Finite Group Databasebeta

← #27437407

Subgroups — #27437407

Order 13568.

1162 conjugacy classes of subgroups.

The subgroup lattice has 534 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#20275664C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20275037C212yesmaximal, complemented
9#20275027C212yesmaximal, complemented
10#20274545C211yesmaximal, characteristic, complemented
11C2×C53×#556C211yesmaximal, 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×#556—24no—
36C53×#556—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#126×C53—22no—
39C2×#124×C53—21no—
40C2×C22⋊Q8×C53C2212yes—
41C2×C4×Q8×C53—21no—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C2×#117×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×C53C2312yes—
82C4⋊Q8×C53—22no—
83C4⋊Q8×C53D414yescomplemented
84#126×C53—216no—
85#124×C53—28no—
86C22⋊Q8×C53—24no—
87C22⋊Q8×C53—24no—
88C4×Q8×C53—28no—
89C4×Q8×C53—22no—
90C4×Q8×C53D414yescomplemented
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93#117×C53—24no—
94#117×C53—24no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2312yes—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53—22no—
103C2×C22⋊C4×C53D412yes—
104C2×C42×C53—22no—
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—44no—
114C2×Q8×C53C2×D412yes—
115C2×D4×C53—22no—
116C2×D4×C53—42no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2411yescharacteristic
130C2×C22⋊D53—44no—
131C2×C22⋊D53—44no—
132C2×C22⋊D53—82no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—22no—
138C2×D212—22no—
139C2×Dic106—21no—
140C2×Dic106—21no—
141C2×Dic106—22no—
142C2×Dic106—22no—
143C4⋊C4×C53—28no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C2×D412yes—
150C4⋊C4×C53C2×D414yes—
151C22⋊C4×C53—28no—
152C22⋊C4×C53—28no—
153C22⋊C4×C53—24no—
154C42×C53—28no—
155C42×C53—24no—
156C42×C53C2×D412yes—
157C42×C53C2×D412yes—
158C22⋊Dic53—44no—
159#11719925—44no—
160#11719924—44no—
161#11719921—44no—
162#11719921—42no—
163#11719921—42no—
164C4⋊Dic53—22no—
165C4⋊Dic53#4114yes—
166#11719919—44no—
167C4×Dic53—81no—
168C2×#88322—22no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53C22×D411yescharacteristic
183C22×D53—41no—
184C22×D53—42no—
185C22×D53—42no—
186C22×D53—41no—
187C22×D53—41no—
188Q8×C53—22no—
189Q8×C53—22no—
190Q8×C53—48no—
191Q8×C53—42no—
192D4×C53—44no—
193C2×C4×C53—22no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—22no—
197C2×C4×C53—24no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—42no—
203C2×C4×C53—44no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53#12012yes—
211C2×C4×C53#12012yes—
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C2×#4112yes—
215C2×C4×C53H(5,𝔽2)12yes—
216C22⋊D53—88no—
217C2×Dic53—24no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—41no—
221C2×Dic53—42no—
222D212—41no—
223D212—41no—
224D212—42no—
225Dic106—41no—
226Dic106—41no—
227Dic106—42no—
228#88322—41no—
229#88322—41no—
230#88322—42no—
231#25580—531nomaximal, Sylow
232C22×C53—42no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—42no—
236C22×C53—42no—
237C22×C53—42no—
238C22×C53—42no—
239C22×C53C2×#12011yescharacteristic
240C22×C53#54612yes—
241C22×C53#57512yes—
242C22×C53#57711yescharacteristic
243C22×C53#57811yescharacteristic
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—82no—
248C2×D53—82no—
249C4×C53—22no—
250C4×C53—22no—
251C4×C53—42no—
252C4×C53—42no—
253C4×C53—44no—
254C4×C53—42no—
255C4×C53#53211yescharacteristic
256C4×C53#53511yescharacteristic
257C4×C53#55012yes—
258Dic53—44no—
259Dic53—81no—
260C2×#556D5311yescharacteristic, complemented
261C2×#489—531no—
262C2×#464—531no—
263#2540—531no—
264#2001—532no—
265#2000—532no—
266#1987—531no—
267#1955—531no—
268#1875—532no—
269#1873—532no—
270#1763—531no—
271C2×C53—82no—
272C2×C53#243111yescharacteristic
273C2×C53#300511yescharacteristic
274C2×C53#301311yescharacteristic
275C2×C53#331112yes—
276C2×C53#331312yes—
277D53—161no—
278C22×Q16—531no—
279C22×SD16—531no—
280#556—24no—
281#556—24no—
282C2×C4⋊Q8C2×D5311yescharacteristic
283C2×#126—22no—
284C2×#124—21no—
285C2×#124—531no—
286C2×C22⋊Q8—532no—
287C2×C22⋊Q8C2×D5312yes—
288C2×#121—532no—
289C2×C4×Q8—21no—
290C2×C4×Q8C2×D5311yescharacteristic
291C2×C4×D4C2×D5311yescharacteristic
292C2×#117C2×D5312yes—
293#489—1064no—
294#464—1064no—
295C2×C4⋊C8—531no—
296C2×#103—532no—
297C2×#103—532no—
298C2×#103—531no—
299C2×#102—532no—
300C2×#102—531no—
301C2×C22⋊C8—532no—
302C2×C22⋊C8—532no—
