Finite Group Databasebeta

← #27394621

Subgroups — #27394621

Order 13568.

834 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
1C2×#16810675C211yesmaximal, characteristic
2#20277111C211yesmaximal, characteristic
3#382×D53C211yesmaximal, characteristic
4#20275485C211yesmaximal, characteristic, complemented
5#20275483C211yesmaximal, characteristic, complemented
6#20275481C211yesmaximal, characteristic, complemented
7#20274909C211yesmaximal, characteristic
8#20274906C211yesmaximal, characteristic
9#20274904C211yesmaximal, characteristic
10#20274901C211yesmaximal, characteristic
11#20274878C211yesmaximal, characteristic, complemented
12#20274876C211yesmaximal, characteristic, complemented
13#20274844C211yesmaximal, characteristic, complemented
14#20274844C211yesmaximal, characteristic, complemented
15C53×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D53C2211yescharacteristic
17C2×C4⋊C4×D53C2211yescharacteristic
18C2×C42×D53C2211yescharacteristic
19#16810675C418yescomplemented
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28C2×#13133129C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810490—21no—
32#16810489C2211yescharacteristic
33#16810488C2211yescharacteristic
34#16810488C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810488C2211yescharacteristic
37#16810487—21no—
38#16810487—21no—
39#16810485—21no—
40#16810480C2211yescharacteristic
41#16810480C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C4×#11719921—21no—
44#16810469—21no—
45#95×D53—21no—
46#95×D53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C4⋊C4×D53—41no—
74C4⋊C4×D53—41no—
75C4⋊C4×D53—41no—
76C4⋊C4×D53—41no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83C42×D53C2×C414yes—
84C42×D53Q814yes—
85#13133151—41no—
86#13133150—41no—
87#13133150—41no—
88#13133149—41no—
89#13133148—41no—
90#13133148—41no—
91C4⋊#88322C2×C414yescomplemented
92#13133129C2×C414yes—
93#13133129C2×C414yes—
94C8×Dic53C2×C414yescomplemented
95C8×Dic53C2×C414yescomplemented
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#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117#107×C53—41no—
118#106×C53—41no—
119C4⋊C8×C53C2×C414yescomplemented
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123C23×D53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53—21no—
143C2×C4×D53—21no—
144C2×C4×D53—41no—
145C2×C4×D53—41no—
146C2×C4×D53—42no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53C4⋊C414yes—
153C2×C4×D53C4⋊C414yes—
154C2×C4×D53C22×C412yes—
155C2×C4×D53C2×Q812yes—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53C22×C412yes—
161C2×C8×C53C22×C412yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C42×C53—42no—
171C42×C53C22×C412yes—
172C42×C53C2×Q812yes—
173#11719921—42no—
174#11719921—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183#11719919—42no—
184#11719919—42no—
185C4×Dic53—42no—
186C4×Dic53C4⋊C414yes—
187C4×Dic53C4⋊C414yes—
188C4×Dic53C22×C412yes—
189C4×Dic53C2×Q812yes—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322C22×C412yes—
195C2×#88322C22×C412yes—
196C23×C53C22×D411yescharacteristic
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53C2×C4⋊C412yes—
202C22×D53#13512yes—
203C22×D53#13512yes—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53C2×C4⋊C412yes—
221C2×C4×C53C2×C4⋊C412yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53H(5,𝔽2)11yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53C2×C4⋊C412yes—
247C2×Dic53C2×C4⋊C412yes—
248C2×Dic53C2×C4⋊C412yes—
249C4×D53—28no—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—44no—
255C4×D53—84no—
256C4×D53—84no—
257C8×C53—22no—
258C8×C53—22no—
259#88322—22no—
260#88322—22no—
261#21983—531nomaximal, Sylow
262C22×C53C22×C4⋊C411yescharacteristic
263C22×C53#53511yescharacteristic
264C22×C53#53611yescharacteristic
265C22×C53C2×#13511yescharacteristic
266C22×C53C2×#13511yescharacteristic
267C22×C53#57511yescharacteristic
268C22×C53#57511yescharacteristic
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53#42714yes—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53#52011yescharacteristic
283C4×C53#52011yescharacteristic
284C4×C53#54411yescharacteristic
285C4×C53#54411yescharacteristic
286Dic53—22no—
287Dic53—22no—
288Dic53—22no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293C2×#431—531no—
294#2283—531no—
295C2×#382—531no—
296#1949—531no—
297#1949—531no—
298#1949—531no—
299#1949D5311yescharacteristic, complemented
300#1853—531no—
301#1853—531no—
302#1852—531no—
303#1852—531no—
304#1846—531no—
305#1846—531no—
306#1835—531no—
307#1835—531no—
308C2×C53#231711yescharacteristic
309C2×C53C2×#42711yescharacteristic
310C2×C53#291111yescharacteristic
311C2×C53#321111yescharacteristic
312C2×C53#321111yescharacteristic
313C2×C53#323211yescharacteristic
314C2×C53#323211yescharacteristic
