Finite Group Databasebeta

← #27367893

Subgroups — #27367893

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284810C211yesmaximal, characteristic, complemented
2#20284802C211yesmaximal, characteristic
3C2×C8×D212C211yesmaximal, characteristic, complemented
4#20274163C212yesmaximal, complemented
5#20274149C212yesmaximal, complemented
6#20274087C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273478C211yesmaximal, characteristic, complemented
9#20273415C211yesmaximal, characteristic, complemented
10#20273410C211yesmaximal, characteristic, complemented
11C53×#3167C211yesmaximal, characteristic, complemented
12C22×C8×D53—21no—
13C2×C4×D212C2211yescharacteristic
14#16810956—21no—
15#16810939—21no—
16#16810910C2212yes—
17#16810900C2212yes—
18#16810825C2212yes—
19#16810815C2212yes—
20#16810773—21no—
21#16810756—21no—
22#16810733—22no—
23#16810721—22no—
24C8×D212—21no—
25C8×D212—21no—
26C8×D212C2212yes—
27C8×D212C2212yes—
28#16810680C2211yescharacteristic
29#16810680C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×#13133129—21no—
32#16810376C2212yescomplemented
33#16810374C2212yescomplemented
34#16810372C2212yescomplemented
35#16810369C2212yescomplemented
36#16810369C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#533C2211yescharacteristic
39C53×#532C2211yescharacteristic
40C53×#498C2211yescharacteristic, complemented
41C53×#492C2211yescharacteristic, complemented
42C53×#487C2211yescharacteristic, complemented
43C53×#486C2211yescharacteristic, complemented
44C53×#465—21no—
45C53×#465—21no—
46C4×SD16×C53—21no—
47C4×SD16×C53—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C22×D212D411yescharacteristic
50C22×C4×D53—21no—
51C2×#11719924—22no—
52C2×#11719922—22no—
53C22×#88322—21no—
54C2×C8×D53—22no—
55C2×C8×D53—22no—
56C2×C8×D53—44no—
57C4×D212—21no—
58C4×D212—21no—
59C4×D212D412yes—
60C4×D212C2312yes—
61C2×C4⋊Dic53D411yescharacteristic
62C2×#11719921—21no—
63#13133150—22no—
64#13133148—22no—
65C4⋊#88322—21no—
66C4⋊#88322C2312yes—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133141—22no—
70#13133138—22no—
71#13133137—22no—
72#13133137—22no—
73C22⋊#88322—21no—
74C22⋊#88322—21no—
75#13133129—22no—
76#13133129—22no—
77#13133128—21no—
78#13133128—21no—
79C2×#41×C53—21no—
80C2×SD16×C53—21no—
81C2×SD16×C53—21no—
82C22×C8×C53D411yescharacteristic
83C22×C8×C53D411yescharacteristic
84C4⋊Q8×C53C2311yescharacteristic
85C4⋊Q8×C53C2311yescharacteristic
86C4⋊Q8×C53C2311yescharacteristic
87#127×C53C2311yescharacteristic
88#125×C53—21no—
89#124×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53—21no—
92#121×C53—21no—
93#121×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98#117×C53—21no—
99C2×C42×C53C2311yescharacteristic
100#106×C53—21no—
101#106×C53—21no—
102#103×C53—21no—
103#103×C53—21no—
104#102×C53—21no—
105#102×C53—21no—
106C4×C8×C53D411yescharacteristic
107C4×C8×C53D411yescharacteristic
108C4×C8×C53C2311yescharacteristic
109C4×C8×C53C2311yescharacteristic
110C23×D53—21no—
111#41×C53—82no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×Dic53—21no—
124C2×D212—21no—
125C2×D212—24no—
126C2×D212C2×D412yes—
127C2×C4×D53—22no—
128C2×C4×D53—22no—
129C2×C4×D53—42no—
130C2×C4×D53—42no—
131SD16×C53—42no—
132SD16×C53—42no—
133C2×C8×C53—22no—
134C2×C8×C53—41no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C42×C53C2×D411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53#4111yescharacteristic
154C42×C53C2411yescharacteristic
155#11719924—44no—
