Finite Group Databasebeta

← #27369291

Subgroups — #27369291

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285667C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3#20283739C211yesmaximal, characteristic, complemented
4#20274154C212yesmaximal, complemented
5#20274140C212yesmaximal, complemented
6#20274078C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273496C211yesmaximal, characteristic, complemented
9#20273421C211yesmaximal, characteristic, complemented
10#20273380C211yesmaximal, characteristic, complemented
11C53×#3110C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811002C2211yescharacteristic, complemented
14#16810997C2211yescharacteristic, complemented
15#16810975C2211yescharacteristic, complemented
16#16810971C2211yescharacteristic, complemented
17#16810936—21no—
18#16810903C2212yes—
19#16810845—22no—
20#16810818C2212yes—
21#16810783—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#16810369C2212yescomplemented
34#16810369C2212yescomplemented
35#16810368C2212yescomplemented
36#16810366C2212yescomplemented
37C53×#533C2211yescharacteristic
38C2×C4×D4×C53C2211yescharacteristic
39C53×#484—21no—
40C53×#474C2212yes—
41C53×#461C2212yes—
42C53×#450—21no—
43C53×#440—22no—
44C53×#432C2211yescharacteristic
45C2×#11719945—21no—
46C2×#11719925—22no—
47C2×#11719924—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
75C22×D4×C53D411yescharacteristic
76C23×C4×C53—21no—
77C2×SD16×C53—22no—
78C4⋊Q8×C53C2311yescharacteristic
79C4⋊Q8×C53C2311yescharacteristic
80#125×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53—21no—
87C4×D4×C53D412yes—
88C4×D4×C53C2312yes—
89#117×C53—21no—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C22⋊C4×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#107×C53—22no—
94C4⋊C8×C53C2312yes—
95#103×C53—22no—
96#102×C53—22no—
97C22⋊C8×C53—21no—
98#97×C53—21no—
99C24×C53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53C2×D412yes—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—82no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—41no—
113#11719945—82no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—21no—
117C2×C4×D53—41no—
118C2×Dic106—21no—
119SD16×C53—44no—
120C2×C8×C53—22no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53C2×D412yes—
129C22⋊C4×C53—22no—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—41no—
132C42×C53—41no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53C2411yescharacteristic
137#11719925—82no—
138#11719924—82no—
139#11719922—44no—
140#11719921—41no—
141#11719921—41no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C2×#88322—22no—
145C2×#88322—22no—
146C2×#88322—82no—
147C2×#88322—82no—
148C2×#88322C2×D412yes—
149C2×#88322C2×D412yes—
150C23×C53—24no—
151C23×C53—22no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53#12111yescharacteristic
155C22×D53—41no—
156Q8×C53—42no—
157Q8×C53—42no—
158D4×C53—24no—
159D4×C53—22no—
160D4×C53#13514yes—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—82no—
170C2×C4×C53—81no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C22×D411yescharacteristic
176C2×C4×C53C2×#4111yescharacteristic
177C22⋊D53—82no—
178C2×Dic53—41no—
179D212—42no—
180D212—42no—
181D212—81no—
182C4×D53—82no—
183Dic106—42no—
184C8×C53—42no—
185#88322—22no—
186#88322—22no—
187#88322—42no—
188#88322—42no—
189#19388—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—24no—
193C22×C53—21no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—84no—
197C22×C53C2×#12111yescharacteristic
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—42no—
206C4×C53—81no—
207C4×C53—81no—
208C4×C53C2×#13512yes—
209C4×C53#57612yes—
210Dic53—81no—
211#3210—531no—
212#3110D5311yescharacteristic, complemented
213#3100—531no—
214#2972—531no—
215#1681—532no—
216#1679—532no—
217#1669—532no—
218#1665—532no—
219#1572—531no—
220#1553—531no—
221#1544—531no—
222C2×C53—41no—
223C2×C53—44no—
224C2×C53#304111yescharacteristic
225C2×C53#305011yescharacteristic
226C2×C53#306911yescharacteristic
227D53—161no—
228#533C2×D5311yescharacteristic
229#532—531no—
230C2×C4×D4C2×D5311yescharacteristic
231#494—531no—
232#492—531no—
233#487—531no—
234#486—531no—
235#484—21no—
236#480—1061no—
237#474—532no—
238#474C2×D5312yes—
239#464—1062no—
240#461C2×D5312yes—
241#459—532no—
242#452—1062no—
243#450—21no—
244#447—1061no—
