Finite Group Databasebeta

← #34100581

Subgroups — #34100581

Order 20096.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#23757780C211yesmaximal, characteristic
2C2×#19097728C211yesmaximal, characteristic, complemented
3C2×#19097698C211yesmaximal, characteristic, complemented
4#23757236C211yesmaximal, characteristic, complemented
5#23757034C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8C22×C4×D157C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097752—21no—
11#19097747C412yescomplemented
12#19097742C412yescomplemented
13#19097739—21no—
14C4×C22⋊D157—22no—
15#19097730C2211yescharacteristic
16#19097728—22no—
17#19097728—22no—
18C2×#14578933—21no—
19C2×#14578931—21no—
20C2×C4×Dic157C2211yescharacteristic
21#19097698—22no—
22#19097683—22no—
23C2×#41×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25C2×C42×C157C2211yescharacteristic
26#104×C157—22no—
27C23×D157D411yescharacteristic
28C2×C22⋊D157—22no—
29C22×Dic157—21no—
30C22×Dic157D411yescharacteristic
31C2×C4×D157—21no—
32C2×C4×D157—21no—
33C2×C4×D157—21no—
34C2×C4×D157—21no—
35C2×C4×D157C2×C412yes—
36C2×C4×D157D412yes—
37#14578938—21no—
38#14578938—21no—
39C2×#11929434—21no—
40C4⋊Dic157C2×C412yes—
41#14578933—24no—
42#14578933—22no—
43C22⋊Dic157—22no—
44C22⋊Dic157—21no—
45#14578931—24no—
46#14578931—22no—
47C4×Dic157—21no—
48C4×Dic157—21no—
49C4×Dic157D412yescomplemented
50#41×C157—21no—
51#41×C157—21no—
52#41×C157—21no—
53#41×C157—21no—
54C2×Q8×C157C2×C411yescharacteristic
55C2×D4×C157—21no—
56C2×D4×C157C2×C411yescharacteristic
57C22×C4×C157—21no—
58C22×C4×C157—21no—
59C22×C4×C157C2311yescharacteristic
60M16×C157—21no—
61M16×C157—21no—
62C2×C8×C157—21no—
63C42×C157—21no—
64C42×C157D412yescomplemented
65C22×D157—21no—
66C22×D157—21no—
67C22×D157—42no—
68C22×D157C22⋊C412yes—
69C22⋊D157—44no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C2×Dic157—42no—
75C2×Dic157C22⋊C412yes—
76C4×D157—22no—
77C4×D157—22no—
78C4×D157—22no—
79C4×D157—22no—
80#11929434—41no—
81C23×C157—21no—
82C23×C157C2×D411yescharacteristic
83Q8×C157—21no—
84Q8×C157—21no—
85D4×C157—21no—
86D4×C157—21no—
87D4×C157—41no—
88D4×C157—41no—
89C2×C4×C157—21no—
90C2×C4×C157—22no—
91C2×C4×C157—42no—
92C2×C4×C157—41no—
93C2×C4×C157—41no—
94C2×C4×C157C22×C411yescharacteristic
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157#4111yescharacteristic
99C2×C4×C157#4111yescharacteristic
100C8×C157—41no—
101C22×C157—22no—
102C22×C157—41no—
103C22×C157—42no—
104C22×C157C2×C22⋊C411yescharacteristic
105C22×C157#12011yescharacteristic
106C22×C157#12311yescharacteristic
107C2×D157—22no—
108C2×D157—22no—
109C2×D157—22no—
110C2×D157—42no—
111C2×D157—42no—
112C4×C157—42no—
113C4×C157—41no—
114C4×C157C4×D411yescharacteristic
115C4×C157C4×D411yescharacteristic
116C4×C157#12111yescharacteristic
117C4×C157#12111yescharacteristic
118Dic157—22no—
119Dic157—42no—
120Dic157—42no—
121C2×C157—21no—
122C2×C157—21no—
123C2×C157—81no—
124C2×C157#38611yescharacteristic
125C2×C157#42112yes—
126D157—42no—
127C157#189711yescharacteristic, complemented, Sylow
128#1897—1571nomaximal, Sylow
