Finite Group Databasebeta

← #34109363

Subgroups — #34109363

Order 20096.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#19097797C212yesmaximal, complemented
2#23757784C211yesmaximal, characteristic, complemented
3#23757424C214yesmaximal, complemented
4#23757407C214yesmaximal, complemented
5#23757278C212yesmaximal, complemented
6#23757237C211yesmaximal, characteristic, complemented
7C23×D4×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic, complemented
9#19097797—28no—
10C2×#14578942C2214yescomplemented
11C2×#14578939C2214yescomplemented
12C2×#14578938C2212yescomplemented
13#19097754—21no—
14#19097750C2212yescomplemented
15#19097745C2212yescomplemented
16#19097740—21no—
17C4×C22⋊D157—22no—
18#19097730C2211yescharacteristic, complemented
19#19097692C2218yescomplemented
20C22⋊#11929434C2214yescomplemented
21C25×C157—21no—
22C22×D4×C157—28no—
23C22×D4×C157—22no—
24C22×D4×C157—21no—
25C22×D4×C157C2214yes—
26C22×D4×C157C2212yes—
27C23×C4×C157C2211yescharacteristic
28#14578954—24no—
29C2×Dic314C2311yescharacteristic, complemented
30C2×D628C2311yescharacteristic, complemented
31C2×C22⋊D157—22no—
32C2×C22⋊D157—21no—
33C2×C4×D157—21no—
34#14578942—28no—
35#14578939—28no—
36#14578938—24no—
37C2×#11929434C2314yescomplemented
38C4⋊Dic157C2312yescomplemented
39#14578933—22no—
40C22⋊Dic157—21no—
41C22⋊Dic157—22no—
42#14578931—22no—
43C4×Dic157—21no—
44C24×C157—28no—
45C24×C157—24no—
46C24×C157—22no—
47C24×C157—48no—
48C24×C157D411yescharacteristic
49C2×D4×C157—28no—
50C2×D4×C157—216no—
51C2×D4×C157—216no—
52C2×D4×C157—24no—
53C2×D4×C157—22no—
54C2×D4×C157—24no—
55C2×D4×C157D418yescomplemented
56C2×D4×C157C2314yes—
57C22×C4×C157—24no—
58C22×C4×C157D412yes—
59C22×C4×C157D411yescharacteristic
60C22×C4×C157C2312yes—
61C22×C4×C157C2311yescharacteristic
62C22×D157—21no—
63Dic314—22no—
64D628—22no—
65C22⋊D157—42no—
66C22⋊D157—44no—
67C2×Dic157—21no—
68C2×Dic157—22no—
69C4×D157—42no—
70#11929434—24no—
71C23×C157—22no—
72C23×C157—24no—
73C23×C157—216no—
74C23×C157—28no—
75C23×C157—24no—
76C23×C157—432no—
77C23×C157—48no—
78C23×C157—48no—
79C23×C157—48no—
80C23×C157C2×D411yescharacteristic
81C23×C157C2×D412yes—
82D4×C157—216no—
83D4×C157—28no—
84D4×C157—24no—
85D4×C157C2×D418yes—
86C2×C4×C157—22no—
87C2×C4×C157—24no—
88C2×C4×C157—24no—
89C2×C4×C157C2×D412yes—
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D414yes—
92C2×C4×C157C2411yescharacteristic
93C22×C157—28no—
94C22×C157—22no—
95C22×C157—22no—
96C22×C157—22no—
97C22×C157—28no—
98C22×C157—416no—
99C22×C157—432no—
100C22×C157—48no—
101C22×C157#12012yes—
102C22×C157AGL(1,ℤ/8)14yescomplemented
103C22×C157C22×D411yescharacteristic
104C2×D157—41no—
105C2×D157—42no—
106C4×C157—22no—
107C4×C157#12012yes—
108C4×C157C22×D412yes—
109Dic157—41no—
110Dic157—42no—
111C2×C157—22no—
112C2×C157—24no—
113C2×C157—48no—
114C2×C157C2×#12011yescharacteristic
115C2×C157C2×AGL(1,ℤ/8)12yes—
116D157—81no—
117C157#300311yescharacteristic, complemented, Sylow
