Finite Group Databasebeta

← #27366398

Subgroups — #27366398

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284719C211yesmaximal, characteristic, complemented
2#20284711C211yesmaximal, characteristic
3C2×C8×D212C211yesmaximal, characteristic, complemented
4#20274163C212yesmaximal, complemented
5#20274149C212yesmaximal, complemented
6#20274135C212yesmaximal, complemented
7#20274121C212yesmaximal, complemented
8#20273463C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10#20273364C211yesmaximal, characteristic, complemented
11C53×#3131C211yesmaximal, characteristic, complemented
12C22×C8×D53—21no—
13C2×C4×D212C2211yescharacteristic
14#16810951—21no—
15#16810934—21no—
16#16810910C2212yes—
17#16810900C2212yes—
18#16810825C2212yes—
19#16810815C2212yes—
20#16810773—21no—
21#16810756—21no—
22C4×#11719924—22no—
23C4×#11719922—22no—
24C8×D212—21no—
25C8×D212—21no—
26C8×D212C2212yes—
27C8×D212C2212yes—
28#16810676C2211yescharacteristic
29#16810676C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×#13133129—21no—
32#16810376C2212yescomplemented
33#16810374C2212yescomplemented
34#16810372C2212yescomplemented
35#16810369C2212yescomplemented
36#16810369C2212yescomplemented
37#16810367C2212yescomplemented
38C53×#533C2211yescharacteristic
39C53×#532C2211yescharacteristic
40C53×#488—21no—
41C53×#488—21no—
42C53×#487C2211yescharacteristic, complemented
43C53×#487C2211yescharacteristic, complemented
44C53×#486C2211yescharacteristic, complemented
45C53×#486C2211yescharacteristic, complemented
46C53×#416—21no—
47C53×#416—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—
77C4×#88322—21no—
78C4×#88322—21no—
79C2×#41×C53—21no—
80C22×C8×C53D411yescharacteristic
81C22×C8×C53D411yescharacteristic
82C4⋊Q8×C53C2311yescharacteristic
83C4⋊Q8×C53C2311yescharacteristic
84C4⋊Q8×C53C2311yescharacteristic
85#127×C53C2311yescharacteristic
86#125×C53—21no—
87#124×C53—21no—
88C22⋊Q8×C53—21no—
89C22⋊Q8×C53—21no—
90#121×C53—21no—
91#121×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96#117×C53—21no—
97C2×C42×C53C2311yescharacteristic
98#103×C53—21no—
99#103×C53—21no—
100#103×C53—21no—
101#103×C53—21no—
102#102×C53—21no—
103#102×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—42no—
112#41×C53—42no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—41no—
120C22×C4×C53—41no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×Dic53—21no—
125C2×D212—21no—
126C2×D212—24no—
127C2×D212C2×D412yes—
128C2×C4×D53—22no—
129C2×C4×D53—22no—
130C2×C4×D53—42no—
131C2×C4×D53—42no—
132C2×C8×C53—22no—
133C2×C8×C53—22no—
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C42×C53—41no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154#11719924—44no—
155#11719922—44no—
156#11719921—22no—
157#11719921—22no—
158C4⋊Dic53—21no—
159C4⋊Dic53C2×D412yes—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—42no—
164C8×D53—44no—
165C8×D53—44no—
166C23×C53—41no—
167C23×C53#12111yescharacteristic
168C22×D53—24no—
169C22×D53—22no—
170C22×D53—42no—
171C22×D53—42no—
172Q8×C53—41no—
173Q8×C53—42no—
174Q8×C53—41no—
175Q8×C53—81no—
176D4×C53—41no—
177D4×C53—42no—
178D4×C53—41no—
179D4×C53—81no—
180D4×C53—82no—
181C2×C4×C53—22no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—82no—
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#13512yes—
191C2×C4×C53AGL(1,ℤ/8)12yes—
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C2×Dic53—22no—
196C2×Dic53—42no—
197D212—24no—
198D212—22no—
199D212#13514yes—
200C4×D53—42no—
201C4×D53—42no—
202C4×D53—44no—
203C8×C53—21no—
204C8×C53—21no—
205C8×C53—21no—
206C8×C53—21no—
207#88322—42no—
208#88322—42no—
209#88322—42no—
210#19110—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—81no—
214C22×C53—81no—
215C22×C53—81no—
216C22×C53#46312yes—
217C22×C53C2×#12111yescharacteristic
218C2×D53—24no—
219C2×D53—48no—
220C2×D53—42no—
221C2×D53—42no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—81no—
226C4×C53—81no—
227C4×C53—81no—
228C4×C53#46312yes—
229C4×C53C2×#13512yes—
230C4×C53C2×#13511yescharacteristic
231C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
232Dic53—42no—
233#3131D5311yescharacteristic, complemented
234#3101—531no—
235#3100—531no—
236C2×C8×D4—531no—
237#1683—532no—
238#1681—532no—
239#1679—532no—
240#1677—532no—
241#1564—531no—
242#1542—531no—
243#1540—531no—
244C2×C53—22no—
245C2×C53—161no—
246C2×C53C2×#46311yescharacteristic
247C2×C53#305011yescharacteristic
248C2×C53#306711yescharacteristic
249D53—44no—
250C23×C8—1061no—
251#533C2×D5311yescharacteristic
252#532C2×D5311yescharacteristic
253C2×C4×D4—531no—
254#488—21no—
255#488—21no—
256#487C2×D5311yescharacteristic, complemented
257#487C2×D5311yescharacteristic, complemented
258#486C2×D5311yescharacteristic, complemented
259#486C2×D5311yescharacteristic, complemented
