Finite Group Databasebeta

← #27449528

Subgroups — #27449528

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279957C211yesmaximal, characteristic, complemented
2#20277790C211yesmaximal, characteristic
3#20276265C211yesmaximal, characteristic, complemented
4#20276238C211yesmaximal, characteristic
5#20276106C211yesmaximal, characteristic, complemented
6#20276099C211yesmaximal, characteristic
7#20275648C211yesmaximal, characteristic
8#20275471C211yesmaximal, characteristic, complemented
9#20275438C211yesmaximal, characteristic, complemented
10#20274971C211yesmaximal, characteristic, complemented
11#20274964C211yesmaximal, characteristic
12#20274728C211yesmaximal, characteristic, complemented
13#20274723C211yesmaximal, characteristic
14#20274612C211yesmaximal, characteristic, complemented
15C53×#1959C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29#16810562—21no—
30#16810562—21no—
31#16810530C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810522—21no—
34#16810522C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810516—21no—
37#16810516—21no—
38#16810498—21no—
39#16810488—21no—
40#16810488C2211yescharacteristic
41#16810488C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C22⋊C4×Dic53—21no—
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#124×C53C2211yescharacteristic
53C2×#103×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#97×C53C2211yescharacteristic
58C53×#384C2211yescharacteristic
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⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53D411yescharacteristic
70C2×C4⋊Dic53D411yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133151—22no—
84#13133150—41no—
85#13133150—41no—
86#13133149—22no—
87C4⋊#88322—22no—
88#13133146—41no—
89#13133146—41no—
90#13133141—41no—
91#13133141—41no—
92#13133138—41no—
93#13133138—41no—
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—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C22×Q8×C53C2311yescharacteristic
107C22×D4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#124×C53—22no—
111#124×C53Q814yes—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#103×C53—22no—
120#103×C53—41no—
121#102×C53—22no—
122#102×C53—41no—
123#97×C53—22no—
124C24×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53C2×Q812yes—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—81no—
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53#4112yes—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—81no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—24no—
168C42×C53C2×Q812yes—
169C42×C53#4112yes—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—81no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—41no—
181C4⋊Dic53—41no—
182C4⋊Dic53—42no—
183C4⋊Dic53—81no—
184C4⋊Dic53—81no—
185C4⋊Dic53#4112yes—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—42no—
191C4×Dic53—42no—
192C4×Dic53—81no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C23×C53—22no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53C22×D411yescharacteristic
207Q8×C53—22no—
208Q8×C53—22no—
209Q8×C53—41no—
210Q8×C53—41no—
211D4×C53—22no—
212D4×C53—22no—
213D4×C53—41no—
214D4×C53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
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×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C22⋊Q812yes—
229C2×C4×C53C22⋊Q812yes—
230C2×C4×C53C22×Q811yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—84no—
261C8×C53—22no—
262C8×C53—41no—
263#88322—41no—
264#88322—41no—
265#26545—531nomaximal, Sylow
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53C2×C22⋊Q811yescharacteristic
274C22×C53#54611yescharacteristic
275C22×C53#55011yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57811yescharacteristic
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—42no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53#52911yescharacteristic
286C4×C53#54011yescharacteristic
287C4×C53#55411yescharacteristic
288C4×C53#55411yescharacteristic
289Dic53—41no—
290Dic53—41no—
291Dic53—41no—
292Dic53—41no—
293Dic53—42no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297#2571—531no—
298#2367—531no—
299#2097—531no—
300#2092—531no—
301#2068—531no—
302#2067—531no—
303#1983—531no—
304#1959D5311yescharacteristic
305#1947—531no—
306#1940—531no—
307#1863—531no—
308#1862—531no—
309#1805—531no—
310#1804—531no—
311#1776—531no—
312C2×C53—82no—
313C2×C53#245411yescharacteristic
314C2×C53#308011yescharacteristic
315C2×C53#308311yescharacteristic
316C2×C53#331311yescharacteristic
317C2×C53#331911yescharacteristic
318C2×C53#340211yescharacteristic