118#3003—1571nomaximal, Sylow
119C23×D4D15711yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1572no—
121#525—1571no—
122#450—1574no—
123#447—1574no—
124#418—1572no—
125#407—1571no—
126C25—21no—
127C2×#41—1571no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D15714yes—
132C22×D4C2×D15712yes—
133C23×C4C2×D15711yescharacteristic
134AGL(1,ℤ/8)—3148no—
135C2×SD16—1574no—
136C2×D8—1574no—
137C2×M16—1572no—
138#123—3141no—
139C22⋊Q8—1572no—
140#121—1572no—
141#120—3141no—
142C4×D4—3142no—
143#117—1571no—
144#102—1578no—
145C22⋊C8—1574no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D15711yescharacteristic
151#41—3144no—
152C2×Q8—1571no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—3142no—
160C2×D4—3141no—
161C2×D4—1571no—
162C2×D4C22⋊D15718yescomplemented
163C2×D4C22×D15714yes—
164C22×C4—24no—
165C22×C4—3141no—
166C22×C4C22⋊D15712yes—
167C22×C4C22⋊D15711yescharacteristic
168C22×C4C22×D15712yes—
169C22×C4C22×D15711yescharacteristic
170SD16—3148no—
171D8—3148no—
172M16—3144no—
173C2×C8—1574no—
174C4⋊C4—3142no—
175C4⋊C4—1572no—
176C22⋊C4—3141no—
177C22⋊C4—3142no—
178C22⋊C4—3142no—
179C42—3141no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—3141no—
190C23C2×C22⋊D15711yescharacteristic
191C23C2×C22⋊D15712yes—
192Q8—3142no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—3142no—
197D4—6282no—
198D4—6284no—
199D4C2×C22⋊D15718yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—3142no—
204C2×C4—3141no—
205C2×C4—6282no—
206C2×C4C2×C22⋊D15712yes—
207C2×C4C2×C22⋊D15711yescharacteristic
208C2×C4C2×C22⋊D15714yes—
209C2×C4C23×D15711yescharacteristic
210C8—3144no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—6281no—
220C22—6282no—
221C22#1909774012yes—
222C22#1909779714yescomplemented
223C22C22×C22⋊D15711yescharacteristic
224C4—22no—
225C4—6281no—
226C4—6282no—
227C4#1909774012yes—
228C4C22×C22⋊D15712yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—12561no—
233C2C2×#1909774011yescharacteristic
234C2C2×#1909779712yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC23×D4×C157C2
socleC22×C157C22×D4
hypercenterC23×D4D157
hypocenterC157#3003
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. #23757784C2semidirect
3. #23757424C2semidirect
4. #23757407C2semidirect
5. #23757278C2semidirect
6. #23757237C2semidirect
7. C23×D4×C157C2semidirect
8. C2×#14578954C22semidirect
10. C2×#14578942C22semidirect
11. C2×#14578939C22semidirect
12. C2×#14578938C22semidirect
14. #19097750C22semidirect
15. #19097745C22semidirect
18. #19097730C22semidirect
19. #19097692C22semidirect
20. C22⋊#11929434C22semidirect
29. C2×Dic314C23semidirect
30. C2×D628C23semidirect
37. C2×#11929434C23semidirect
38. C4⋊Dic157C23semidirect
55. C2×D4×C157D4semidirect
102. C22×C157AGL(1,ℤ/8)semidirect
117. C157#3003semidirect
119. C23×D4D157semidirect
162. C2×D4C22⋊D157semidirect
222. C22#19097797semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
#237577842
#237574242
#237574072
#237572782
#237572372
C23×D4×C1572
#3003157