Finite Group Databasebeta

← #27369295

Subgroups — #27369295

Order 13568.

684 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284802C211yesmaximal, characteristic, complemented
2#20284711C211yesmaximal, characteristic, complemented
3#20283738C211yesmaximal, characteristic, complemented
4#20274163C212yesmaximal, complemented
5#20274121C212yesmaximal, complemented
6#20274067C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273420C211yesmaximal, characteristic, complemented
10#20273377C211yesmaximal, characteristic, complemented
11C53×#3207C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810939—21no—
14#16810934—21no—
15#16810910C2212yes—
16#16810900C2212yes—
17#16810815C2212yes—
18#16810815C2212yes—
19#16810756—21no—
20#16810756—21no—
21#16810733—22no—
22C4×#11719922—22no—
23C8×D212C2212yes—
24C8×D212C2212yes—
25#16810688—22no—
26#16810680C2211yescharacteristic
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29C22⋊C8×D53—22no—
30#16810376C2212yescomplemented
31#16810374C2212yescomplemented
32#16810372C2212yescomplemented
33#16810369C2212yescomplemented
34#16810367C2212yescomplemented
35#16810367C2212yescomplemented
36C53×#532C2211yescharacteristic
37C53×#532C2211yescharacteristic
38C53×#493C2211yescharacteristic, complemented
39C53×#492C2211yescharacteristic, complemented
40C53×#487C2211yescharacteristic, complemented
41C53×#486C2211yescharacteristic, complemented
42C53×#463—22no—
43C53×#449—22no—
44C53×#431C2211yescharacteristic
45C22×D212D411yescharacteristic
46C22×C4×D53—21no—
47C2×#11719922—22no—
48C2×#11719922—22no—
49C2×C8×D53—22no—
50C2×C8×D53—22no—
51C4×D212—21no—
52C4×D212—21no—
53C4×D212D412yes—
54C4×D212C2312yes—
55C2×C4⋊Dic53D411yescharacteristic
56C2×#11719921—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133138—22no—
63#13133138—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69#13133129—22no—
70#13133129—22no—
71#13133128—21no—
72C4×#88322—21no—
73C4×#88322—21no—
74C2×#41×C53—21no—
75C2×#41×C53—21no—
76C2×SD16×C53—22no—
77C2×D8×C53—22no—
78C4⋊Q8×C53C2311yescharacteristic
79C4⋊Q8×C53C2311yescharacteristic
80#127×C53C2311yescharacteristic
81#127×C53C2311yescharacteristic
82#124×C53—21no—
83#124×C53—21no—
84#121×C53—21no—
85#121×C53—21no—
86#121×C53—21no—
87#121×C53—21no—
88C4×D4×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C2×C42×C53C2311yescharacteristic
93C4⋊C8×C53D412yes—
94#103×C53—22no—
95#102×C53—22no—
96C22⋊C8×C53D412yes—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C23×D53—21no—
100#41×C53—82no—
101#41×C53—82no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C22×C4×C53—41no—
113C22×C4×C53—41no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×Dic53—21no—
118C2×D212—21no—
119C2×D212—24no—
120C2×D212C2×D412yes—
121C2×C4×D53—22no—
122C2×C4×D53—22no—
123C2×C4×D53—82no—
124SD16×C53—82no—
125D8×C53—82no—
126C2×C8×C53C2×D412yes—
127C2×C8×C53C2×D412yes—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140#11719922—44no—
141#11719922—44no—
142#11719921—22no—
143#11719921—22no—
144C4⋊Dic53—21no—
145C4⋊Dic53C2×D412yes—
146C2×#88322—22no—
147C2×#88322—22no—
148C2×#88322—22no—
149C8×D53—82no—
150C8×D53—82no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53#12111yescharacteristic
154C22×D53—22no—
155C22×D53—24no—
156C22×D53—81no—
157C22×D53—81no—
158Q8×C53—42no—
159Q8×C53—81no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—81no—
165D4×C53—81no—
166D4×C53—82no—
167D4×C53—82no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—82no—
174C2×C4×C53—82no—
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C22×D411yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C2×Dic53—22no—
182C2×Dic53—81no—
183D212—24no—
184D212—22no—
185D212#13514yes—
186C4×D53—42no—
187C4×D53—42no—
188C4×D53—82no—
189C8×C53—22no—
190C8×C53—22no—
191#88322—42no—
192#88322—42no—
193#88322—42no—
194#19389—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53C2×#12111yescharacteristic
205C2×D53—24no—
206C2×D53—42no—
207C2×D53—42no—
208C2×D53—84no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53C2×#13512yes—
216C4×C53#57612yes—
217Dic53—42no—
218#3207D5311yescharacteristic, complemented
219#3109—531no—
220#3100—531no—
221#2972—531no—
222#1683—532no—
223#1677—532no—
224#1667—532no—
225#1665—532no—
226#1572—531no—
227#1553—531no—
228#1543—531no—
229C2×C53—41no—
230C2×C53—161no—
231C2×C53—161no—
232C2×C53#304111yescharacteristic
233C2×C53#305011yescharacteristic
234C2×C53#306811yescharacteristic
235D53—44no—
236#532C2×D5311yescharacteristic
237#532C2×D5311yescharacteristic
238C2×C4×D4—531no—
239#493C2×D5311yescharacteristic, complemented
240#492C2×D5311yescharacteristic, complemented