156#11719922—44no—
157#11719921—22no—
158#11719921—22no—
159C4⋊Dic53—21no—
160C4⋊Dic53C2×D412yes—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322—42no—
165C8×D53—44no—
166C8×D53—44no—
167C23×C53—41no—
168C23×C53#12111yescharacteristic
169C22×D53—24no—
170C22×D53—22no—
171C22×D53—42no—
172C22×D53—42no—
173Q8×C53—42no—
174Q8×C53—41no—
175Q8×C53—41no—
176Q8×C53—81no—
177D4×C53—42no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—81no—
181D4×C53—82no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—82no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C2×Dic53—22no—
197C2×Dic53—42no—
198D212—24no—
199D212—22no—
200D212#13514yes—
201C4×D53—44no—
202C4×D53—42no—
203C4×D53—42no—
204C8×C53—21no—
205C8×C53—21no—
206C8×C53—21no—
207C8×C53AGL(1,ℤ/8)12yes—
208#88322—42no—
209#88322—42no—
210#88322—42no—
211#19257—531nomaximal, Sylow
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—81no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53#47112yes—
218C22×C53C2×#12111yescharacteristic
219C2×D53—24no—
220C2×D53—48no—
221C2×D53—42no—
222C2×D53—42no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53C2×#13512yes—
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C4×C53#57511yescharacteristic
233Dic53—42no—
234#3167D5311yescharacteristic, complemented
235#3110—531no—
236#3109—531no—
237C2×C8×D4—531no—
238#1683—532no—
239#1681—532no—
240#1671—532no—
241#1665—532no—
242#1568—531no—
243#1552—531no—
244#1551—531no—
245C2×C53—22no—
246C2×C53—161no—
247C2×C53#303611yescharacteristic
248C2×C53#305011yescharacteristic
249C2×C53C2×#47111yescharacteristic
250D53—44no—
251C23×C8—1061no—
252#533C2×D5311yescharacteristic
253#532C2×D5311yescharacteristic
254C2×C4×D4—531no—
255#498C2×D5311yescharacteristic, complemented
256#492C2×D5311yescharacteristic, complemented
257#487C2×D5311yescharacteristic, complemented
258#486C2×D5311yescharacteristic, complemented
259#484—1061no—
260#481—1061no—
261#476—532no—
262#474—532no—
263#465—21no—
264#465—21no—
265#461—532no—
266#459—532no—
267#450—1061no—
268#447—1061no—
269#442—1062no—
270#440—1062no—
271C4×SD16—21no—
272C4×SD16—21no—
273C8×D4—532no—
274C8×D4—532no—
275C8×D4—1061no—
276C8×D4—1061no—
277#432—531no—
278#432—531no—
279C2×C4⋊C8—531no—
280C2×C22⋊C8—1061no—
281C2×C4×C8C2×D5311yescharacteristic
282#335—532no—
283#334—532no—
284#333—532no—
285#332—532no—
286#332—532no—
287#331—532no—
288C53#1925711yescharacteristic, complemented, Sylow
289C2×#41—21no—
290C22×D4—531no—
291C23×C4—1061no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—1062no—
295C2×D8—1062no—
296C22×C8—1061no—
297C22×C8—1062no—
298C22×C8—1062no—
299C22×C8—2124no—
300C22×C8C22⋊D5311yescharacteristic
301C22×C8C22⋊D5311yescharacteristic
302C4⋊Q8C22×D5311yescharacteristic
303C4⋊Q8C22×D5311yescharacteristic
304C4⋊Q8C22×D5311yescharacteristic
305#127C22×D5311yescharacteristic
306#125—21no—
307#124—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310#121—21no—
311#121—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—532no—
317C4×D4—532no—
318C4×D4—1061no—
319C4×D4—1061no—
320#117—21no—
321C2×C4⋊C4—531no—
322C2×C22⋊C4—1061no—
323C2×C42C22×D5311yescharacteristic
324#107—1062no—
325#106—21no—
326#106—21no—
327#106—1062no—
328C4⋊C8—532no—
329C4⋊C8—532no—
330C4⋊C8—532no—
331C4⋊C8—1061no—
332#103—21no—
333#103—21no—
334#103—1062no—
335#102—21no—
336#102—21no—
337#102—1062no—
338#102—1062no—
339#102—1062no—
340C22⋊C8—1061no—
341C22⋊C8—1061no—
342C22⋊C8—1062no—
