Finite Group Databasebeta

← #27369316

Subgroups — #27369316

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285843C211yesmaximal, characteristic, complemented
2#20284179C211yesmaximal, characteristic
3#20283739C211yesmaximal, characteristic, complemented
4#20274126C214yesmaximal, complemented
5#20274092C214yesmaximal, complemented
6#20273485C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic, complemented
9C2×C53×#476C211yesmaximal, characteristic, complemented
10C2×#13133243—21no—
11#16811173C2211yescharacteristic
12#16811019C2212yescomplemented
13#16810971C2212yescomplemented
14#16810941—22no—
15#16810910—22no—
16#16810838—22no—
17#16810828C2212yes—
18#16810818C2212yes—
19D4×#88322—22no—
20D4×#88322C2214yes—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810622—21no—
25C2×C22⋊#88322—22no—
26#16810373C2214yescomplemented
27#16810372C2214yescomplemented
28#16810370C2212yescomplemented
29#16810366C2212yescomplemented
30C2×C4⋊Q8×C53C2211yescharacteristic
31C2×C4×D4×C53C2211yescharacteristic
32C53×#476—22no—
33C53×#476C2214yes—
34C2×#106×C53—21no—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#102×C53—21no—
37C2×#11719945—21no—
38#13133243—44no—
39C2×#11719923—22no—
40C2×#11719922—22no—
41C2×#11719917—21no—
42C22×#88322—24no—
43#13133217C2311yescharacteristic
44#13133214C2311yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×D212—21no—
49C4×D212—21no—
50#13133148—24no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133140—22no—
56#13133137—22no—
57#13133137—24no—
58C22⋊#88322—24no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61C4×#88322—21no—
62C4×#88322C2312yes—
63C22×Q8×C53—21no—
64C22×D4×C53D411yescharacteristic
65C23×C4×C53—21no—
66C22×C8×C53—21no—
67C4⋊Q8×C53—21no—
68C4⋊Q8×C53—21no—
69C4⋊Q8×C53—21no—
70C4⋊Q8×C53C2312yes—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53D412yes—
74C4×D4×C53C2312yes—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80#106×C53—24no—
81C4⋊C8×C53—21no—
82C4⋊C8×C53C2312yes—
83#102×C53—24no—
84C24×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—41no—
87C2×Q8×C53—82no—
88C2×D4×C53—21no—
89C2×D4×C53—22no—
90C2×D4×C53SD1614yes—
91C2×D4×C53C2×D412yes—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—41no—
96C22×C4×C53—82no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C2×C22⋊D53—41no—
101#11719945—42no—
102#11719945—42no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108C2×C8×C53—22no—
109C2×C8×C53—42no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53C2×D412yes—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53C2411yescharacteristic
122#11719923—44no—
123#11719922—44no—
124#11719921—41no—
125#11719921—41no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128#11719917—42no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—84no—
132C2×#88322C2×D414yes—
133C23×C53—22no—
134C23×C53—24no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53#12111yescharacteristic
138C22×D53—41no—
139Q8×C53—82no—
140Q8×C53—82no—
141D4×C53—22no—
142D4×C53—24no—
143D4×C53C2×SD1614yes—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—44no—
148C2×C4×C53—42no—
149C2×C4×C53—82no—
150C2×C4×C53—81no—
151C2×C4×C53—82no—
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53C2×SD1612yes—
156C2×C4×C53#13512yes—
157C2×C4×C53C22×D411yescharacteristic
158C2×C4×C53C22×D411yescharacteristic
159C2×C4×C53C2×#4111yescharacteristic
160C22⋊D53—82no—
161C2×Dic53—41no—
162D212—41no—
163D212—41no—
164D212—42no—
165D212—81no—
166C4×D53—82no—
167Dic106—41no—
168Dic106—41no—
169C8×C53—42no—
170#88322—24no—
171#88322—42no—
172#88322—42no—
173#19391—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—24no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—84no—
180C22×C53#46512yes—
181C22×C53C2×#12111yescharacteristic
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—42no—
189C4×C53—82no—
190C4×C53#46512yes—
191C4×C53C22×SD1611yescharacteristic
192C4×C53C2×#13511yescharacteristic
193C4×C53#57712yes—
194Dic53—81no—
195#3227—531no—
196C2×#476D5311yescharacteristic, complemented
197#3036—531no—
198#2972—531no—
199#1677—534no—
200#1671—534no—
201#1570—531no—
202#1548—531no—
203#1540—531no—
204C2×C53—22no—
205C2×C53—44no—