241#487C2×D5311yescharacteristic, complemented
242#486C2×D5311yescharacteristic, complemented
243#481—1061no—
244#480—1061no—
245#476—532no—
246#474—532no—
247#463—22no—
248#459—532no—
249#459—532no—
250#449—22no—
251#447—1061no—
252#447—1061no—
253#442—1062no—
254C4×D8—1062no—
255C8×D4—532no—
256C8×D4—532no—
257C8×D4—1062no—
258#432—531no—
259#431—531no—
260#431—531no—
261#431C2×D5311yescharacteristic
262C2×C22⋊C8—1062no—
263#335—532no—
264#334—532no—
265#333—532no—
266#332—532no—
267#331—532no—
268#331—532no—
269C53#1938911yescharacteristic, complemented, Sylow
270C2×#41—21no—
271C2×#41—21no—
272C22×D4—531no—
273C23×C4—1061no—
274C2×SD16—22no—
275C2×D8—22no—
276C2×D8—1062no—
277C2×D8—1062no—
278C22×C8—1062no—
279C22×C8—1062no—
280C4⋊Q8C22×D5311yescharacteristic
281C4⋊Q8C22×D5311yescharacteristic
282#127C22×D5311yescharacteristic
283#127C22×D5311yescharacteristic
284#124—21no—
285#124—21no—
286#121—21no—
287#121—21no—
288#121—21no—
289#121—21no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—532no—
295C4×D4—532no—
296C4×D4—1061no—
297C4×D4—1061no—
298C2×C4⋊C4—531no—
299C2×C22⋊C4—1061no—
300C2×C42C22×D5311yescharacteristic
301#107—1062no—
302#106—1062no—
303C4⋊C8—532no—
304C4⋊C8—532no—
305C4⋊C8—532no—
306C4⋊C8C22⋊D5312yes—
307#103—22no—
308#102—22no—
309#102—1062no—
310#102—1062no—
311#102—1062no—
312#102—1062no—
313C22⋊C8—1061no—
314C22⋊C8—1061no—
315C22⋊C8—1061no—
316C22⋊C8—1062no—
317C22⋊C8—1062no—
318C22⋊C8C22⋊D5312yes—
319#97—1061no—
320C4×C8—1061no—
321C4×C8—1061no—
322C4×C8C22×D5311yescharacteristic
323C4×C8C22×D5311yescharacteristic
324C24—1061no—
325#41—82no—
326#41—82no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—532no—
338C2×D4—1064no—
339C2×D4—1061no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—1061no—
343C22×C4—1062no—
344C22×C4—1062no—
345C22×C4—4242no—
346C22×C4D4×D5311yescharacteristic
347C22×C4#1171994711yescharacteristic
348C22×C4C2×C22⋊D5311yescharacteristic
349SD16—82no—
350D8—82no—
351D8—2124no—
352D8—2124no—
353C2×C8—1062no—
354C2×C8—1062no—
355C2×C8—1062no—
356C2×C8—4242no—
357C2×C8—4242no—
358C2×C8C2×C22⋊D5312yes—
359C2×C8C2×C22⋊D5312yes—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—532no—
365C4⋊C4—1061no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—1062no—
371C22⋊C4—1062no—
372C42D4×D5311yescharacteristic
373C42#1171994711yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C23×D5311yescharacteristic
376C23—41no—
377C23—41no—
378C23—1064no—
379C23—1062no—
380C23—4241no—
381C23—4241no—
382C23#1313318811yescharacteristic
383Q8—42no—
384Q8—81no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—81no—
390D4—81no—
391D4—82no—
392D4—82no—
393D4—534no—
394D4—1064no—
395D4—1062no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—1062no—
404C2×C4—4242no—
405C2×C4—4241no—
406C2×C4—2122no—
407C2×C4—2122no—
408C2×C4#1313318811yescharacteristic
409C2×C4#1313318811yescharacteristic
410C2×C4#1313318811yescharacteristic
411C2×C4C2×D4×D5311yescharacteristic
412C2×C4C22×C22⋊D5311yescharacteristic
413C2×C4C2×#1171994711yescharacteristic
414C8—22no—
415C8—22no—
416C8—2122no—
417C8—2122no—
418C8—2122no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—1064no—
429C22—4244no—
430C22—2122no—
431C22—2122no—
432C22C2×#1313318811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—81no—
438C4—81no—
439C4—2122no—
440C4#135×D5312yes—
441C4#1681142612yes—
442C2—41no—
443C2—161no—
444C2—161no—
445C2—2124no—
446C2#2028421411yescharacteristic
447C2#2028429211yescharacteristic
448C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3207C2
socleC22×C53C2×#121
hypercenter#3207D53
hypocenterC53#19389
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3207

Complements

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

NH ≅ G/Ntype
1. #20284802C2semidirect
2. #20284711C2semidirect
3. #20283738C2semidirect
4. #20274163C2semidirect
5. #20274121C2semidirect
6. #20274067C2semidirect
7. #20274053C2semidirect
8. #20273495C2semidirect
9. #20273420C2semidirect
10. #20273377C2semidirect
11. C53×#3207C2semidirect
30. #16810376C22semidirect
31. #16810374C22semidirect
32. #16810372C22semidirect
33. #16810369C22semidirect
34. #16810367C22semidirect
35. #16810367C22semidirect
38. C53×#493C22semidirect
39. C53×#492C22semidirect
40. C53×#487C22semidirect
41. C53×#486C22semidirect
218. #3207D53semidirect
239. #493C2×D53semidirect
240. #492C2×D53semidirect
241. #487C2×D53semidirect
242. #486C2×D53semidirect
269. C53#19389semidirect

Maximal subgroups

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

subgroupindex
#202848022
#202847112
#202837382
#202741632
#202741212
#202740672
#202740532
#202734952
#202734202
#202733772
C53×#32072
#1938953