319C2×C53#340311yescharacteristic
320C2×#124C2×D5311yescharacteristic
321C2×C4×Q8—531no—
322C2×C4×D4—531no—
323C2×#107—531no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#106—531no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×#103—531no—
331C2×#103—531no—
332C2×#103C2×D5311yescharacteristic
333C2×#103C2×D5311yescharacteristic
334C2×#102—531no—
335C2×#102—531no—
336C2×#102C2×D5311yescharacteristic
337C2×#102C2×D5311yescharacteristic
338C2×#97C2×D5311yescharacteristic
339#400—1061no—
340#400—1061no—
341#391—531no—
342#390—531no—
343#389—531no—
344#389—531no—
345#389—1061no—
346#388—1061no—
347#388—1061no—
348#385—1061no—
349#384C2×D5311yescharacteristic
350#382—531no—
351#382—531no—
352#382—1061no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C22⋊C4—1061no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#336—531no—
362#336—531no—
363C53#2654511yescharacteristic, complemented, Sylow
364C22×Q8C22×D5311yescharacteristic
365C22×D4C22×D5311yescharacteristic
366C23×C4—1061no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371#124—22no—
372#124Dic10614yes—
373C4×Q8—1062no—
374C4×Q8—2121no—
375C4×D4—1062no—
376C4×D4—2121no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—1061no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42C22×D5311yescharacteristic
401#107—1062no—
402#107—2121no—
403#107—2121no—
404#106—1062no—
405C4⋊C8—1062no—
406C4⋊C8—2121no—
407C4⋊C8—2121no—
408#103—22no—
409#103—41no—
410#103—2121no—
411#103—2121no—
412#102—22no—
413#102—41no—
414#102—2121no—
415#102—2121no—
416#97—22no—
417#95—531no—
418#95—531no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429C24—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8C2×Dic10612yes—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C2×Dic10612yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—4241no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8#1171994712yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—81no—
477C4⋊C4—532no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4—2121no—
488C4⋊C4—2121no—
489C4⋊C4—2122no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—4241no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—1061no—
496C42—1061no—
497C42—1061no—
498C42—1061no—
499C42—4241no—
500C42—2122no—
501C42—2122no—
502C42C2×Dic10612yes—
503C42#1171994712yes—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D5311yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516D4—22no—
517D4—22no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1062no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—4244no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4#1313319412yes—
559C2×C4#1313319412yes—
560C2×C4C22×Dic10611yescharacteristic
561C2×C4#41×D5311yescharacteristic
562C2×C4C2×#1171994711yescharacteristic
563C2×C4C2×#1171994711yescharacteristic
564C2×C4#1313326211yescharacteristic
565C2×C4#1313326311yescharacteristic
566C8—22no—
567C8—41no—
568C8—2121no—
569C8—2121no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C2×#1313319411yescharacteristic
578C22#1681126611yescharacteristic
579C22#1681129311yescharacteristic
580C22#1681141911yescharacteristic
581C22#1681142211yescharacteristic
582C22#1681142711yescharacteristic
583C22#1681143911yescharacteristic
584C4—22no—
585C4—22no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—4241no—
590C4—4241no—
591C4—4241no—
592C4—2121no—
593C4—2121no—
594C4—2121no—
595C4—2121no—
596C4—2122no—
597C4#1681115411yescharacteristic
598C4#1681122611yescharacteristic
599C4#1681133011yescharacteristic
600C4#1681133211yescharacteristic
601C2—82no—
602C2#2027859011yescharacteristic
603C2#2028453511yescharacteristic
604C2#2028457011yescharacteristic
605C2#2028678011yescharacteristic
606C2#2028687011yescharacteristic
607C2#2028779111yescharacteristic
608C2#2028780011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1959C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1959D53
hypocenterC53#26545
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1959

Complements

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

NH ≅ G/Ntype
1. #20279957C2semidirect
3. #20276265C2semidirect
5. #20276106C2semidirect
8. #20275471C2semidirect
9. #20275438C2semidirect
10. #20274971C2semidirect
12. #20274728C2semidirect
14. #20274612C2semidirect
363. C53#26545semidirect

Maximal subgroups

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

subgroupindex
#202799572
#202777902
#202762652
#202762382
#202761062
#202760992
#202756482
#202754712
#202754382
#202749712
#202749642
#202747282
#202747232
#202746122
C53×#19592
#2654553