Finite Group Databasebeta

← #27457163

Subgroups — #27457163

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279137C211yesmaximal, characteristic
2#20276123C211yesmaximal, characteristic, complemented
3#20276096C211yesmaximal, characteristic, complemented
4#20275915C211yesmaximal, characteristic
5#20275915C211yesmaximal, characteristic
6#20275473C211yesmaximal, characteristic, complemented
7#20275450C211yesmaximal, characteristic, complemented
8#20275330C211yesmaximal, characteristic
9#20275304C211yesmaximal, characteristic, complemented
10#20275286C211yesmaximal, characteristic, complemented
11#20274969C211yesmaximal, characteristic, complemented
12#20274914C211yesmaximal, characteristic
13#20274810C211yesmaximal, characteristic
14#20274545C211yesmaximal, characteristic, complemented
15C53×#2362C211yesmaximal, characteristic
16C2×#13133210—21no—
17C2×#13133150C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29#16810563—21no—
30#16810548—21no—
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810508—21no—
34#16810493C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C22⋊Q8×C53C2211yescharacteristic
49C2×C4×Q8×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#388—21no—
54C53×#388—21no—
55C53×#387—21no—
56C4×C4⋊C4×C53—21no—
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133210—44no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C4×Dic53—21no—
75#13133150—22no—
76#13133150—22no—
77#13133150—22no—
78#13133148—41no—
79C4⋊#88322—22no—
80#13133141—22no—
81#13133141—22no—
82#13133141—22no—
83#13133141—41no—
84C22⋊#88322—22no—
85C22⋊#88322—41no—
86#13133128—41no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92C22×Q8×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—22no—
95C22⋊Q8×C53—22no—
96C4×Q8×C53—22no—
97C4×Q8×C53—41no—
98#117×C53—22no—
99#117×C53—41no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C22⋊C4×C53D411yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53D411yescharacteristic
120#95×C53D411yescharacteristic
121C24×C53C2×D411yescharacteristic
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53#4112yes—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C4⋊C4×C53—24no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—42no—
154C4⋊C4×C53—42no—
155C4⋊C4×C53#4112yes—
156C4⋊C4×C53#4112yes—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—41no—
160C22⋊C4×C53—42no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—42no—
166C42×C53—42no—
167C42×C53—42no—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—42no—
179C4⋊Dic53—42no—
180#11719919—42no—
181#11719919—42no—
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#4112yes—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53C22×D411yescharacteristic
203Q8×C53—22no—
204Q8×C53—22no—
205Q8×C53—41no—
206Q8×C53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—22no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
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—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
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×C53AGL(1,ℤ/8)12yes—
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×Dic53—41no—
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—41no—
256#88322—22no—
257#88322—41no—
258#88322—41no—
259#88322—41no—
260#27058—531nomaximal, Sylow
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53#52511yescharacteristic
269C22×C53#54611yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53#57511yescharacteristic
273C22×C53#57811yescharacteristic
274C22×C53#57811yescharacteristic
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—42no—
280C4×C53—41no—
281C4×C53—42no—
282C4×C53—41no—
283C4×C53—81no—
284C4×C53#52911yescharacteristic
285C4×C53#53911yescharacteristic
286C4×C53#55311yescharacteristic
287C4×C53#55611yescharacteristic
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2496—531no—
293#2362D5311yescharacteristic
294#2072—531no—
295#2066—531no—
296#2036—531no—
297#2036—531no—
298#1947—531no—
299#1943—531no—
300#1924—531no—
301#1921—531no—
302#1918—531no—
303#1863—531no—
304#1854—531no—
305#1825—531no—
306#1763—531no—
307C2×C53—81no—
308C2×C53#246611yescharacteristic
309C2×C53#311511yescharacteristic
310C2×C53#312711yescharacteristic
311C2×C53#331311yescharacteristic
312C2×C53#333011yescharacteristic
313C2×C53#334311yescharacteristic
314C2×C53#334911yescharacteristic
315C2×#126—1061no—
316C2×C22⋊Q8C2×D5311yescharacteristic
317C2×C4×Q8C2×D5311yescharacteristic
318C2×#117C2×D5311yescharacteristic
319C2×#107—531no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#106—531no—
323C2×C4⋊C8—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×#103—531no—
328C2×C22⋊C8—531no—
329C2×C22⋊C8—531no—
330C2×#97—531no—
331#400—1061no—
332#397—1061no—
333#391C2×D5311yescharacteristic
334#389—531no—
335#389—531no—
336#389C2×D5311yescharacteristic
337#388—21no—
338#388—21no—
339#387—21no—
340#387—1061no—
341#384—531no—
342#380—531no—
343#380—531no—
344C4×C4⋊C4—21no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4C2×D5311yescharacteristic
347C4×C22⋊C4C2×D5311yescharacteristic
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#336—531no—
357#336—531no—
358C53#2705811yescharacteristic, complemented, Sylow
359C22×Q8C22×D5311yescharacteristic
360C23×C4C22×D5311yescharacteristic
361C22×C8—531no—
362C22×C8—531no—
363C22×C8—531no—
364C22×C8—531no—
365#126—2124no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—22no—
369C4×Q8—41no—
370#117—22no—
371#117—41no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4C22×D5311yescharacteristic
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⋊C4C22⋊D5311yescharacteristic
391C2×C22⋊C4C22⋊D5311yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—1061no—
396C2×C42C22×D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398#107—1062no—
399#107—1062no—
400#107—1062no—
401#106—2121no—
402C4⋊C8—1062no—
403#103—1062no—
404#103—1062no—
405#103—1062no—
406#103—2121no—
407C22⋊C8—1062no—
408C22⋊C8—2121no—
409#97—2121no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95C22⋊D5311yescharacteristic
420#95C22⋊D5311yescharacteristic
421C24C2×C22⋊D5311yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1171994512yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C23×D5311yescharacteristic
448C2×C8—532no—
449C2×C8—1061no—
450C2×C8—1061no—
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—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—2122no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4#1171994512yes—
480C4⋊C4#1171994712yes—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—2122no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22×C22⋊D5311yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
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—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
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—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4#1313324212yes—
550C2×C4#41×D5311yescharacteristic
551C2×C4C2×#1171994711yescharacteristic
552C2×C4C2×#1171994511yescharacteristic
553C2×C4C2×#1171994511yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326411yescharacteristic
556C8—1062no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1681113411yescharacteristic
568C22#1681126711yescharacteristic
569C22#1681127211yescharacteristic
570C22C2×#1313324211yescharacteristic
571C22#1681142111yescharacteristic
572C22#1681143811yescharacteristic
573C22#1681143811yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—4241no—
587C4#1681115311yescharacteristic
588C4#1681121511yescharacteristic
589C4#1681132311yescharacteristic
590C4#1681134311yescharacteristic
591C2—81no—
592C2#2027874511yescharacteristic
593C2#2028486611yescharacteristic
594C2#2028501511yescharacteristic
595C2#2028678111yescharacteristic
596C2#2028702011yescharacteristic
597C2#2028715511yescharacteristic
598C2#2028723911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2362C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2362D53
hypocenterC53#27058
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #20276123C2semidirect
3. #20276096C2semidirect
6. #20275473C2semidirect
7. #20275450C2semidirect
9. #20275304C2semidirect
10. #20275286C2semidirect
11. #20274969C2semidirect
14. #20274545C2semidirect
358. C53#27058semidirect

Maximal subgroups

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

subgroupindex
#202791372
#202761232
#202760962
#202759152
#202754732
#202754502
#202753302
#202753042
#202752862
#202749692
#202749142
#202748102
#202745452
C53×#23622
#2705853