303C2×#97—531no—
304#399—1061no—
305#392—1061no—
306#390—1062no—
307C4×C4⋊C4—531no—
308#340—532no—
309#340—532no—
310#340—532no—
311C53#2558011yescharacteristic, complemented, Sylow
312C22×Q8—21no—
313C22×Q8—531no—
314C22×Q8C22×D5311yescharacteristic
315C22×D4—531no—
316C22×D4—1062no—
317C22×D4C22⋊D5311yescharacteristic
318C23×C4C22×D5312yes—
319C2×Q16—1062no—
320C2×Q16—2122no—
321C2×SD16—1062no—
322C2×SD16—2122no—
323C22×C8—531no—
324C22×C8—531no—
325C22×C8—532no—
326C4⋊Q8—22no—
327C4⋊Q8C22⋊D5314yescomplemented
328#126—216no—
329#124—28no—
330#124—1062no—
331#124—1062no—
332C22⋊Q8—24no—
333C22⋊Q8—24no—
334C22⋊Q8—1064no—
335C22⋊Q8—2122no—
336#121—1064no—
337#121—2122no—
338C4×Q8—28no—
339C4×Q8—22no—
340C4×Q8C22⋊D5314yescomplemented
341C4×D4—22no—
342C4×D4—22no—
343#117—24no—
344#117—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—532no—
349C2×C4⋊C4—1062no—
350C2×C4⋊C4C22⋊D5311yescharacteristic
351C2×C4⋊C4C22×D5311yescharacteristic
352C2×C4⋊C4C22×D5312yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—1062no—
356C2×C22⋊C4—1062no—
357C2×C22⋊C4—1061no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4C22⋊D5312yes—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—1061no—
363C2×C42C22×D5311yescharacteristic
364C4⋊C8—1062no—
365#103—1064no—
366#103—1062no—
367#103—1064no—
368#102—1062no—
369#102—1064no—
370C22⋊C8—2122no—
371C22⋊C8—2122no—
372#97—1062no—
373#95—1061no—
374C24—1061no—
375C24C2×C22⋊D5312yes—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—1061no—
382C2×Q8—1061no—
383C2×Q8—1062no—
384C2×Q8—1062no—
385C2×Q8C2×C22⋊D5312yes—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—1062no—
391C2×D4—1062no—
392C2×D4—4242no—
393C2×D4—2124no—
394C2×D4—2124no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—1062no—
405C22×C4—1061no—
406C22×C4C2×C22⋊D5312yes—
407C22×C4C2×C22⋊D5311yescharacteristic
408C22×C4C2×C22⋊D5311yescharacteristic
409C22×C4C23×D5311yescharacteristic
410Q16—2124no—
411SD16—2124no—
412C2×C8—1062no—
413C2×C8—1061no—
414C2×C8—1061no—
415C2×C8—1062no—
416C2×C8—1062no—
417C2×C8—1062no—
418C2×C8—1062no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—534no—
426C4⋊C4—1062no—
427C4⋊C4—2124no—
428C4⋊C4C2×C22⋊D5312yes—
429C4⋊C4C2×C22⋊D5314yes—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—2124no—
434C22⋊C4—2124no—
435C22⋊C4—2122no—
436C22⋊C4—2122no—
437C42—28no—
438C42—24no—
439C42—4241no—
440C42C2×C22⋊D5312yes—
441C42C2×C22⋊D5312yes—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—2121no—
450C23—2122no—
451C23—2121no—
452C23—2121no—
453C23—2122no—
454C23C22×C22⋊D5311yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—48no—
458Q8—42no—
459Q8—2122no—
460Q8—2121no—
461Q8—2121no—
462D4—44no—
463D4—4248no—
464D4—2122no—
465D4—2121no—
466D4—2121no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—1064no—
485C2×C4—2121no—
486C2×C4—2122no—
487C2×C4—2122no—
488C2×C4—2122no—
489C2×C4#1313318612yes—
490C2×C4#1313318612yes—
491C2×C4C22×C22⋊D5311yescharacteristic
492C2×C4C22×C22⋊D5311yescharacteristic
493C2×C4#41×D5312yes—
494C2×C4#1313326312yes—
495C8—2122no—
496C8—2121no—
497C8—2121no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—4242no—
506C22—4241no—
507C22—4242no—
508C22—4241no—
509C22—4241no—
510C22C2×#1313318611yescharacteristic
511C22#1681126112yes—
512C22#1681142112yes—
513C22#1681143111yescharacteristic
514C22#1681143811yescharacteristic
515C4—22no—
516C4—22no—
517C4—44no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—4241no—
522C4—2124no—
523C4#1681116911yescharacteristic
524C4#1681118711yescharacteristic
525C4#1681129212yes—
526C2—82no—
527C2—8481no—
528C2#2027833011yescharacteristic
529C2#2028395711yescharacteristic
530C2#2028403011yescharacteristic
531C2#2028674512yes—
532C2#2028677512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#556C2
socleC23×C53C22×D4
prime radical#20279654C2
hypercenterC2×#556D53
hypocenterC53#25580
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#556

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. #20275664C2semidirect
7. #20275514C2semidirect
8. #20275037C2semidirect
9. #20275027C2semidirect
10. #20274545C2semidirect
11. C2×C53×#556C2semidirect
83. C4⋊Q8×C53D4semidirect
90. C4×Q8×C53D4semidirect
260. C2×#556D53semidirect
311. C53#25580semidirect
327. C4⋊Q8C22⋊D53semidirect
340. 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
#202756642
#202755142
#202750372
#202750272
#202745452
C2×C53×#5562
#2558053