Finite Group Databasebeta

← #27360399

Subgroups — #27360399

Order 13568.

982 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285095C211yesmaximal, characteristic, complemented
2#20284814C212yesmaximal, complemented
3#20273943C218yesmaximal, complemented
4#20273441C211yesmaximal, characteristic, complemented
5#20273158C212yesmaximal, complemented
6C2×C53×#555C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16810991—22no—
9#16810973C2214yescomplemented
10#16810958—22no—
11#16810903C2214yescomplemented
12#16810828C2214yescomplemented
13#16810774—22no—
14#16810723—24no—
15#16810680C2212yes—
16#16810632—22no—
17C4×#11719917C2211yescharacteristic
18#16810372C2214yescomplemented
19D4⋊#88322C2218yescomplemented
20C53×#555—22no—
21C53×#555—24no—
22C53×#555C2214yes—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×#124×C53—22no—
25C2×#121×C53—22no—
26C2×C4×D4×C53C2212yes—
27C2×#117×C53—22no—
28C2×#11719923—24no—
29C2×#11719917—21no—
30#13133217C2312yes—
31#13133209—21no—
32#13133196—22no—
33C4×Dic106—22no—
34#13133176—21no—
35#13133150—24no—
36C4⋊#88322C2314yes—
37#13133141—28no—
38#13133140—24no—
39#13133138—24no—
40C22⋊#88322—22no—
41#13133128—21no—
42#13133128—22no—
43C4×#88322C2312yes—
44C22×Q8×C53D412yes—
45C22×D4×C53—22no—
46C22×D4×C53D412yes—
47C23×C4×C53—22no—
48C4⋊Q8×C53—22no—
49C4⋊Q8×C53D412yes—
50C4⋊Q8×C53C2312yes—
51#124×C53—28no—
52#124×C53—28no—
53#121×C53—28no—
54#121×C53—28no—
55C4×D4×C53—22no—
56C4×D4×C53—22no—
57C4×D4×C53D414yes—
58C4×D4×C53C2314yes—
59#117×C53—28no—
60#117×C53—28no—
61C2×C4⋊C4×C53—22no—
62C2×C4⋊C4×C53D412yes—
63C2×C22⋊C4×C53—24no—
64C2×C22⋊C4×C53—22no—
65C2×C42×C53—22no—
66C2×C42×C53C2311yescharacteristic
67C24×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53—44no—
70C2×Q8×C53C2×D414yes—
71C2×D4×C53—28no—
72C2×D4×C53—22no—
73C2×D4×C53—44no—
74C2×D4×C53—84no—
75C2×D4×C53C2×D414yes—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—24no—
79C22×C4×C53—24no—
80C22×C4×C53—48no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C2×Dic106—22no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53C2×D414yes—
88C22⋊C4×C53—216no—
89C22⋊C4×C53—24no—
90C22⋊C4×C53—24no—
91C42×C53—28no—
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D412yes—
94C42×C53C2411yescharacteristic
95C22⋊Dic53—41no—
96#11719923—48no—
97C4⋊Dic53—22no—
98C4⋊Dic53—22no—
99#11719919—42no—
100C4×Dic53—41no—
101#11719917—42no—
102C2×#88322—22no—
103C2×#88322—24no—
104C23×C53—24no—
105C23×C53—44no—
106C23×C53—44no—
107C23×C53—44no—
108C23×C53#12011yescharacteristic
109Q8×C53—44no—
110Q8×C53—82no—
111D4×C53—24no—
112D4×C53—24no—
113D4×C53—44no—
114D4×C53—88no—
115C2×C4×C53—22no—
116C2×C4×C53—24no—
117C2×C4×C53—24no—
118C2×C4×C53—24no—
119C2×C4×C53—48no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53—48no—
123C2×C4×C53#12012yes—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53C22×D411yescharacteristic
126C2×C4×C53C22×D412yes—
127C2×Dic53—42no—
128Dic106—44no—
129#88322—42no—
130#88322—44no—
131#18552—531nomaximal, Sylow
132C22×C53—22no—
133C22×C53—44no—
134C22×C53—44no—
135C22×C53—44no—
136C22×C53—88no—
137C22×C53#45512yes—
138C22×C53C2×#12011yescharacteristic
139C4×C53—22no—
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—44no—
143C4×C53—44no—
144C4×C53#57514yes—
145Dic53—82no—
146C2×#555D5311yescharacteristic
