacteristic
597C2#2028655011yescharacteristic
598C2#2028690111yescharacteristic
599C2#2028736011yescharacteristic
600C2#2028747711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2367C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2367D53
hypocenterC53#26760
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #20279253C2semidirect
6. #20275394C2semidirect
7. #20275346C2semidirect
8. #20275300C2semidirect
9. #20275293C2semidirect
10. #20275274C2semidirect
11. #20274983C2semidirect
13. #20274730C2semidirect
359. C53#26760semidirect

Maximal subgroups

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

subgroupindex
#202792532
#202761282
#202760892
#202759512
#202753942
#202753462
#202753002
#202752932
#202752742
#202749832
#202749762
#202747302
C4×#131331382
C53×#23672
#2676053