Finite Group Databasebeta

← #27360397

Subgroups — #27360397

Order 13568.

982 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#16811339C211yesmaximal, characteristic
2#20284810C212yesmaximal, complemented
3#20273940C218yesmaximal, complemented
4#20273441C211yesmaximal, characteristic, complemented
5#20273159C212yesmaximal, complemented
6C53×#3139C211yesmaximal, characteristic, complemented
7#16811339—24no—
8#16811339—22no—
9#16811339C2214yes—
10C2×#13133206—22no—
11C2×#13133192—22no—
12C2×C4×D212C2212yes—
13C2×#13133175—22no—
14#16810956—22no—
15#16810900C2214yescomplemented
16#16810825C2214yescomplemented
17#16810773—22no—
18#16810721—24no—
19#16810680C2212yes—
20#16810372C2214yescomplemented
21#16810352C2218yescomplemented
22C53×#533C2211yescharacteristic
23C2×C4⋊Q8×C53C2211yescharacteristic
24C53×#491—22no—
25C53×#487C2214yescomplemented
26C53×#419—22no—
27C4×M16×C53C2211yescharacteristic
28C22×D212—22no—
29C22×D212D412yes—
30C22×C4×D53—22no—
31C2×#11719924—24no—
32#13133206—28no—
33#13133206—28no—
34#13133192—28no—
35#13133192—28no—
36C4×D212—22no—
37C4×D212—22no—
38C4×D212D414yes—
39C4×D212C2314yes—
40#13133175—28no—
41#13133175—28no—
42C2×C4⋊Dic53D412yes—
43C2×#11719921—24no—
44C2×#11719921—22no—
45C2×C4×Dic53—22no—
46#13133150—24no—
47C4⋊#88322C2314yes—
48#13133141—24no—
49#13133137—24no—
50C22⋊#88322—22no—
51#13133128—22no—
52C22×Q8×C53D412yes—
53C2×M16×C53—21no—
54C4⋊Q8×C53—22no—
55C4⋊Q8×C53D412yes—
56C4⋊Q8×C53C2312yes—
57C4⋊Q8×C53C2312yes—
58#125×C53—21no—
59C22⋊Q8×C53—22no—
60C4×Q8×C53—22no—
61#117×C53—21no—
62C2×C4⋊C4×C53—22no—
63C2×C42×C53C2311yescharacteristic
64#103×C53—28no—
65#97×C53—21no—
66C4×C8×C53C2312yes—
67C23×D53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53—22no—
70C2×Q8×C53—44no—
71C2×Q8×C53C2×D414yes—
72C22×C4×C53—22no—
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—22no—
76C2×D212—22no—
77C2×D212—28no—
78C2×D212—44no—
79C2×D212—84no—
80C2×D212C2×D414yes—
81C2×C4×D53—24no—
82C2×C4×D53—24no—
83C2×C4×D53—48no—
84M16×C53—42no—
85C2×C8×C53—22no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—42no—
91C22⋊C4×C53—41no—
92C42×C53—41no—
93C42×C53C2×D412yes—
94C42×C53C2×D411yescharacteristic
95C42×C53C2411yescharacteristic
96#11719924—48no—
97#11719921—216no—
98#11719921—24no—
99#11719921—24no—
100C4⋊Dic53—22no—
101C4⋊Dic53C2×D414yes—
102C4×Dic53—28no—
103C2×#88322—24no—
104C23×C53#12011yescharacteristic
105C22×D53—24no—
106C22×D53—44no—
107C22×D53—44no—
108C22×D53—44no—
109Q8×C53—44no—
110Q8×C53—44no—
111Q8×C53—82no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—44no—
116C2×C4×C53—42no—
117C2×C4×C53#12012yes—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53C22×D412yes—
120C2×C4×C53C22×D411yescharacteristic
121C2×Dic53—24no—
122C2×Dic53—44no—
123D212—24no—
124D212—24no—
125D212—44no—
126D212—88no—
127C4×D53—48no—
128C4×D53—48no—
129C8×C53—42no—
130#88322—44no—
131#18552—531nomaximal, Sylow
132C22×C53—22no—
133C22×C53#45512yes—
134C22×C53C2×#12011yescharacteristic
135C2×D53—44no—
136C2×D53—44no—
137C2×D53—44no—
138C2×D53—88no—
139C4×C53—22no—
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—44no—
143C4×C53—82no—
144C4×C53#57514yes—
145Dic53—44no—
146C2×#555—531no—
147#3139D5311yescharacteristic, complemented
148#3110—532no—
149#1646—538no—
150#1558—531no—
151#1490—532no—
152C2×C53—22no—
153C2×C53#300512yes—
