Finite Group Databasebeta

← #27369302

Subgroups — #27369302

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#20285644C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274168C212yesmaximal, complemented
5#20274126C212yesmaximal, complemented
6#20274068C212yesmaximal, complemented
7#20274054C212yesmaximal, complemented
8#20273496C211yesmaximal, characteristic, complemented
9#20273421C211yesmaximal, characteristic, complemented
10#20273377C211yesmaximal, characteristic, complemented
11C53×#3109C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810999C2211yescharacteristic, complemented
14#16810995C2211yescharacteristic, complemented
15#16810975C2211yescharacteristic, complemented
16#16810971C2211yescharacteristic, complemented
17#16810936—21no—
18#16810903C2212yes—
19#16810838—22no—
20#16810818C2212yes—
21#16810766—22no—
22#16810757—21no—
23C4×#11719922—22no—
24D4×#88322—22no—
25D4×#88322C2212yes—
26D4×#88322C2212yes—
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29#16810676C2211yescharacteristic
30C2×C22⋊#88322—22no—
31#16810375C2212yescomplemented
32#16810373C2212yescomplemented
33#16810372C2212yescomplemented
34#16810368C2212yescomplemented
35#16810367C2212yescomplemented
36#16810366C2212yescomplemented
37C53×#532C2211yescharacteristic
38C2×C4×D4×C53C2211yescharacteristic
39C53×#481—21no—
40C53×#476C2212yes—
41C53×#459C2212yes—
42C53×#447—21no—
43C53×#442—22no—
44C53×#432C2211yescharacteristic
45C2×#11719945—21no—
46C2×#11719923—22no—
47C2×#11719922—22no—
48C2×#11719922—22no—
49C22×#88322—22no—
50C22×#88322—22no—
51#13133217C2311yescharacteristic
52#13133214C2311yescharacteristic
53#13133205—21no—
54#13133191—21no—
55#13133191—21no—
56C4×D212—21no—
57C4×D212—21no—
58#13133150—22no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133140—22no—
63#13133138—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68C22⋊#88322—21no—
69C22⋊#88322—21no—
70C22⋊#88322D412yes—
71C4×#88322—21no—
72C4×#88322—21no—
73C4×#88322C2311yescharacteristic
74C4×#88322C2311yescharacteristic
75C2×#41×C53—21no—
76C22×D4×C53D411yescharacteristic
77C23×C4×C53—21no—
78C2×D8×C53—22no—
79C4⋊Q8×C53C2311yescharacteristic
80#127×C53C2311yescharacteristic
81#124×C53—21no—
82#121×C53—21no—
83#121×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53—21no—
87C4×D4×C53—21no—
88C4×D4×C53D412yes—
89C4×D4×C53C2312yes—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C22⋊C4×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#106×C53—22no—
94C4⋊C8×C53C2312yes—
95#102×C53—22no—
96#102×C53—22no—
97C22⋊C8×C53—21no—
98#97×C53—21no—
99C24×C53—21no—
100#41×C53—82no—
101C2×Q8×C53—21no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—41no—
108C2×D4×C53C2×D412yes—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—41no—
113C22×C4×C53—82no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—41no—
118#11719945—82no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—21no—
122C2×C4×D53—41no—
123C2×Dic106—21no—
124D8×C53—44no—
125C2×C8×C53—22no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53C2×D412yes—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136C42×C53#4111yescharacteristic
137C42×C53C2411yescharacteristic
138#11719923—82no—
139#11719922—44no—
140#11719922—82no—
141#11719921—41no—
142#11719921—41no—
143C4⋊Dic53—21no—
144C4⋊Dic53—21no—
145C2×#88322—22no—
146C2×#88322—22no—
147C2×#88322—82no—
148C2×#88322—82no—
149C2×#88322C2×D412yes—
150C2×#88322C2×D412yes—
151C23×C53—24no—
152C23×C53—22no—
153C23×C53—41no—
154C23×C53—81no—
155C23×C53—81no—
156C23×C53#12111yescharacteristic
157C22×D53—41no—
158Q8×C53—81no—
159D4×C53—24no—
160D4×C53—22no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—81no—
164D4×C53—82no—
165D4×C53#13514yes—
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—41no—
173C2×C4×C53—82no—
174C2×C4×C53—82no—
175C2×C4×C53—81no—
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53C22×D411yescharacteristic
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C2×#4111yescharacteristic
182C22⋊D53—82no—
183C2×Dic53—41no—
184D212—42no—
185D212—42no—
186D212—81no—
187C4×D53—82no—
188Dic106—42no—
189C8×C53—42no—
190#88322—22no—
191#88322—22no—
192#88322—42no—
193#88322—42no—
194#19389—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—24no—
198C22×C53—21no—
199C22×C53—42no—
200C22×C53—42no—
201C22×C53—84no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53C2×#12111yescharacteristic
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—81no—
215C4×C53C2×#13512yes—
216C4×C53#57612yes—
217Dic53—81no—
218#3207—531no—
219#3109D5311yescharacteristic, complemented
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—44no—
