Finite Group Databasebeta

← #27418127

Subgroups — #27418127

Order 13568.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic, complemented
2C2×#16810733C211yesmaximal, characteristic, complemented
3#20279616C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275711C212yesmaximal, complemented
6#20275689C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20275032C212yesmaximal, complemented
9#20274908C212yesmaximal, complemented
10#20274725C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic, complemented
12C22×#11719922C2211yescharacteristic
13C2×#13133191C2212yes—
14C2×C4×D212C2211yescharacteristic
15#16810910—24no—
16#16810733—24no—
17C2×#13133148C2212yes—
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2212yes—
22C2×#13133137C2212yes—
23C2×#13133137C2211yescharacteristic
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yes—
26C2×#13133128C2211yescharacteristic
27#16810549—21no—
28#16810531—21no—
29#16810499—22no—
30#16810489C2211yescharacteristic
31#16810483C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53—21no—
36C53×#547—24no—
37C53×#547—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—22no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2212yes—
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—22no—
47C22×C4×D53—21no—
48C2×#11719922—22no—
49C2×#11719922—22no—
50C2×#11719922—22no—
51C22×#88322C2311yescharacteristic
52C22×#88322C2312yes—
53C22×#88322C2311yescharacteristic
54#13133191—24no—
55#13133191—42no—
56C4×D212—22no—
57C4×D212—22no—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2312yes—
60C2×C22⋊Dic53—22no—
61C2×#11719921—22no—
62C2×#11719921—21no—
63#13133148—24no—
64#13133148—22no—
65C4⋊#88322—22no—
66#13133138—22no—
67#13133138—42no—
68#13133137—24no—
69#13133137—22no—
70C22⋊#88322—24no—
71C22⋊#88322—42no—
72#13133128—22no—
73#13133124—21no—
74#13133124—22no—
75#13133124—21no—
76C2×#41×C53—28no—
77C2×#41×C53—48no—
78C22×Q8×C53—21no—
79C22×D4×C53—22no—
80C22×D4×C53D411yescharacteristic
81C22×D4×C53C2311yescharacteristic
82C23×C4×C53—22no—
83C23×C4×C53—21no—
84C23×C4×C53C2312yes—
85C4⋊Q8×C53—22no—
86C4⋊Q8×C53D414yescomplemented
87#123×C53—216no—
88C22⋊Q8×C53—28no—
89#121×C53—24no—
90#121×C53—24no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53D414yescomplemented
95C2×C4⋊C4×C53—24no—
96C2×C4⋊C4×C53—24no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2312yes—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—22no—
104C2×C22⋊C4×C53D412yes—
105C2×C42×C53C2311yescharacteristic
106C24×C53—21no—
107C24×C53C2×D412yes—
108C23×D53—21no—
109#41×C53—432no—
110C2×Q8×C53—24no—
111C2×Q8×C53—44no—
112C2×D4×C53—24no—
113C2×D4×C53—28no—
114C2×D4×C53—22no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—44no—
121C2×D4×C53—44no—
122C2×D4×C53C2×D412yes—
123C22×C4×C53—28no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—44no—
134C22×C4×C53—44no—
135C22×C4×C53—44no—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C2×C22⋊D53—44no—
141C2×C22⋊D53—44no—
142C2×C22⋊D53—82no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212#4112yescomplemented
151C2×C4×D53—42no—
152C2×C4×D53—82no—
153C4⋊C4×C53—28no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53C2×D412yes—
159C4⋊C4×C53C2×D414yes—
160C22⋊C4×C53—28no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C42×C53C2×D412yes—
164C42×C53C2×D412yes—
165C22⋊Dic53—82no—
166#11719922—28no—
167#11719921—44no—
168#11719921—42no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—21no—
179C2×#88322—22no—