343C22⋊C8—1062no—
344#97—1061no—
345#97—1061no—
346C4×C8C22⋊D5311yescharacteristic
347C4×C8C22⋊D5311yescharacteristic
348C4×C8C22×D5311yescharacteristic
349C4×C8C22×D5311yescharacteristic
350C24—1061no—
351#41—82no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—532no—
360C2×D4—1064no—
361C2×D4—1061no—
362C22×C4—41no—
363C22×C4—1061no—
364C22×C4—1062no—
365C22×C4—1062no—
366C22×C4—2122no—
367C22×C4—2122no—
368C22×C4D4×D5311yescharacteristic
369C22×C4#1171994711yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371SD16—42no—
372SD16—42no—
373SD16—2124no—
374D8—2124no—
375C2×C8—22no—
376C2×C8—41no—
377C2×C8—1062no—
378C2×C8—1062no—
379C2×C8—1062no—
380C2×C8—2124no—
381C2×C8—2122no—
382C2×C8—2124no—
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C2×C22⋊D5311yescharacteristic
385C2×C8C2×C22⋊D5311yescharacteristic
386C2×C8C2×C22⋊D5311yescharacteristic
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—532no—
396C4⋊C4—1061no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—1062no—
401C22⋊C4—1062no—
402C42—41no—
403C42D4×D5311yescharacteristic
404C42#1171994711yescharacteristic
405C42C2×C22⋊D5311yescharacteristic
406C42C23×D5311yescharacteristic
407C23—41no—
408C23—1064no—
409C23—1062no—
410C23—2122no—
411C23—2122no—
412C23#1313318811yescharacteristic
413Q8—42no—
414Q8—41no—
415Q8—41no—
416Q8—81no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—81no—
421D4—82no—
422D4—534no—
423D4—1064no—
424D4—1062no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—1062no—
434C2×C4—2124no—
435C2×C4—2122no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4#1313318811yescharacteristic
439C2×C4#1313318811yescharacteristic
440C2×C4#1313318811yescharacteristic
441C2×C4C2×D4×D5311yescharacteristic
442C2×C4C22×C22⋊D5311yescharacteristic
443C2×C4C2×#1171994711yescharacteristic
444C8—21no—
445C8—21no—
446C8—21no—
447C8—2122no—
448C8—2122no—
449C8—2122no—
450C8#1313324212yes—
451C22—21no—
452C22—21no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—1064no—
457C22—2128no—
458C22—2122no—
459C22—2122no—
460C22#1681088312yes—
461C22C2×#1313318811yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—2122no—
470C4#135×D5312yes—
471C4C2×#1313324211yescharacteristic
472C4#1681142111yescharacteristic
473C2—22no—
474C2—161no—
475C2—2124no—
476C2#2028417511yescharacteristic
477C2#2028429211yescharacteristic
478C2C2×#1681088311yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3167C2
socleC22×C53C2×#121
prime radical#20284802C2
hypercenter#3167D53
hypocenterC53#19257
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20284810C2semidirect
3. C2×C8×D212C2semidirect
4. #20274163C2semidirect
5. #20274149C2semidirect
6. #20274087C2semidirect
7. #20274053C2semidirect
8. #20273478C2semidirect
9. #20273415C2semidirect
10. #20273410C2semidirect
11. C53×#3167C2semidirect
32. #16810376C22semidirect
33. #16810374C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
36. #16810369C22semidirect
37. #16810367C22semidirect
40. C53×#498C22semidirect
41. C53×#492C22semidirect
42. C53×#487C22semidirect
43. C53×#486C22semidirect
234. #3167D53semidirect
255. #498C2×D53semidirect
256. #492C2×D53semidirect
257. #487C2×D53semidirect
258. #486C2×D53semidirect
288. C53#19257semidirect

Maximal subgroups

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

subgroupindex
#202848102
#202848022
C2×C8×D2122
#202741632
#202741492
#202740872
#202740532
#202734782
#202734152
#202734102
C53×#31672
#1925753