129#525—1571no—
130C2×#117—1571no—
131C2×#104—1571no—
132C2×#104D15711yescharacteristic, complemented
133#407—1571no—
134#339—1571no—
135#339—1571no—
136C2×#41C2×D15711yescharacteristic
137C23×C4—1571no—
138C2×M16—1571no—
139C2×M16C2×D15711yescharacteristic
140#123—3141no—
141C22⋊Q8—1572no—
142#121—1572no—
143#120—3141no—
144C4×D4—3142no—
145#117—3142no—
146#117—3142no—
147#117—1571no—
148C2×C4⋊C4—3141no—
149C2×C22⋊C4—3141no—
150C2×C42—1571no—
151C2×C42C2×D15711yescharacteristic
152#104—22no—
153#104—3142no—
154C22⋊C8—3142no—
155C24—1571no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160C2×Q8C4×D15711yescharacteristic
161C2×D4—21no—
162C2×D4—3142no—
163C2×D4C4×D15711yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—3141no—
167C22×C4—3141no—
168C22×C4—3141no—
169C22×C4—3141no—
170C22×C4—3141no—
171C22×C4—1571no—
172C22×C4—1572no—
173C22×C4—1572no—
174C22×C4C22×D15711yescharacteristic
175M16—21no—
176M16—21no—
177M16—3141no—
178M16—3141no—
179C2×C8—21no—
180C2×C8—3141no—
181C4⋊C4—3144no—
182C4⋊C4—3142no—
183C4⋊C4—1572no—
184C22⋊C4—3144no—
185C22⋊C4—3142no—
186C22⋊C4—3141no—
187C22⋊C4—3142no—
188C42—21no—
189C42—3141no—
190C42—3141no—
191C42—1572no—
192C42C22⋊D15712yescomplemented
193C23—21no—
194C23—3141no—
195C23—3141no—
196C23—6282no—
197C23—1572no—
198C23D4×D15711yescharacteristic
199Q8—21no—
200Q8—21no—
201D4—21no—
202D4—21no—
203D4—41no—
204D4—41no—
205D4—6284no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—3141no—
212C2×C4—3141no—
213C2×C4—3142no—
214C2×C4—3142no—
215C2×C4—3142no—
216C2×C4—3142no—
217C2×C4—3142no—
218C2×C4—3142no—
219C2×C4—6282no—
220C2×C4—1572no—
221C2×C4C2×C4×D15711yescharacteristic
222C2×C4D4×D15711yescharacteristic
223C2×C4D4×D15711yescharacteristic
224C2×C4C2×C22⋊D15711yescharacteristic
225C2×C4#1457895311yescharacteristic
226C2×C4#1457895411yescharacteristic
227C8—41no—
228C8—6281no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—3142no—
233C22—3142no—
234C22—3142no—
235C22—6282no—
236C22—6282no—
237C22C22⋊C4×D15711yescharacteristic
238C22#1909773911yescharacteristic
239C22#1909775311yescharacteristic
240C4—42no—
241C4—41no—
242C4—3142no—
243C4—6282no—
244C4—6282no—
245C4#1909773111yescharacteristic
246C4C4×C22⋊D15711yescharacteristic
247C4#1909774211yescharacteristic
248C4#1909774411yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—6282no—
253C2#2375714911yescharacteristic
254C2#2375728812yes—
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757780C2
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1897
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#19097728C2semidirect
3. C2×#19097698C2semidirect
4. #23757236C2semidirect
5. #23757034C2semidirect
6. #23757033C2semidirect
7. C2×#104×C157C2semidirect
11. #19097747C4semidirect
12. #19097742C4semidirect
49. C4×Dic157D4semidirect
64. C42×C157D4semidirect
127. C157#1897semidirect
132. C2×#104D157semidirect
192. C42C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237577802
C2×#190977282
C2×#190976982
#237572362
#237570342
#237570332
C2×#104×C1572
#1897157