Finite Group Databasebeta

← #27369288

Subgroups — #27369288

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#20285673C211yesmaximal, characteristic, complemented
2#20284814C211yesmaximal, characteristic, complemented
3#20284715C211yesmaximal, characteristic
4#20274152C212yesmaximal, complemented
5#20274138C212yesmaximal, complemented
6#20274080C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273418C211yesmaximal, characteristic, complemented
10#20273379C211yesmaximal, characteristic, complemented
11C53×#2972C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16811003C2211yescharacteristic, complemented
15#16810996C2211yescharacteristic, complemented
16#16810974C2211yescharacteristic, complemented
17#16810970C2211yescharacteristic, complemented
18#16810958—21no—
19#16810936—21no—
20#16810903C2212yes—
21#16810903C2212yes—
22#16810846—22no—
23#16810828C2212yes—
24#16810818C2212yes—
25#16810787—22no—
26#16810774—21no—
27#16810757—21no—
28#16810723—22no—
29C4×#11719922—22no—
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810376C2212yescomplemented
33#16810374C2212yescomplemented
34#16810370C2212yescomplemented
35#16810370C2212yescomplemented
36#16810368C2212yescomplemented
37#16810366C2212yescomplemented
38C2×C4×D4×C53C2211yescharacteristic
39C8×D4×C53—22no—
40C8×D4×C53C2212yes—
41C8×D4×C53C2212yes—
42C53×#431C2211yescharacteristic
43C53×#431C2211yescharacteristic
44C2×C22⋊C8×C53—22no—
45C2×#11719945—21no—
46C2×Dic212—22no—
47C2×#11719913—22no—
48C2×#11719923—22no—
49C2×#11719922—22no—
50#13133217C2311yescharacteristic
51#13133217C2311yescharacteristic
52#13133217C2311yescharacteristic
53#13133214C2311yescharacteristic
54#13133209—21no—
55#13133205—21no—
56#13133196—21no—
57#13133196—21no—
58#13133191—21no—
59#13133191—21no—
60C4×Dic106—21no—
61C4×Dic106—21no—
62C4×D212—21no—
63C4×D212—21no—
64#13133176—21no—
65#13133150—22no—
66#13133150—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133142—22no—
70#13133140—22no—
71#13133139—22no—
72#13133138—22no—
73#13133138—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77#13133128—21no—
78C4×#88322—21no—
79C22×D4×C53D411yescharacteristic
80C23×C4×C53—21no—
81C22×C8×C53—22no—
82C22×C8×C53—22no—
83C4×D4×C53—21no—
84C4×D4×C53—21no—
85C4×D4×C53D412yes—
86C4×D4×C53C2312yes—
87C2×C4⋊C4×C53D411yescharacteristic
88C2×C22⋊C4×C53—21no—
89C2×C42×C53C2311yescharacteristic
90C4⋊C8×C53D412yes—
91C4⋊C8×C53C2312yes—
92C22⋊C8×C53—22no—
93C22⋊C8×C53—22no—
94C22⋊C8×C53—21no—
95C22⋊C8×C53D412yes—
96C4×C8×C53—21no—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C24×C53—21no—
100C2×D4×C53—24no—
101C2×D4×C53—21no—
102C2×D4×C53C2×D412yes—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—82no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×C22⋊D53—41no—
111#11719945—82no—
112C2×D212—21no—
113C2×D212—21no—
114C2×D212—21no—
115C2×C4×D53—41no—
116C2×Dic106—21no—
117C2×Dic106—21no—
118C2×Dic106—21no—
119C2×C8×C53—22no—
120C2×C8×C53—82no—
121C2×C8×C53—82no—
122C2×C8×C53C2×D412yes—
123C2×C8×C53C2×D412yes—
124C4⋊C4×C53—21no—
125C4⋊C4×C53C2×D412yes—
126C22⋊C4×C53—22no—
127C22⋊C4×C53—22no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132C22⋊Dic53—41no—
133#11719923—44no—
134#11719922—44no—
135#11719921—41no—
136#11719921—41no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139C4⋊Dic53—21no—
140C4⋊Dic53—21no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143#11719919—41no—
144#11719919—41no—
145C4×Dic53—41no—
146C2×#88322—22no—
147C2×#88322—22no—
148Dic212—82no—
149#11719913—82no—
150C23×C53—24no—
151C23×C53—22no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53#12111yescharacteristic
155C22×D53—41no—
156D4×C53—24no—
157D4×C53—22no—
158D4×C53#13514yes—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—82no—
166C2×C4×C53—81no—
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C2×#4111yescharacteristic
173C22⋊D53—82no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177D212—42no—
178D212—42no—
179D212—81no—
180C4×D53—82no—
181Dic106—42no—
182Dic106—42no—
183Dic106—42no—
184C8×C53—22no—
185C8×C53—22no—
186C8×C53—42no—
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×C53C2×#13512yes—
207C4×C53#57612yes—
208Dic53—81no—
209Dic53—81no—
210Dic53—81no—
211#3210—531no—
212#3110—531no—
213#3100—531no—
214#2972D5311yescharacteristic, complemented
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#533—531no—
229#532—531no—