180C2×#88322—21no—
181C2×#88322#4112yescomplemented
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53AGL(1,ℤ/8)14yes—
194C23×C53C22×D411yescharacteristic
195C22×D53—22no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—42no—
199C22×D53—42no—
200Q8×C53—48no—
201D4×C53—28no—
202D4×C53—22no—
203D4×C53—22no—
204D4×C53—416no—
205D4×C53—41no—
206D4×C53—41no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—48no—
215C2×C4×C53—416no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53#12012yes—
224C2×C4×C53#12012yes—
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53H(5,𝔽2)11yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C22⋊D53—88no—
232C2×Dic53—41no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237D212—22no—
238D212—22no—
239D212—41no—
240D212—41no—
241C4×D53—84no—
242#88322—22no—
243#88322—41no—
244#88322—42no—
245#24128—531nomaximal, Sylow
246C22×C53—24no—
247C22×C53—24no—
248C22×C53—24no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—82no—
257C22×C53—82no—
258C22×C53C2×#12011yescharacteristic
259C22×C53#54611yescharacteristic
260C22×C53#54711yescharacteristic
261C22×C53C2×AGL(1,ℤ/8)12yes—
262C22×C53#57712yes—
263C2×D53—42no—
264C2×D53—44no—
265C2×D53—82no—
266C2×D53—81no—
267C2×D53—81no—
268C4×C53—22no—
269C4×C53—22no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53—44no—
273C4×C53#52511yescharacteristic
274C4×C53#53711yescharacteristic
275C4×C53#54811yescharacteristic
276C4×C53#55011yescharacteristic
277Dic53—81no—
278Dic53—82no—
279C2×#547D5311yescharacteristic, complemented
280C2×#476—531no—
281C2×#442—531no—
282#2535—531no—
283#2034—532no—
284#1999—532no—
285#1993—531no—
286#1955—531no—
287#1874—532no—
288#1853—532no—
289#1804—531no—
290C2×C53—44no—
291C2×C53—82no—
292C2×C53#240711yescharacteristic
293C2×C53#300311yescharacteristic
294C2×C53#301011yescharacteristic
295C2×C53#330612yes—
296C2×C53#334512yes—
297D53—82no—
298C22×#41—21no—
299C22×D8—531no—
300#547—24no—
301#547—24no—
302C2×C4⋊Q8C2×D5311yescharacteristic
303C2×#123—22no—
304C2×C22⋊Q8—21no—
305C2×#121—532no—
306C2×#121C2×D5312yes—
307C2×C4×D4—531no—
308C2×C4×D4C2×D5311yescharacteristic
309C2×C4×D4C2×D5311yescharacteristic
310C22×C4⋊C4C2×D5312yes—
311#476—1064no—
312#442—1064no—
313C2×#106—531no—
314C2×#106—532no—
315C2×C4⋊C8—531no—
316C2×#102—531no—
317C2×#102—531no—
318C2×#102—532no—
319C2×#102—532no—
320C2×C22⋊C8—532no—
321C2×C22⋊C8—532no—
322C2×#97—531no—
323#397—1061no—
324#392—1061no—
325#386—1062no—
326#382—531no—
327#380—532no—
328#340—532no—
329#340—532no—
330#340—532no—
331C53#2412811yescharacteristic, complemented, Sylow
332C2×#41—28no—
333C2×#41—48no—
334C22×Q8—21no—
335C22×D4—22no—
336C22×D4—531no—
337C22×D4—1062no—
338C22×D4C22⋊D5311yescharacteristic
339C22×D4C22×D5311yescharacteristic
340C23×C4—22no—
341C23×C4—21no—
342C23×C4—1061no—
343C23×C4C22×D5312yes—
344C2×D8—1062no—
345C2×D8—1062no—
346C2×D8—1062no—
347C22×C8—531no—
348C22×C8—531no—
349C22×C8—532no—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D5314yescomplemented
352#123—216no—
353C22⋊Q8—28no—
354#121—24no—
355#121—24no—
356#121—1064no—
357#121—2122no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—1062no—
362C4×D4—1062no—
363C4×D4C22⋊D5314yescomplemented
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—532no—
370C2×C4⋊C4C22⋊D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C4⋊C4C22×D5312yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1061no—
377C2×C22⋊C4—1062no—
378C2×C22⋊C4C22⋊D5312yes—
