Finite Group Databasebeta

← #27456818

Subgroups — #27456818

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#20279126C211yesmaximal, characteristic
2#20276133C211yesmaximal, characteristic, complemented
3#20276110C211yesmaximal, characteristic, complemented
4#20275915C211yesmaximal, characteristic
5#20275908C211yesmaximal, characteristic
6#20275483C211yesmaximal, characteristic, complemented
7#20275445C211yesmaximal, characteristic, complemented
8#20275323C211yesmaximal, characteristic
9#20275297C211yesmaximal, characteristic, complemented
10#20275286C211yesmaximal, characteristic, complemented
11#20274969C211yesmaximal, characteristic, complemented
12#20274903C211yesmaximal, characteristic
13#20274801C211yesmaximal, characteristic
14C4×#13133141C211yesmaximal, characteristic, complemented
15C53×#2360C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810564—21no—
29#16810551—21no—
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810511—21no—
33#16810490—21no—
34#16810489C2211yescharacteristic
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
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#387—21no—
55C53×#385—21no—
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C22⋊C4×C53—21no—
58C4×C22⋊C4×C53C2211yescharacteristic
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×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53—21no—
72#13133150—22no—
73#13133150—22no—
74#13133150—41no—
75#13133148—22no—
76C4⋊#88322—22no—
77#13133141—22no—
78#13133141—22no—
79#13133141—22no—
80#13133141—41no—
81C22⋊#88322—22no—
82C22⋊#88322—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
92C23×C4×C53—21no—
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—22no—
95C22⋊Q8×C53—22no—
96C4×Q8×C53—22no—
97C4×Q8×C53—41no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—41no—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C22⋊C4×C53D411yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53#4112yes—
127C22×C4×C53—21no—
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—42no—
138C22×C4×C53—42no—
139C22×C4×C53—81no—
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—24no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—81no—
158C4⋊C4×C53#4112yes—
159C4⋊C4×C53#4112yes—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—42no—
168C42×C53—81no—
169C22⋊Dic53—42no—
170C22⋊Dic53—81no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179#11719919—81no—
180#11719919—81no—
181C4×Dic53—42no—
182C2×#88322—21no—
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#4112yes—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53#13512yes—
201C23×C53C22×D411yescharacteristic
202Q8×C53—22no—
203Q8×C53—22no—
204Q8×C53—41no—
205Q8×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—21no—
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—41no—
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—84no—
237C2×C4×C53#13712yes—
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#27035—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53#52511yescharacteristic
269C22×C53#54611yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57611yescharacteristic
274C22×C53#57811yescharacteristic
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—41no—
282C4×C53—42no—
283C4×C53—81no—
284C4×C53#52911yescharacteristic
285C4×C53#53911yescharacteristic
286C4×C53#55311yescharacteristic
287C4×C53#55611yescharacteristic
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2495—531no—
293#2360D5311yescharacteristic
294#2074—531no—
295#2068—531no—
296#2036—531no—
297#2035—531no—
298#1949—531no—
299#1942—531no—
300#1923—531no—
301#1920—531no—
302#1918—531no—
303#1863—531no—
304#1852—531no—
305#1823—531no—
306C4×#103—531no—
307C2×C53—42no—
308C2×C53#246611yescharacteristic
309C2×C53#310811yescharacteristic
310C2×C53#312411yescharacteristic
311C2×C53#329711yescharacteristic
312C2×C53#331211yescharacteristic
313C2×C53#333611yescharacteristic
314C2×C53#335711yescharacteristic
315C2×C22⋊Q8C2×D5311yescharacteristic
316C2×C4×Q8C2×D5311yescharacteristic
317C22×C4⋊C4C2×D5311yescharacteristic
318C2×#107—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#106—531no—
322C2×C4⋊C8—531no—
323C2×#103—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×C22⋊C8—531no—
328C2×C22⋊C8—531no—
329C2×C4×C8—531no—
330#400—1061no—
331#397—1061no—
332#391C2×D5311yescharacteristic
333#389—21no—
334#389—531no—
335#389—531no—
336#389C2×D5311yescharacteristic
337#388—1061no—
338#387—21no—
339#385—21no—
340#383—1061no—
341#382—531no—
342#380—531no—
343#380—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4C2×D5311yescharacteristic
346C4×C22⋊C4—21no—
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#2703511yescharacteristic, complemented, Sylow
359C22×Q8C22×D5311yescharacteristic
360C23×C4—21no—
361C23×C4C22×D5311yescharacteristic
362C22×C8—531no—
363C22×C8—531no—
364C22×C8—531no—
365C22×C8—531no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C4×Q8—22no—
369C4×Q8—41no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—41no—
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⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4C22⋊D5311yescharacteristic
389C2×C22⋊C4C22⋊D5311yescharacteristic
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395#107—1062no—
396#107—1062no—
397#107—2121no—
398#106—1062no—
399C4⋊C8—1062no—
400#103—1062no—
401#103—1062no—
402#103—1062no—
403#103—2121no—
404C22⋊C8—1062no—
405C22⋊C8—2121no—
406C4×C8—2121no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95C22⋊D5311yescharacteristic
419#95C22⋊D5311yescharacteristic
420C24C2×C22⋊D5311yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1171994512yes—
427C22×C4—21no—
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—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×C4C23×D5311yescharacteristic
450C2×C8—532no—
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—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—4241no—
479C4⋊C4—4241no—
480C4⋊C4#1171994512yes—
481C4⋊C4#1171994712yes—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—4241no—
486C22⋊C4—2122no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—81no—
493C42—2122no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1313323712yes—
501C23C22×C22⋊D5311yescharacteristic
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
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—84no—
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#1313324812yes—
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—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1681113411yescharacteristic
568C22#1681126711yescharacteristic
569C22#1681127211yescharacteristic
570C22C2×#1313323711yescharacteristic
571C22C2×#1313324811yescharacteristic
572C22#1681142611yescharacteristic
573C22#1681143811yescharacteristic
574C4—22no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—4241no—
587C4#1681115311yescharacteristic
588C4#1681121511yescharacteristic
589C4#1681132311yescharacteristic
590C4#1681134311yescharacteristic
591C2—42no—
592C2#2027874511yescharacteristic
593C2#2028479511yescharacteristic
594C2#2028497711yescharacteristic
595C2#2028656511yescharacteristic
596C2#2028676611yescharacteristic
597C2#2028707711yescharacteristic
598C2#2028734111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2360C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2360D53
hypocenterC53#27035
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #20276133C2semidirect
3. #20276110C2semidirect
6. #20275483C2semidirect
7. #20275445C2semidirect
9. #20275297C2semidirect
10. #20275286C2semidirect
11. #20274969C2semidirect
14. C4×#13133141C2semidirect
358. C53#27035semidirect

Maximal subgroups

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

subgroupindex
#202791262
#202761332
#202761102
#202759152
#202759082
#202754832
#202754452
#202753232
#202752972
#202752862
#202749692
#202749032
#202748012
C4×#131331412
C53×#23602
#2703553