Finite Group Databasebeta

← #34100583

Subgroups — #34100583

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#23757785C211yesmaximal, characteristic
2C2×#19097728C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic, complemented
4C2×#19097697C211yesmaximal, characteristic, complemented
5#23757236C211yesmaximal, characteristic, complemented
6#23757032C211yesmaximal, characteristic, complemented
7#339×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×C4×D157C2211yescharacteristic
10C2×#14578938C2211yescharacteristic
11#19097755—21no—
12#19097751C412yescomplemented
13#19097746C412yescomplemented
14#19097741—21no—
15C4×D628—22no—
16#19097728—22no—
17#19097728—22no—
18C2×#14578933—21no—
19C2×#14578931—21no—
20C2×C4×Dic157C2211yescharacteristic
21#19097699—22no—
22#19097697—22no—
23#19097683—22no—
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26C2×C42×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32C2×Dic314C2×C411yescharacteristic
33C2×D628—22no—
34C2×D628C2×C411yescharacteristic
35C2×C22⋊D157—21no—
36C22×Dic157—21no—
37C22×Dic157D411yescharacteristic
38C2×C4×D157—21no—
39C2×C4×D157—21no—
40C2×C4×D157—21no—
41C2×C4×D157—21no—
42C2×C4×D157—21no—
43C2×C4×D157C2×C412yes—
44C2×C4×D157D412yes—
45#14578938—21no—
46#14578938—21no—
47C2×#11929434—21no—
48C4⋊Dic157—22no—
49#14578933—24no—
50#14578931—24no—
51#14578931—22no—
52#14578931—22no—
53C4×Dic157—21no—
54C4×Dic157D412yescomplemented
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57M16×C157—21no—
58M16×C157—21no—
59C2×C8×C157—21no—
60C4⋊C4×C157C2×C412yes—
61C22⋊C4×C157—21no—
62C42×C157—21no—
63C42×C157—21no—
64C42×C157D412yescomplemented
65C22×D157—21no—
66C22×D157—21no—
67C22×D157—21no—
68C22×D157—42no—
69C22×D157C22⋊C412yes—
70Dic314—21no—
71Dic314—21no—
72D628—21no—
73D628—21no—
74D628—44no—
75C22⋊D157—41no—
76C22⋊D157—41no—
77C2×Dic157—21no—
78C2×Dic157—21no—
79C2×Dic157—21no—
80C2×Dic157—22no—
81C2×Dic157—42no—
82C2×Dic157C22⋊C412yes—
83C4×D157—22no—
84C4×D157—22no—
85C4×D157—22no—
86C4×D157—22no—
87C4×D157—41no—
88C4×D157—41no—
89#11929434—41no—
90C23×C157C2×D411yescharacteristic
91C2×C4×C157—22no—
92C2×C4×C157—22no—
93C2×C4×C157—42no—
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×C157C2×C22⋊C411yescharacteristic
103C22×C157#12011yescharacteristic
104C22×C157#12311yescharacteristic
105C2×D157—22no—
106C2×D157—22no—
107C2×D157—22no—
108C2×D157—42no—
109C2×D157—42no—
110C2×D157—41no—
111C2×D157—42no—
112C4×C157—42no—
113C4×C157—42no—
114C4×C157C4×D411yescharacteristic
115C4×C157C4×D411yescharacteristic
116C4×C157#12111yescharacteristic
117C4×C157#12111yescharacteristic
118Dic157—22no—
119Dic157—42no—
120Dic157—41no—
121C2×C157—21no—
122C2×C157—21no—
123C2×C157#38611yescharacteristic
124C2×C157#42112yes—
125D157—42no—
126D157—81no—
127C157#189711yescharacteristic, complemented, Sylow
128#1897—1571nomaximal, Sylow
129#525—1571no—
130C2×#117—1571no—
131C2×#104—1571no—
132C2×#104—1571no—
133#407—1571no—
134#339—1571no—
135#339D15711yescharacteristic, complemented
136C2×#41—1571no—
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#117C2×D15711yescharacteristic
148C2×C4⋊C4—3141no—
149C2×C22⋊C4—3141no—
150C2×C42—1571no—
151C2×C42C2×D15711yescharacteristic
152#104—3142no—
153#104—3142no—
154C22⋊C8—3142no—
155C24—1571no—
156#41—3141no—
157#41—3141no—
158#41—3141no—
159#41—3141no—
160C2×Q8—1571no—
161C2×D4—3142no—
162C2×D4—3141no—
163C2×D4—1571no—
164C22×C4—21no—
165C22×C4—3141no—
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⋊C4C4×D15712yes—
184C22⋊C4—21no—
185C22⋊C4—3144no—
186C22⋊C4—3142no—
187C22⋊C4—3142no—
188C42—21no—
189C42—21no—
190C42—3141no—
191C42—1572no—
192C42D62812yescomplemented
193C23—3141no—
194C23—3141no—
195C23—3141no—
196C23—6282no—
197C23—1572no—
198C23D4×D15711yescharacteristic
199Q8—3141no—
200Q8—3141no—
201D4—3141no—
202D4—3141no—
203D4—6284no—
204D4—6281no—
205D4—6281no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3141no—
210C2×C4—3141no—
211C2×C4—3142no—
212C2×C4—3142no—
213C2×C4—3142no—
214C2×C4—3142no—
215C2×C4—3141no—
216C2×C4—3142no—
217C2×C4—6282no—
218C2×C4—6281no—
219C2×C4—6281no—
220C2×C4—1572no—
221C2×C4C2×C4×D15711yescharacteristic
222C2×C4D4×D15711yescharacteristic
223C2×C4D4×D15711yescharacteristic
224C2×C4C2×D62811yescharacteristic
225C2×C4#1457895411yescharacteristic
226C2×C4#1457895511yescharacteristic
227C8—41no—
228C8—6281no—
229C22—22no—
230C22—3142no—
231C22—3142no—
232C22—3142no—
233C22—6282no—
234C22—6282no—
235C22—6281no—
236C22—6282no—
237C22C22⋊C4×D15711yescharacteristic
238C22#1909774111yescharacteristic
239C22#1909775611yescharacteristic
240C4—42no—
241C4—42no—
242C4—3142no—
243C4—6282no—
244C4—6281no—
245C4#1909773311yescharacteristic
246C4C4×D62811yescharacteristic
247C4#1909774411yescharacteristic
248C4#1909774611yescharacteristic
249C2—21no—
250C2—21no—
251C2—6282no—
252C2—12561no—
253C2#2375715111yescharacteristic
254C2#2375728812yes—
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757785C2
cosocleC22×C4×C157C23
hypercenter#339D157
hypocenterC157#1897
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#19097728C2semidirect
3. C2×#19097699C2semidirect
4. C2×#19097697C2semidirect
5. #23757236C2semidirect
6. #23757032C2semidirect
7. #339×C157C2semidirect
12. #19097751C4semidirect
13. #19097746C4semidirect
54. C4×Dic157D4semidirect
64. C42×C157D4semidirect
127. C157#1897semidirect
135. #339D157semidirect
192. C42D628semidirect

Maximal subgroups

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

subgroupindex
#237577852
C2×#190977282
C2×#190976992
C2×#190976972
#237572362
#237570322
#339×C1572
#1897157