206C2×C53#304211yescharacteristic
207C2×C53C2×#46511yescharacteristic
208C2×C53#306811yescharacteristic
209D53—161no—
210C2×AGL(1,ℤ/8)—1061no—
211#532—531no—
212C2×C4⋊Q8C2×D5311yescharacteristic
213C2×C4×D4C2×D5311yescharacteristic
214#498—532no—
215#486—532no—
216#481—1062no—
217#476—22no—
218#476—1062no—
219#476C2×D5314yes—
220#463—1062no—
221#461—532no—
222#459—532no—
223C8×D4—534no—
224C8×D4—1062no—
225#431—531no—
226#431—531no—
227C2×#106—21no—
228#423—531no—
229C2×C4⋊C8C2×D5311yescharacteristic
230#417—1061no—
231C2×#102—21no—
232C2×C22⋊C8—1062no—
233#334—534no—
234#333—534no—
235#332—532no—
236#331—532no—
237C53#1939111yescharacteristic, complemented, Sylow
238C2×#41—1061no—
239C22×Q8—21no—
240C22×D4C22⋊D5311yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—2124no—
243C2×SD16—1062no—
244C2×D8—1062no—
245C2×M16—1061no—
246C22×C8—21no—
247C22×C8—1064no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—531no—
252C4⋊Q8C22×D5312yes—
253#127—531no—
254#124—1061no—
255#121—1061no—
256#121—1061no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—1061no—
260C4×D4—1061no—
261C4×D4C22⋊D5312yes—
262C4×D4C22×D5312yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D5311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D5311yescharacteristic
268#106—24no—
269#106—1064no—
270C4⋊C8—21no—
271C4⋊C8—532no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—1061no—
275C4⋊C8C22×D5312yes—
276#103—1062no—
277#102—24no—
278#102—1064no—
279#102—1062no—
280C22⋊C8—532no—
281C22⋊C8—1064no—
282C22⋊C8—1061no—
283C4×C8—532no—
284C4×C8—1061no—
285C24—21no—
286#41—2122no—
287#41—2122no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—1061no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—2121no—
298C2×D4#1171992414yes—
299C2×D4C2×C22⋊D5312yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—2121no—
306C22×C4D4×D5311yescharacteristic
307C22×C4#1171994711yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—2124no—
310D8—2124no—
311M16—2122no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—534no—
315C2×C8—1062no—
316C2×C8—1062no—
317C2×C8—4244no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4C2×C22⋊D5312yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42D4×D5311yescharacteristic
331C42#1171994711yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C23×D5311yescharacteristic
334C23—22no—
335C23—24no—
336C23—81no—
337C23—81no—
338C23—2121no—
339C23#1313318911yescharacteristic
340Q8—82no—
341Q8—82no—
342Q8—2121no—
343Q8—2121no—
344D4—24no—
345D4—22no—
346D4—4241no—
347D4—4242no—
348D4—2121no—
349D4—2121no—
350D4—2122no—
351D4C2×#1171992414yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—4242no—
361C2×C4—2121no—
362C2×C4#1313318911yescharacteristic
363C2×C4#1313318911yescharacteristic
364C2×C4#1313318911yescharacteristic
365C2×C4C2×#1171992412yes—
366C2×C4#1313323712yes—
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C22×C22⋊D5311yescharacteristic
369C2×C4C2×#1171994711yescharacteristic
370C8—42no—
371C8—1064no—
372C8—2122no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—4241no—
381C22—4241no—
382C22—4241no—
383C22#1681085412yes—
384C22C2×#1313318911yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—82no—
390C4—4241no—
391C4#1681085412yes—
392C4C22×#1171992411yescharacteristic
393C4C2×#1313323711yescharacteristic
394C4#1681142912yes—
395C2—22no—
396C2—44no—
397C2—8481no—
398C2#2028422411yescharacteristic
399C2C2×#1681085411yescharacteristic
400C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#476C2
socleC22×C53C2×#121
prime radical#20284179C2
hypercenterC2×#476D53
hypocenterC53#19391
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #20285843C2semidirect
3. #20283739C2semidirect
4. #20274126C2semidirect
5. #20274092C2semidirect
6. #20273485C2semidirect
7. #20273396C2semidirect
8. #20273365C2semidirect
9. C2×C53×#476C2semidirect
12. #16811019C22semidirect
13. #16810971C22semidirect
26. #16810373C22semidirect
27. #16810372C22semidirect
28. #16810370C22semidirect
29. #16810366C22semidirect
196. C2×#476D53semidirect
237. C53#19391semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202841792
#202837392
#202741262
#202740922
#202734852
#202733962
#202733652
C2×C53×#4762
#1939153