Finite Group Databasebeta

← #27437396

Subgroups — #27437396

Order 13568.

1206 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810985C211yesmaximal, characteristic, complemented
2C2×#16810845C211yesmaximal, characteristic, complemented
3#20279654C211yesmaximal, characteristic
4#20275723C212yesmaximal, complemented
5#20275716C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20275032C212yesmaximal, complemented
9#20275022C212yesmaximal, complemented
10#20274545C211yesmaximal, characteristic, complemented
11C2×C53×#554C211yesmaximal, characteristic, complemented
12C22×#11719925C2211yescharacteristic
13C22×#11719924C2211yescharacteristic
14C2×#13133205C2211yescharacteristic
15C2×#13133196C2212yes—
16C2×#13133191C2212yes—
17#16810985—24no—
18#16810845—24no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133141C2212yes—
21C2×#13133140C2212yes—
22C2×#13133140C2211yescharacteristic
23C2×#13133137C2212yes—
24C2×#13133137C2211yescharacteristic
25C2×C22⋊#88322C2212yes—
26C2×C22⋊#88322C2212yes—
27C2×#13133128C2211yescharacteristic
28#16810558—21no—
29#16810531—21no—
30#16810523—22no—
31C4×C4⋊Dic53C2211yescharacteristic
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C53×#554—24no—
36C53×#554—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#125×C53—21no—
39C2×C22⋊Q8×C53—22no—
40C2×C22⋊Q8×C53C2212yes—
41C2×C4×Q8×C53C2211yescharacteristic
42C2×C4×D4×C53—21no—
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2212yes—
45C22×Dic106C2311yescharacteristic
46C22×D212C2311yescharacteristic
47C22×C22⋊D53—22no—
48C2×#11719925—22no—
49C2×#11719925—42no—
50C2×#11719924—22no—
51C2×#11719924—42no—
52C22×#88322C2312yes—
53C22×#88322C2311yescharacteristic
54C22×#88322C2311yescharacteristic
55#13133205—22no—
56#13133205—22no—
57#13133196—24no—
58#13133196—42no—
59#13133191—24no—
60#13133191—42no—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—22no—
63C2×C22⋊Dic53—22no—
64C2×#11719921—21no—
65C2×#11719921—22no—
66C2×#11719921—21no—
67C2×C4×Dic53—21no—
68C4⋊#88322—22no—
69#13133141—24no—
70#13133140—22no—
71#13133140—24no—
72#13133137—22no—
73#13133137—24no—
74C22⋊#88322—42no—
75C22⋊#88322—42no—
76#13133128—22no—
77#13133124—21no—
78C22×Q8×C53—21no—
79C22×Q8×C53C2311yescharacteristic
80C22×D4×C53D411yescharacteristic
81C23×C4×C53—22no—
82C23×C4×C53C2312yes—
83C4⋊Q8×C53—22no—
84C4⋊Q8×C53D414yescomplemented
85#125×C53—28no—
86C22⋊Q8×C53—216no—
87C22⋊Q8×C53—24no—
88C22⋊Q8×C53—24no—
89C4×Q8×C53—22no—
90C4×Q8×C53D414yescomplemented
91C4×D4×C53—28no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—24no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2312yes—
103C2×C22⋊C4×C53—22no—
104C2×C22⋊C4×C53D412yes—
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C24×C53C2×D412yes—
108C23×D53—21no—
109C2×Q8×C53—24no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—22no—
113C2×Q8×C53—81no—
114C2×Q8×C53—81no—
115C2×Q8×C53C2×D412yes—
116C2×D4×C53—24no—
117C2×D4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—28no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—82no—
129C22×C4×C53—82no—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—44no—
135C2×C22⋊D53—44no—
136C2×C22⋊D53—82no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—22no—
142C2×D212—22no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—22no—
146C2×Dic106—22no—
147C4⋊C4×C53—28no—
148C4⋊C4×C53—28no—
149C4⋊C4×C53—24no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53C2×D412yes—
155C4⋊C4×C53C2×D414yes—
156C22⋊C4×C53—28no—
157C22⋊C4×C53—24no—