154C2×C53C2×#45511yescharacteristic
155D53—84no—
156#555—534no—
157#555—1064no—
158#555—1062no—
159#533C2×D5311yescharacteristic
160C2×C4⋊Q8C2×D5311yescharacteristic
161C2×#124—1062no—
162C2×#121—1062no—
163C2×C4×D4—532no—
164C2×#117—1062no—
165#491—22no—
166#487C2×D5314yescomplemented
167#484—1062no—
168#474—534no—
169#461—534no—
170#450—1062no—
171#440—1064no—
172#432—532no—
173#419—22no—
174C4×M16C2×D5311yescharacteristic
175#333—534no—
176D4⋊C8—538no—
177C53#1855211yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5312yes—
179C22×D4—532no—
180C22×D4—1062no—
181C23×C4—1062no—
182C2×SD16—1064no—
183C2×M16—21no—
184C4⋊Q8—22no—
185C4⋊Q8C22⋊D5312yes—
186C4⋊Q8C22×D5312yes—
187C4⋊Q8C22×D5312yes—
188#125—21no—
189#124—1068no—
190#124—1068no—
191C22⋊Q8—22no—
192#121—1068no—
193#121—1068no—
194C4×Q8—22no—
195C4×D4—534no—
196C4×D4—534no—
197C4×D4—1062no—
198C4×D4—1062no—
199#117—21no—
200#117—1068no—
201#117—1068no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—532no—
204C2×C22⋊C4—1064no—
205C2×C22⋊C4—1062no—
206C2×C42—1062no—
207C2×C42C22×D5311yescharacteristic
208#107—1064no—
209C4⋊C8—534no—
210#103—28no—
211#103—1064no—
212#102—1064no—
213C22⋊C8—1062no—
214#97—21no—
215#97—1062no—
216C4×C8C22×D5312yes—
217C24—1062no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—44no—
221C2×Q8C2×C22⋊D5314yes—
222C2×D4—534no—
223C2×D4—1068no—
224C2×D4—1062no—
225C2×D4—4244no—
226C2×D4—2124no—
227C22×C4—22no—
228C22×C4—1062no—
229C22×C4—1064no—
230C22×C4—1064no—
231C22×C4—2128no—
232C22×C4D4×D5312yes—
233C22×C4C2×C22⋊D5311yescharacteristic
234SD16—2128no—
235M16—42no—
236C2×C8—22no—
237C2×C8—1064no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—534no—
244C4⋊C4—1062no—
245C22⋊C4—41no—
246C22⋊C4—10616no—
247C22⋊C4—1064no—
248C22⋊C4—1064no—
249C42—41no—
250C42—1068no—
251C42D4×D5312yes—
252C42C2×C22⋊D5311yescharacteristic
253C42C23×D5311yescharacteristic
254C23—1064no—
255C23—2124no—
256C23—2124no—
257C23—2124no—
258C23#1313318511yescharacteristic
259Q8—44no—
260Q8—44no—
261Q8—82no—
262D4—1064no—
263D4—1064no—
264D4—4248no—
265D4—2124no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—1064no—
272C2×C4—2128no—
273C2×C4—2124no—
274C2×C4—2128no—
275C2×C4#1313318511yescharacteristic
276C2×C4#1313318512yes—
277C2×C4C2×D4×D5312yes—
278C2×C4C22×C22⋊D5311yescharacteristic
279C8—42no—
280C8—2124no—
281C22—22no—
282C22—4248no—
283C22—2124no—
284C22—2124no—
285C22—2124no—
286C22#1681080012yes—
287C22C2×#1313318511yescharacteristic
288C4—22no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—82no—
293C4—2124no—
294C4#1681142214yes—
295C2—22no—
296C2—4244no—
297C2#2028395612yes—
298C2C2×#1681080011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3139C2
socleC22×C53C2×#120
prime radicalC2×#16811339C2
hypercenter#3139D53
hypocenterC53#18552
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #20284810C2semidirect
3. #20273940C2semidirect
4. #20273441C2semidirect
5. #20273159C2semidirect
6. C53×#3139C2semidirect
15. #16810900C22semidirect
16. #16810825C22semidirect
20. #16810372C22semidirect
21. #16810352C22semidirect
25. C53×#487C22semidirect
147. #3139D53semidirect
166. #487C2×D53semidirect
177. C53#18552semidirect

Maximal subgroups

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

subgroupindex
C2×#168113392
#202848102
#202739402
#202734412
#202731592
C53×#31392
#1855253