230C2×C4×D4C2×D5311yescharacteristic
231#494—531no—
232#492—531no—
233#487—531no—
234#486—531no—
235#484—1061no—
236#480—1061no—
237#474—532no—
238#474—532no—
239#464—1062no—
240#461—532no—
241#459—532no—
242#452—1062no—
243#450—1061no—
244#447—1061no—
245#440—1062no—
246C4×D8—1062no—
247C8×D4—22no—
248C8×D4C2×D5312yes—
249C8×D4C2×D5312yes—
250#432—531no—
251#431—531no—
252#431C2×D5311yescharacteristic
253#431C2×D5311yescharacteristic
254C2×C22⋊C8—22no—
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×D4D21211yescharacteristic
264C23×C4—21no—
265C2×Q16—1062no—
266C2×SD16—1062no—
267C2×SD16—1062no—
268C2×D8—1062no—
269C22×C8—22no—
270C22×C8—22no—
271C4⋊Q8—531no—
272C4⋊Q8—531no—
273C4⋊Q8—531no—
274#127—531no—
275#125—1061no—
276#124—1061no—
277C22⋊Q8—1061no—
278C22⋊Q8—1061no—
279#121—1061no—
280#121—1061no—
281C4×Q8—1061no—
282C4×Q8—1061no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—1061no—
286C4×D4—1061no—
287C4×D4D21212yes—
288C4×D4C22×D5312yes—
289#117—1061no—
290C2×C4⋊C4D21211yescharacteristic
291C2×C22⋊C4—21no—
292C2×C42C22×D5311yescharacteristic
293#107—1062no—
294#107—1062no—
295C4⋊C8—532no—
296C4⋊C8—532no—
297C4⋊C8C22⋊D5312yes—
298C4⋊C8C22×D5312yes—
299#103—1062no—
300#103—1062no—
301#102—1062no—
302#102—1062no—
303#102—1062no—
304#102—1062no—
305C22⋊C8—21no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—1061no—
309C22⋊C8—1061no—
310C22⋊C8C22⋊D5312yes—
311#97—1061no—
312C4×C8—21no—
313C4×C8—1061no—
314C4×C8C22×D5311yescharacteristic
315C4×C8C22×D5311yescharacteristic
316C24—21no—
317#41—4242no—
318C2×Q8—1061no—
319C2×Q8—1061no—
320C2×Q8—1061no—
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—1061no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—2121no—
327C2×D4C2×D21212yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—82no—
332C22×C4—2121no—
333C22×C4C2×D21211yescharacteristic
334C22×C4#1171994511yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336Q16—4242no—
337SD16—4242no—
338SD16—2124no—
339D8—2124no—
340C2×C8—22no—
341C2×C8—82no—
342C2×C8—82no—
343C2×C8—1062no—
344C2×C8—1062no—
345C2×C8C2×C22⋊D5312yes—
346C2×C8C2×C22⋊D5312yes—
347C4⋊C4—21no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4—1061no—
353C4⋊C4—1061no—
354C4⋊C4—2121no—
355C4⋊C4—2121no—
356C4⋊C4C2×D21212yes—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C42—2121no—
363C42C2×D21211yescharacteristic
364C42#1171994511yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C23×D5311yescharacteristic
367C23—24no—
368C23—22no—
369C23—81no—
370C23—81no—
371C23—2121no—
372C23#1313319111yescharacteristic
373Q8—2122no—
374Q8—2122no—
375Q8—2122no—
376D4—24no—
377D4—22no—
378D4—4241no—
379D4—4242no—
380D4—2122no—
381D4—2122no—
382D4#1313323914yes—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—4242no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4#1313319111yescharacteristic
396C2×C4#1313319111yescharacteristic
397C2×C4#1313319111yescharacteristic
398C2×C4C22×C22⋊D5311yescharacteristic
399C2×C4C22×D21211yescharacteristic
400C2×C4C2×#1171994511yescharacteristic
401C8—22no—
402C8—22no—
403C8—42no—
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×#1313319111yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—4241no—
423C4—4241no—
424C4—4241no—
425C4C2×#1313323912yes—
426C4#1681142612yes—
427C2—41no—
428C2—44no—
429C2—8481no—
430C2#2028421811yescharacteristic
431C2#2028429611yescharacteristic
432C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2972C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenter#2972D53
hypocenterC53#19388
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #20285673C2semidirect
2. #20284814C2semidirect
4. #20274152C2semidirect
5. #20274138C2semidirect
6. #20274080C2semidirect
7. #20274056C2semidirect
8. #20273495C2semidirect
9. #20273418C2semidirect
10. #20273379C2semidirect
11. C53×#2972C2semidirect
14. #16811003C22semidirect
15. #16810996C22semidirect
16. #16810974C22semidirect
17. #16810970C22semidirect
32. #16810376C22semidirect
33. #16810374C22semidirect
34. #16810370C22semidirect
35. #16810370C22semidirect
36. #16810368C22semidirect
37. #16810366C22semidirect
214. #2972D53semidirect
261. C53#19388semidirect

Maximal subgroups

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

subgroupindex
#202856732
#202848142
#202847152
#202741522
#202741382
#202740802
#202740562
#202734952
#202734182
#202733792
C53×#29722
#1938853