147#3139—531no—
148#3110—532no—
149#1646—538no—
150#1558—531no—
151#1490—532no—
152C2×C53—22no—
153C2×C53—84no—
154C2×C53#300512yes—
155C2×C53C2×#45511yescharacteristic
156#555—22no—
157#555—24no—
158#555C2×D5314yes—
159#533—531no—
160C2×C4⋊Q8C2×D5311yescharacteristic
161C2×#124—22no—
162C2×#121—22no—
163C2×C4×D4C2×D5312yes—
164C2×#117—22no—
165#491—1062no—
166#487—534no—
167#484—1062no—
168#474—534no—
169#461—534no—
170#450—1062no—
171#440—1064no—
172#432—532no—
173#419—1062no—
174C4×M16—531no—
175#333—534no—
176D4⋊C8—538no—
177C53#1855211yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5312yes—
179C22×D4—22no—
180C22×D4C22⋊D5312yes—
181C23×C4—22no—
182C2×SD16—1064no—
183C2×M16—1061no—
184C4⋊Q8—22no—
185C4⋊Q8—532no—
186C4⋊Q8C22⋊D5312yes—
187C4⋊Q8C22×D5312yes—
188#125—1061no—
189#124—28no—
190#124—28no—
191C22⋊Q8—1062no—
192#121—28no—
193#121—28no—
194C4×Q8—1062no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4C22⋊D5314yes—
198C4×D4C22×D5314yes—
199#117—28no—
200#117—28no—
201#117—1061no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4C22⋊D5312yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42C22×D5311yescharacteristic
208#107—1064no—
209C4⋊C8—534no—
210#103—1068no—
211#103—1064no—
212#102—1064no—
213C22⋊C8—1062no—
214#97—1061no—
215#97—1062no—
216C4×C8—532no—
217C24—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—1062no—
221C2×Q8C2×C22⋊D5314yes—
222C2×D4—28no—
223C2×D4—22no—
224C2×D4—44no—
225C2×D4—84no—
226C2×D4C2×C22⋊D5314yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—48no—
232C22×C4C2×C22⋊D5312yes—
233C22×C4C2×C22⋊D5311yescharacteristic
234SD16—2128no—
235M16—2122no—
236C2×C8—1062no—
237C2×C8—1064no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—1062no—
242C4⋊C4—1062no—
243C4⋊C4—2122no—
244C4⋊C4C2×C22⋊D5314yes—
245C22⋊C4—216no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—2121no—
249C42—28no—
250C42—2121no—
251C42C2×C22⋊D5311yescharacteristic
252C42C2×C22⋊D5312yes—
253C42C23×D5311yescharacteristic
254C23—24no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23#1313318611yescharacteristic
259Q8—44no—
260Q8—82no—
261Q8—2124no—
262D4—24no—
263D4—24no—
264D4—44no—
265D4—88no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—2122no—
275C2×C4#1313318611yescharacteristic
276C2×C4#1313318612yes—
277C2×C4C22×C22⋊D5311yescharacteristic
278C2×C4C22×C22⋊D5312yes—
279C8—2122no—
280C8—2124no—
281C22—22no—
282C22—44no—
283C22—44no—
284C22—44no—
285C22—88no—
286C22#1681080112yes—
287C22C2×#1313318611yescharacteristic
288C4—22no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—44no—
293C4—4242no—
294C4#1681142114yes—
295C2—22no—
296C2—84no—
297C2#2028395712yes—
298C2C2×#1681080111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC2×C53×#555C2
socleC22×C53C2×#120
hypercenterC2×#555D53
hypocenterC53#18552
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #20285095C2semidirect
2. #20284814C2semidirect
3. #20273943C2semidirect
4. #20273441C2semidirect
5. #20273158C2semidirect
9. #16810973C22semidirect
11. #16810903C22semidirect
12. #16810828C22semidirect
18. #16810372C22semidirect
19. D4⋊#88322C22semidirect
177. C53#18552semidirect

Maximal subgroups

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

subgroupindex
#202850952
#202848142
#202739432
#202734412
#202731582
C2×C53×#5552
#1855253