158C42×C53—24no—
159C42×C53C2×D412yes—
160C42×C53C2×D412yes—
161C22⋊Dic53—44no—
162#11719925—44no—
163#11719924—44no—
164#11719921—44no—
165#11719921—42no—
166#11719921—42no—
167C4⋊Dic53—22no—
168C4⋊Dic53#4114yes—
169#11719919—44no—
170C4×Dic53—81no—
171C2×#88322—22no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53#13714yes—
185C23×C53C22×D411yescharacteristic
186C22×D53—41no—
187C22×D53—42no—
188C22×D53—42no—
189C22×D53—41no—
190C22×D53—41no—
191Q8×C53—22no—
192Q8×C53—22no—
193Q8×C53—42no—
194Q8×C53—84no—
195D4×C53—28no—
196C2×C4×C53—22no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—42no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—88no—
215C2×C4×C53#12012yes—
216C2×C4×C53#12012yes—
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C2×#4112yes—
220C2×C4×C53H(5,𝔽2)12yes—
221C22⋊D53—88no—
222C2×Dic53—24no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—41no—
226C2×Dic53—42no—
227D212—41no—
228D212—41no—
229D212—42no—
230Dic106—41no—
231Dic106—41no—
232Dic106—42no—
233#88322—41no—
234#88322—41no—
235#88322—42no—
236#25579—531nomaximal, Sylow
237C22×C53—24no—
238C22×C53—24no—
239C22×C53—24no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53C2×#12011yescharacteristic
245C22×C53#54612yes—
246C22×C53C2×#13712yes—
247C22×C53#57611yescharacteristic
248C22×C53#57711yescharacteristic
249C2×D53—81no—
250C2×D53—81no—
251C2×D53—81no—
252C2×D53—82no—
253C2×D53—82no—
254C4×C53—22no—
255C4×C53—22no—
256C4×C53—42no—
257C4×C53—42no—
258C4×C53—42no—
259C4×C53—44no—
260C4×C53#53211yescharacteristic
261C4×C53#53511yescharacteristic
262C4×C53#55012yes—
263Dic53—44no—
264Dic53—81no—
265C2×#554D5311yescharacteristic, complemented
266C2×#489—531no—
267C2×#464—531no—
268#2540—531no—
269#2001—532no—
270#2000—532no—
271#1988—531no—
272#1955—531no—
273#1874—532no—
274#1872—532no—
275#1763—531no—
276C2×C53—44no—
277C2×C53#243111yescharacteristic
278C2×C53#300411yescharacteristic
279C2×C53#301211yescharacteristic
280C2×C53#331412yes—
281C2×C53#331612yes—
282D53—161no—
283C22×Q16—531no—
284C22×SD16—531no—
285#554—24no—
286#554—24no—
287C2×C4⋊Q8C2×D5311yescharacteristic
288C2×#125—21no—
289C2×#124—531no—
290C2×C22⋊Q8—22no—
291C2×C22⋊Q8—532no—
292C2×C22⋊Q8C2×D5312yes—
293C2×#121—532no—
294C2×C4×Q8C2×D5311yescharacteristic
295C2×C4×D4—21no—
296C2×C4×D4C2×D5311yescharacteristic
297C22×C4⋊C4C2×D5312yes—
298#489—1064no—
299#464—1064no—
300C2×C4⋊C8—531no—
301C2×#103—532no—
302C2×#103—532no—
303C2×#103—531no—
304C2×#102—532no—
305C2×#102—531no—
306C2×C22⋊C8—532no—
307C2×C22⋊C8—532no—
308C2×#97—531no—
309#399—1061no—
310#392—1061no—
311#390—1062no—
312C4×C4⋊C4—531no—
313#340—532no—
314#340—532no—
315#340—532no—
316C53#2557911yescharacteristic, complemented, Sylow
317C22×Q8—21no—
318C22×Q8—531no—
319C22×Q8C22×D5311yescharacteristic
320C22×D4—531no—
321C22×D4—1062no—
322C22×D4C22⋊D5311yescharacteristic
323C23×C4—22no—
324C23×C4C22×D5312yes—
325C2×Q16—1062no—
326C2×Q16—2122no—
327C2×SD16—1062no—
328C2×SD16—2122no—
329C22×C8—531no—
330C22×C8—531no—
331C22×C8—532no—
332C4⋊Q8—22no—
333C4⋊Q8C22⋊D5314yescomplemented
334#125—28no—
335#124—1062no—
336#124—1062no—
337C22⋊Q8—216no—
338C22⋊Q8—24no—
339C22⋊Q8—24no—
340C22⋊Q8—1064no—
341C22⋊Q8—2122no—
342#121—1064no—
343#121—2122no—
344C4×Q8—22no—
345C4×Q8C22⋊D5314yescomplemented
346C4×D4—28no—
347C4×D4—22no—
348C4×D4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—532no—
356C2×C4⋊C4—1062no—
357C2×C4⋊C4C22⋊D5311yescharacteristic
358C2×C4⋊C4C22×D5311yescharacteristic