231C2×C53—161no—
232C2×C53#304111yescharacteristic
233C2×C53#305011yescharacteristic
234C2×C53#306811yescharacteristic
235D53—161no—
236#532—531no—
237#532C2×D5311yescharacteristic
238C2×C4×D4C2×D5311yescharacteristic
239#493—531no—
240#492—531no—
241#487—531no—
242#486—531no—
243#481—21no—
244#480—1061no—
245#476C2×D5312yes—
246#474—532no—
247#463—1062no—
248#459—532no—
249#459C2×D5312yes—
250#449—1062no—
251#447—21no—
252#447—1061no—
253#442—22no—
254C4×D8—1062no—
255C8×D4—532no—
256C8×D4—532no—
257C8×D4—1062no—
258#432C2×D5311yescharacteristic
259#431—531no—
260#431—531no—
261#431—531no—
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—1061no—
272C22×D4C22⋊D5311yescharacteristic
273C23×C4—21no—
274C2×SD16—1062no—
275C2×D8—22no—
276C2×D8—1062no—
277C2×D8—1062no—
278C22×C8—1062no—
279C22×C8—1062no—
280C4⋊Q8—531no—
281C4⋊Q8C22×D5311yescharacteristic
282#127—531no—
283#127C22×D5311yescharacteristic
284#124—21no—
285#124—1061no—
286#121—21no—
287#121—21no—
288#121—1061no—
289#121—1061no—
290C4×D4—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—1061no—
295C4×D4—1061no—
296C4×D4C22⋊D5312yes—
297C4×D4C22×D5312yes—
298C2×C4⋊C4C22⋊D5311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C42C22×D5311yescharacteristic
301#107—1062no—
302#106—22no—
303C4⋊C8—532no—
304C4⋊C8—532no—
305C4⋊C8—532no—
306C4⋊C8C22×D5312yes—
307#103—1062no—
308#102—22no—
309#102—22no—
310#102—1062no—
311#102—1062no—
312#102—1062no—
313C22⋊C8—21no—
314C22⋊C8—532no—
315C22⋊C8—1062no—
316C22⋊C8—1062no—
317C22⋊C8—1061no—
318C22⋊C8—1061no—
319#97—21no—
320C4×C8—531no—
321C4×C8—531no—
322C4×C8—1061no—
323C4×C8—1061no—
324C24—21no—
325#41—82no—
326#41—4242no—
327C2×Q8—21no—
328C2×Q8—1061no—
329C2×D4—24no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—1061no—
336C2×D4—1061no—
337C2×D4—1061no—
338C2×D4—2121no—
339C2×D4C2×C22⋊D5312yes—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—82no—
345C22×C4—2121no—
346C22×C4D4×D5311yescharacteristic
347C22×C4#1171994711yescharacteristic
348C22×C4C2×C22⋊D5311yescharacteristic
349SD16—4242no—
350D8—44no—
351D8—4242no—
352D8—2124no—
353C2×C8—22no—
354C2×C8—532no—
355C2×C8—532no—
356C2×C8—1062no—
357C2×C8—1062no—
358C2×C8—4242no—
359C2×C8—4242no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—1061no—
364C4⋊C4—1061no—
365C4⋊C4C2×C22⋊D5312yes—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—2121no—
371C22⋊C4—2121no—
372C42D4×D5311yescharacteristic
373C42#1171994711yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C23×D5311yescharacteristic
376C23—24no—
377C23—22no—
378C23—41no—
379C23—81no—
380C23—81no—
381C23—2121no—
382C23#1313318911yescharacteristic
383Q8—81no—
384Q8—2122no—
385D4—24no—
386D4—22no—
387D4—42no—
388D4—42no—
389D4—81no—
390D4—82no—
391D4—4241no—
392D4—4242no—
393D4—2122no—
394D4—2122no—
395D4#1313323714yes—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—4242no—
407C2×C4—2121no—
408C2×C4#1313318911yescharacteristic
409C2×C4#1313318911yescharacteristic
410C2×C4#1313318911yescharacteristic
411C2×C4C2×D4×D5311yescharacteristic
412C2×C4C22×C22⋊D5311yescharacteristic
413C2×C4C2×#1171994711yescharacteristic
414C8—42no—
415C8—1062no—
416C8—1062no—
417C8—2122no—
418C8—2122no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—24no—
423C22—42no—
424C22—42no—
425C22—84no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—4241no—
430C22—4241no—
431C22—4241no—
432C22C2×#1313318911yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—81no—
439C4—4241no—
440C4C2×#1313323712yes—
441C4#1681142712yes—
442C2—41no—
443C2—44no—
444C2—161no—
445C2—8481no—
446C2#2028422011yescharacteristic
447C2#2028429311yescharacteristic
448C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3109C2
socleC22×C53C2×#121
hypercenter#3109D53
hypocenterC53#19389
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #20285644C2semidirect
2. #20284715C2semidirect
3. #20283739C2semidirect
4. #20274168C2semidirect
5. #20274126C2semidirect
6. #20274068C2semidirect
7. #20274054C2semidirect
8. #20273496C2semidirect
9. #20273421C2semidirect
10. #20273377C2semidirect
11. C53×#3109C2semidirect
13. #16810999C22semidirect
14. #16810995C22semidirect
15. #16810975C22semidirect
16. #16810971C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810372C22semidirect
34. #16810368C22semidirect
35. #16810367C22semidirect
36. #16810366C22semidirect
219. #3109D53semidirect
269. C53#19389semidirect

Maximal subgroups

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

subgroupindex
#202856442
#202847152
#202837392
#202741682
#202741262
#202740682
#202740542
#202734962
#202734212
#202733772
C53×#31092
#1938953