315D53—24no—
316C22×C4⋊C4—531no—
317C22×C4⋊C4—531no—
318C22×C42—531no—
319#431—538no—
320C2×#107—531no—
321C2×#107—531no—
322C2×#107—531no—
323C2×#107C2×D5311yescharacteristic
324C2×#106—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×#106C2×D5311yescharacteristic
328C2×C4⋊C8—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330C2×C22⋊C8—531no—
331C2×C22⋊C8—531no—
332C2×C4×C8—531no—
333C2×C4×C8—531no—
334#383—1061no—
335#382—531no—
336#382—531no—
337#382—531no—
338#382—531no—
339#382—531no—
340#382—1061no—
341#382—1061no—
342#382C2×D5311yescharacteristic
343#381—1061no—
344#380—531no—
345#380—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4C2×D5311yescharacteristic
348C4×C22⋊C4—1061no—
349#376—1061no—
350C2×#95—1061no—
351C2×#95—1061no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340C2×D5311yescharacteristic
359#340C2×D5311yescharacteristic
360C53#2198311yescharacteristic, complemented, Sylow
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—531no—
364C23×C4—1061no—
365C23×C4—1061no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8C22×D5311yescharacteristic
369C22×C8C22×D5311yescharacteristic
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—2121no—
377C2×C4⋊C4—2121no—
378C2×C4⋊C4—2121no—
379C2×C4⋊C4—2121no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—21no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—534no—
391C2×C42—534no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394#107—41no—
395#107—2121no—
396#107—2121no—
397#107—2121no—
398#106—41no—
399#106—2121no—
400#106—2121no—
401#106—2121no—
402C4⋊C8—534no—
403C4⋊C8C4×D5314yescomplemented
404C22⋊C8—534no—
405C22⋊C8—534no—
406C4×C8—534no—
407C4×C8—534no—
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#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424C24—531no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—534no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—534no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—2121no—
447C22×C4—2121no—
448C22×C4—2122no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4D4×D5311yescharacteristic
455C22×C4D4×D5311yescharacteristic
456C22×C4C23×D5311yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—532no—
462C2×C8—532no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8C2×C4×D5312yes—
468C2×C8C2×C4×D5312yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—42no—
490C42—534no—
491C42—534no—
492C42—532no—
493C42—532no—
494C42—2122no—
495C42C2×C4×D5312yes—
496C42Q8×D5312yes—
497C23—532no—
498C23—532no—
499C23—532no—
500C23—1061no—
501C23—1061no—
502C23—1061no—
503C23—1061no—
504C23C2×D4×D5311yescharacteristic
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—1068no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—4244no—
536C2×C4—4244no—
537C2×C4—2124no—
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—2121no—
550C2×C4C4⋊C4×D5312yes—
551C2×C4C4⋊C4×D5312yes—
552C2×C4C22×C4×D5311yescharacteristic
553C2×C4C2×Q8×D5311yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326311yescharacteristic
556C2×C4#1313326411yescharacteristic
557C2×C4#1313326511yescharacteristic
558C8—22no—
559C8—22no—
560C8—1062no—
561C8—1062no—
562C22—534no—
563C22—1062no—
564C22—1062no—
565C22—1062no—
566C22—1062no—
567C22—1062no—
568C22—1062no—
569C22C2×C4⋊C4×D5311yescharacteristic
570C22#1681118911yescharacteristic
571C22#1681119611yescharacteristic
572C22#135×D5311yescharacteristic
573C22#135×D5311yescharacteristic
574C22#1681141911yescharacteristic
575C22#1681142211yescharacteristic
576C4—22no—
577C4—22no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—1062no—
583C4—1062no—
584C4—1062no—
585C4—4241no—
586C4—4241no—
587C4—4241no—
588C4—4241no—
589C4#1681110211yescharacteristic
590C4#1681110511yescharacteristic
591C4#1681125011yescharacteristic
592C4#1681125311yescharacteristic
593C2—1064no—
594C2#2027733711yescharacteristic
595C2#427×D5311yescharacteristic
596C2#2028330211yescharacteristic
597C2#2028567811yescharacteristic
598C2#2028568411yescharacteristic
599C2#2028588111yescharacteristic
600C2#2028588711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53#21983
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #20275485C2semidirect
5. #20275483C2semidirect
6. #20275481C2semidirect
11. #20274878C2semidirect
12. #20274876C2semidirect
13. #20274844C2semidirect
14. #20274844C2semidirect
15. C53×#1949C2semidirect
19. #16810675C4semidirect
91. C4⋊#88322C2×C4semidirect
94. C8×Dic53C2×C4semidirect
95. C8×Dic53C2×C4semidirect
119. C4⋊C8×C53C2×C4semidirect
299. #1949D53semidirect
360. C53#21983semidirect
403. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106752
#202771112
#382×D532
#202754852
#202754832
#202754812
#202749092
#202749062
#202749042
#202749012
#202748782
#202748762
#202748442
C53×#19492
#2198353