260#483—1061no—
261#480—1061no—
262#476—532no—
263#474—532no—
264#461—532no—
265#459—532no—
266#450—1061no—
267#447—1061no—
268C4×SD16—1062no—
269C4×D8—1062no—
270C8×D4—532no—
271C8×D4—532no—
272C8×D4—1061no—
273C8×D4—1061no—
274#431—531no—
275#431—531no—
276C2×C4⋊C8—531no—
277#416—21no—
278#416—21no—
279C2×C22⋊C8—1061no—
280C2×C4×C8C2×D5311yescharacteristic
281#335—532no—
282#334—532no—
283#333—532no—
284#332—532no—
285#332—532no—
286#331—532no—
287C53#1911011yescharacteristic, complemented, Sylow
288C2×#41—21no—
289C22×D4—531no—
290C23×C4—1061no—
291C2×SD16—1062no—
292C2×D8—1062no—
293C22×C8—1061no—
294C22×C8—1062no—
295C22×C8—1062no—
296C22×C8—2124no—
297C22×C8C22⋊D5311yescharacteristic
298C22×C8C22⋊D5311yescharacteristic
299C4⋊Q8C22×D5311yescharacteristic
300C4⋊Q8C22×D5311yescharacteristic
301C4⋊Q8C22×D5311yescharacteristic
302#127C22×D5311yescharacteristic
303#125—21no—
304#124—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307#121—21no—
308#121—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—532no—
314C4×D4—532no—
315C4×D4—1061no—
316C4×D4—1061no—
317#117—21no—
318C2×C4⋊C4—531no—
319C2×C22⋊C4—1061no—
320C2×C42C22×D5311yescharacteristic
321#107—1062no—
322#106—1062no—
323C4⋊C8—532no—
324C4⋊C8—532no—
325C4⋊C8—532no—
326C4⋊C8—1061no—
327#103—21no—
328#103—21no—
329#103—21no—
330#103—21no—
331#103—1062no—
332#102—21no—
333#102—21no—
334#102—21no—
335#102—21no—
336#102—1062no—
337#102—1062no—
338#102—1062no—
339C22⋊C8—1061no—
340C22⋊C8—1061no—
341C22⋊C8—1062no—
342C22⋊C8—1062no—
343C4×C8—1061no—
344C4×C8—1061no—
345C4×C8C22⋊D5311yescharacteristic
346C4×C8C22⋊D5311yescharacteristic
347C4×C8C22×D5311yescharacteristic
348C4×C8C22×D5311yescharacteristic
349C24—1061no—
350#41—42no—
351#41—42no—
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—2124no—
372D8—2124no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—1062no—
376C2×C8—1062no—
377C2×C8—1062no—
378C2×C8—2124no—
379C2×C8—2124no—
380C2×C8—2122no—
381C2×C8C2×C22⋊D5311yescharacteristic
382C2×C8C2×C22⋊D5311yescharacteristic
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C2×C22⋊D5311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—532no—
394C4⋊C4—1061no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—1062no—
399C22⋊C4—1062no—
400C42—41no—
401C42D4×D5311yescharacteristic
402C42#1171994711yescharacteristic
403C42C2×C22⋊D5311yescharacteristic
404C42C23×D5311yescharacteristic
405C23—41no—
406C23—1064no—
407C23—1062no—
408C23—2122no—
409C23—2122no—
410C23#1313318811yescharacteristic
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415D4—42no—
416D4—41no—
417D4—41no—
418D4—81no—
419D4—82no—
420D4—534no—
421D4—1064no—
422D4—1062no—
423C2×C4—22no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—1062no—
430C2×C4—2122no—
431C2×C4—2124no—
432C2×C4—2122no—
433C2×C4—2122no—
434C2×C4#1313318811yescharacteristic
435C2×C4#1313318811yescharacteristic
436C2×C4#1313318811yescharacteristic
437C2×C4#1313323712yes—
438C2×C4#1313324212yes—
439C2×C4C2×D4×D5311yescharacteristic
440C2×C4C22×C22⋊D5311yescharacteristic
441C2×C4C2×#1171994711yescharacteristic
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—2122no—
447C8—2122no—
448C8—2122no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—1064no—
455C22—2128no—
456C22—2122no—
457C22—2122no—
458C22#1681083512yes—
459C22C2×#1313318811yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—2122no—
467C4#1681083512yes—
468C4#135×D5312yes—
469C4C2×#1313323711yescharacteristic
470C4C2×#1313324211yescharacteristic
471C2—22no—
472C2—161no—
473C2—2124no—
474C2C2×#1681083511yescharacteristic
475C2#2028429211yescharacteristic
476C2#2028443111yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3131C2
socleC22×C53C2×#121
prime radical#20284711C2
hypercenter#3131D53
hypocenterC53#19110
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #20284719C2semidirect
3. C2×C8×D212C2semidirect
4. #20274163C2semidirect
5. #20274149C2semidirect
6. #20274135C2semidirect
7. #20274121C2semidirect
8. #20273463C2semidirect
9. #20273372C2semidirect
10. #20273364C2semidirect
11. C53×#3131C2semidirect
32. #16810376C22semidirect
33. #16810374C22semidirect
34. #16810372C22semidirect
35. #16810369C22semidirect
36. #16810369C22semidirect
37. #16810367C22semidirect
42. C53×#487C22semidirect
43. C53×#487C22semidirect
44. C53×#486C22semidirect
45. C53×#486C22semidirect
233. #3131D53semidirect
256. #487C2×D53semidirect
257. #487C2×D53semidirect
258. #486C2×D53semidirect
259. #486C2×D53semidirect
287. C53#19110semidirect

Maximal subgroups

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

subgroupindex
#202847192
#202847112
C2×C8×D2122
#202741632
#202741492
#202741352
#202741212
#202734632
#202733722
#202733642
C53×#31312
#1911053