245#440—22no—
246C4×D8—1062no—
247C8×D4—532no—
248C8×D4—532no—
249C8×D4—1062no—
250#432C2×D5311yescharacteristic
251#431—531no—
252#431—531no—
253#431—531no—
254C2×C22⋊C8—1062no—
255#335—532no—
256#334—532no—
257#332—532no—
258#332—532no—
259#332—532no—
260#331—532no—
261C53#1938811yescharacteristic, complemented, Sylow
262C2×#41—1061no—
263C22×D4C22⋊D5311yescharacteristic
264C23×C4—21no—
265C2×Q16—1062no—
266C2×SD16—22no—
267C2×SD16—1062no—
268C2×D8—1062no—
269C22×C8—1062no—
270C22×C8—1062no—
271C4⋊Q8—531no—
272C4⋊Q8C22×D5311yescharacteristic
273C4⋊Q8C22×D5311yescharacteristic
274#127—531no—
275#125—21no—
276#124—1061no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279#121—1061no—
280#121—1061no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—1061no—
286C4×D4—1061no—
287C4×D4C22⋊D5312yes—
288C4×D4C22×D5312yes—
289#117—21no—
290C2×C4⋊C4C22⋊D5311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C42C22×D5311yescharacteristic
293#107—22no—
294#107—1062no—
295C4⋊C8—532no—
296C4⋊C8—532no—
297C4⋊C8—532no—
298C4⋊C8C22×D5312yes—
299#103—22no—
300#103—1062no—
301#102—22no—
302#102—1062no—
303#102—1062no—
304#102—1062no—
305C22⋊C8—21no—
306C22⋊C8—532no—
307C22⋊C8—1062no—
308C22⋊C8—1062no—
309C22⋊C8—1061no—
310C22⋊C8—1061no—
311#97—21no—
312C4×C8—531no—
313C4×C8—531no—
314C4×C8—1061no—
315C4×C8—1061no—
316C24—21no—
317#41—4242no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—1061no—
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—1061no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—2121no—
327C2×D4C2×C22⋊D5312yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—82no—
332C22×C4—2121no—
333C22×C4D4×D5311yescharacteristic
334C22×C4#1171994711yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336Q16—4242no—
337SD16—44no—
338SD16—4242no—
339D8—2124no—
340C2×C8—22no—
341C2×C8—532no—
342C2×C8—532no—
343C2×C8—1062no—
344C2×C8—1062no—
345C2×C8—4242no—
346C2×C8—4242no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—1061no—
355C4⋊C4—1061no—
356C4⋊C4C2×C22⋊D5312yes—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—41no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C42—41no—
363C42D4×D5311yescharacteristic
364C42#1171994711yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C23×D5311yescharacteristic
367C23—24no—
368C23—22no—
369C23—81no—
370C23—81no—
371C23—2121no—
372C23#1313318911yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—2122no—
376D4—24no—
377D4—22no—
378D4—4241no—
379D4—4242no—
380D4—2122no—
381D4—2122no—
382D4#1313323714yes—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—4242no—
394C2×C4—2121no—
395C2×C4#1313318911yescharacteristic
396C2×C4#1313318911yescharacteristic
397C2×C4#1313318911yescharacteristic
398C2×C4C2×D4×D5311yescharacteristic
399C2×C4C22×C22⋊D5311yescharacteristic
400C2×C4C2×#1171994711yescharacteristic
401C8—42no—
402C8—1062no—
403C8—1062no—
404C8—2122no—
405C8—2122no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—24no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—4241no—
414C22—4241no—
415C22—4241no—
416C22C2×#1313318911yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—81no—
423C4—81no—
424C4—4241no—
425C4C2×#1313323712yes—
426C4#1681142712yes—
427C2—41no—
428C2—44no—
429C2—8481no—
430C2#2028422011yescharacteristic
431C2#2028429311yescharacteristic
432C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3110C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenter#3110D53
hypocenterC53#19388
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #20285667C2semidirect
3. #20283739C2semidirect
4. #20274154C2semidirect
5. #20274140C2semidirect
6. #20274078C2semidirect
7. #20274058C2semidirect
8. #20273496C2semidirect
9. #20273421C2semidirect
10. #20273380C2semidirect
11. C53×#3110C2semidirect
13. #16811002C22semidirect
14. #16810997C22semidirect
15. #16810975C22semidirect
16. #16810971C22semidirect
31. #16810375C22semidirect
32. #16810373C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
35. #16810368C22semidirect
36. #16810366C22semidirect
212. #3110D53semidirect
261. C53#19388semidirect

Maximal subgroups

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

subgroupindex
#202856672
#202847152
#202837392
#202741542
#202741402
#202740782
#202740582
#202734962
#202734212
#202733802
C53×#31102
#1938853