379C2×C42C22×D5311yescharacteristic
380#106—1062no—
381#106—1064no—
382C4⋊C8—1062no—
383#102—1062no—
384#102—1062no—
385#102—1064no—
386#102—2122no—
387C22⋊C8—1064no—
388C22⋊C8—2122no—
389#97—1062no—
390#95—1062no—
391#95—1061no—
392#95—1061no—
393C24—21no—
394C24—1061no—
395C24C2×C22⋊D5312yes—
396#41—432no—
397C2×Q8—24no—
398C2×Q8—44no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—532no—
410C2×D4—1061no—
411C2×D4—1061no—
412C2×D4—1061no—
413C2×D4—1061no—
414C2×D4—1061no—
415C2×D4—4242no—
416C2×D4—2124no—
417C2×D4—2124no—
418C2×D4C2×C22⋊D5312yes—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—1062no—
433C22×C4—1061no—
434C22×C4—4242no—
435C22×C4—2122no—
436C22×C4C2×C22⋊D5312yes—
437C22×C4C2×C22⋊D5311yescharacteristic
438C22×C4C2×C22⋊D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440D8—1068no—
441C2×C8—532no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1062no—
446C2×C8—1062no—
447C2×C8—1061no—
448C2×C8—1062no—
449C2×C8—1061no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1062no—
458C4⋊C4—1062no—
459C4⋊C4C2×C22⋊D5312yes—
460C4⋊C4C2×C22⋊D5314yes—
461C22⋊C4—28no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—4242no—
465C22⋊C4—2124no—
466C22⋊C4—2122no—
467C42C2×C22⋊D5312yes—
468C42C2×C22⋊D5312yes—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—1062no—
481C23—2121no—
482C23—2121no—
483C23—2122no—
484C23—2122no—
485C23#1313324314yes—
486C23C22×C22⋊D5311yescharacteristic
487Q8—48no—
488D4—28no—
489D4—22no—
490D4—22no—
491D4—416no—
492D4—41no—
493D4—41no—
494D4—1062no—
495D4—1062no—
496D4—4248no—
497D4—2121no—
498D4—2121no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—416no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—4244no—
516C2×C4—2121no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4#1313318612yes—
522C2×C4#1313318612yes—
523C2×C4C22×C22⋊D5311yescharacteristic
524C2×C4C22×C22⋊D5311yescharacteristic
525C2×C4#41×D5311yescharacteristic
526C2×C4#41×D5311yescharacteristic
527C2×C4#1313326311yescharacteristic
528C2×C4#1313326411yescharacteristic
529C8—1062no—
530C8—2122no—
531C8—2121no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—4241no—
545C22—4241no—
546C22—4242no—
547C22—2122no—
548C22—2124no—
549C22C2×#1313318611yescharacteristic
550C22#1681126111yescharacteristic
551C22#1681127211yescharacteristic
552C22C2×#1313324312yes—
553C22#1681143112yes—
554C4—22no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—4241no—
560C4—4242no—
561C4#1681113011yescharacteristic
562C4#1681120111yescharacteristic
563C4#1681128011yescharacteristic
564C4#1681129211yescharacteristic
565C2—44no—
566C2—82no—
567C2—4242no—
568C2#2027805411yescharacteristic
569C2#2028394211yescharacteristic
570C2#2028400111yescharacteristic
571C2#2028667012yes—
572C2#2028718512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#24128
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
2. C2×#16810733C2semidirect
3. #20279616C2semidirect
4. #20275901C2semidirect
5. #20275711C2semidirect
6. #20275689C2semidirect
7. #20275514C2semidirect
8. #20275032C2semidirect
9. #20274908C2semidirect
10. #20274725C2semidirect
11. C2×C53×#547C2semidirect
86. C4⋊Q8×C53D4semidirect
94. C4×D4×C53D4semidirect
150. C2×D212#41semidirect
181. C2×#88322#41semidirect
279. C2×#547D53semidirect
331. C53#24128semidirect
351. C4⋊Q8C22⋊D53semidirect
363. C4×D4C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168107332
#202796162
#202759012
#202757112
#202756892
#202755142
#202750322
#202749082
#202747252
C2×C53×#5472
#2412853