359C2×C4⋊C4C22×D5312yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—1062no—
362C2×C22⋊C4—1062no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4C22⋊D5312yes—
366C2×C42—21no—
367C2×C42—1061no—
368C2×C42C22×D5311yescharacteristic
369C4⋊C8—1062no—
370#103—1064no—
371#103—1062no—
372#103—1064no—
373#102—1062no—
374#102—1064no—
375C22⋊C8—2122no—
376C22⋊C8—2122no—
377#97—1062no—
378#95—1061no—
379C24—1061no—
380C24C2×C22⋊D5312yes—
381C2×Q8—24no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—1061no—
388C2×Q8—1061no—
389C2×Q8—1062no—
390C2×Q8—1062no—
391C2×Q8C2×C22⋊D5312yes—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—1061no—
395C2×D4—1061no—
396C2×D4—1062no—
397C2×D4—1062no—
398C2×D4—4242no—
399C2×D4—2124no—
400C2×D4—2124no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—1062no—
414C22×C4—1061no—
415C22×C4C2×C22⋊D5312yes—
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C23×D5311yescharacteristic
419Q16—2124no—
420SD16—2124no—
421C2×C8—1062no—
422C2×C8—1061no—
423C2×C8—1061no—
424C2×C8—1062no—
425C2×C8—1062no—
426C2×C8—1062no—
427C2×C8—1062no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—534no—
436C4⋊C4—1062no—
437C4⋊C4—2124no—
438C4⋊C4C2×C22⋊D5312yes—
439C4⋊C4C2×C22⋊D5314yes—
440C22⋊C4—28no—
441C22⋊C4—24no—
442C22⋊C4—2124no—
443C22⋊C4—2124no—
444C22⋊C4—2122no—
445C22⋊C4—2122no—
446C42—24no—
447C42—4241no—
448C42C2×C22⋊D5312yes—
449C42C2×C22⋊D5312yes—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—2121no—
457C23—2122no—
458C23—2121no—
459C23—2121no—
460C23—2122no—
461C23#1313324814yes—
462C23C22×C22⋊D5311yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—84no—
467Q8—2122no—
468Q8—2121no—
469Q8—2121no—
470D4—28no—
471D4—4248no—
472D4—2122no—
473D4—2121no—
474D4—2121no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—88no—
494C2×C4—1064no—
495C2×C4—2121no—
496C2×C4—2122no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4#1313318612yes—
500C2×C4#1313318612yes—
501C2×C4C22×C22⋊D5311yescharacteristic
502C2×C4C22×C22⋊D5311yescharacteristic
503C2×C4#41×D5312yes—
504C2×C4#1313326312yes—
505C8—2122no—
506C8—2121no—
507C8—2121no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—4241no—
516C22—4241no—
517C22—4242no—
518C22—4242no—
519C22—4241no—
520C22C2×#1313318611yescharacteristic
521C22#1681126112yes—
522C22C2×#1313324812yes—
523C22#1681142611yescharacteristic
524C22#1681143411yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—44no—
531C4—4241no—
532C4—2124no—
533C4#1681116911yescharacteristic
534C4#1681118711yescharacteristic
535C4#1681129212yes—
536C2—44no—
537C2—8481no—
538C2#2027833011yescharacteristic
539C2#2028394911yescharacteristic
540C2#2028401911yescharacteristic
541C2#2028679012yes—
542C2#2028682012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#554C2
socleC23×C53C22×D4
prime radical#20279654C2
hypercenterC2×#554D53
hypocenterC53#25579
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16810985C2semidirect
2. C2×#16810845C2semidirect
4. #20275723C2semidirect
5. #20275716C2semidirect
6. #20275668C2semidirect
7. #20275514C2semidirect
8. #20275032C2semidirect
9. #20275022C2semidirect
10. #20274545C2semidirect
11. C2×C53×#554C2semidirect
84. C4⋊Q8×C53D4semidirect
90. C4×Q8×C53D4semidirect
265. C2×#554D53semidirect
316. C53#25579semidirect
333. C4⋊Q8C22⋊D53semidirect
345. C4×Q8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109852
C2×#168108452
#202796542
#202757232
#202757162
#202756682
#202755142
#202750322
#202750222
#202745452
C2×C53×#5542
#2557953