Finite Group Databasebeta

← #34100962

Subgroups — #34100962

Order 20096.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#23757819C211yesmaximal, characteristic
2#23757296C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic, complemented
4C2×#19097698C211yesmaximal, characteristic, complemented
5C2×#19097697C211yesmaximal, characteristic, complemented
6C42×Dic157C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097769C411yescharacteristic, complemented
11#19097767C411yescharacteristic, complemented
12#19097760C411yescharacteristic, complemented
13#19097758C411yescharacteristic, complemented
14#19097743—22no—
15C4×C22⋊D157—22no—
16C2×C4×Dic157—22no—
17C2×C4×Dic157C2211yescharacteristic
18C2×C4×Dic157C2211yescharacteristic
19#19097700—22no—
20#19097699—22no—
21#19097698—22no—
22#19097697—22no—
23C2×#41×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25C2×C42×C157C2211yescharacteristic
26#104×C157—22no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31C2×Dic314C2×C411yescharacteristic
32C2×D628C2×C411yescharacteristic
33C2×C22⋊D157—22no—
34C2×C22⋊D157—21no—
35C22×Dic157—21no—
36C22×Dic157D412yes—
37C2×C4×D157—21no—
38#14578938—21no—
39#14578938—21no—
40C2×#11929434—21no—
41#14578933—22no—
42C22⋊Dic157—22no—
43#14578931—22no—
44C4×Dic157—28no—
45C4×Dic157—21no—
46C4×Dic157C2×C411yescharacteristic
47C4×Dic157C2×C411yescharacteristic
48C4×Dic157D412yes—
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—
66Dic314—21no—
67Dic314—21no—
68D628—21no—
69D628—21no—
70C22⋊D157—41no—
71C22⋊D157—41no—
72C22⋊D157—44no—
73C2×Dic157—24no—
74C2×Dic157—22no—
75C2×Dic157—24no—
76C2×Dic157—21no—
77C2×Dic157C22⋊C412yes—
78C2×Dic157C22⋊C412yes—
79C4×D157—41no—
80C4×D157—41no—
81#11929434—41no—
82C23×C157—21no—
83C23×C157C2×D411yescharacteristic
84Q8×C157—21no—
85Q8×C157—21no—
86D4×C157—21no—
87D4×C157—21no—
88D4×C157—41no—
89D4×C157—41no—
90C2×C4×C157—24no—
91C2×C4×C157—22no—
92C2×C4×C157—21no—
93C2×C4×C157—41no—
94C2×C4×C157—41no—
95C2×C4×C157C22×C411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157#4111yescharacteristic
100C2×C4×C157#4111yescharacteristic
101C8×C157—41no—
102C22×C157—22no—
103C22×C157—41no—
104C22×C157—42no—
105C22×C157C2×C22⋊C411yescharacteristic
106C22×C157#12111yescharacteristic
107C22×C157#12111yescharacteristic
108C2×D157—41no—
109C2×D157—42no—
110C4×C157—24no—
111C4×C157—41no—
112C4×C157C4×D411yescharacteristic
113C4×C157C4×D411yescharacteristic
114C4×C157#12411yescharacteristic
115C4×C157#12711yescharacteristic
116Dic157—24no—
117Dic157—22no—
118Dic157—41no—
119Dic157#10414yescomplemented
120C2×C157—21no—
121C2×C157—21no—
122C2×C157—81no—
123C2×C157#39011yescharacteristic
124C2×C157C2×#10412yes—
125D157—81no—
126C157#196211yescharacteristic, complemented, Sylow
127#1962—1571nomaximal, Sylow
128#532—1571no—
129#423—1571no—
130C2×#104—1571no—
131C2×#104—1571no—
132C2×#104—1571no—
133C2×#104D15711yescharacteristic, complemented
134C43—1571no—
135C2×#41—1571no—
136C2×#41C2×D15711yescharacteristic
137C2×M16—1571no—
138C2×M16C2×D15711yescharacteristic
139C4⋊Q8—1571no—
140#127—1571no—
141#124—1571no—
142#124—1571no—
143#121—3142no—
144C4×D4—3142no—
145C2×C42—3142no—
146C2×C42—1571no—
147C2×C42—1571no—
148C2×C42C2×D15711yescharacteristic
149C4⋊C8—3142no—
150#104—22no—
151#104—3142no—
152#104—3142no—
153#104—3142no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—3141no—
159#41—3141no—
160#41—3141no—
161#41—3141no—
162C2×Q8—1571no—
163C2×Q8C4×D15711yescharacteristic
164C2×D4—21no—
165C2×D4—3141no—
166C2×D4—3142no—
167C2×D4—1571no—
168C2×D4C4×D15711yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—3141no—
172C22×C4—3141no—
173C22×C4—1572no—
174C22×C4C22×D15711yescharacteristic
175M16—21no—
176M16—21no—
177M16—3141no—
178M16—3141no—
179C2×C8—21no—
180C2×C8—3141no—
181C4⋊C4—3142no—
182C22⋊C4—3142no—
183C22⋊C4—3142no—
184C42—21no—
185C42—3148no—
186C42—3141no—
187C42—1572no—
188C42—1571no—
189C42—1571no—
190C42—1572no—
191C42C22⋊D15712yescomplemented
192C23—21no—
193C23—3141no—
194C23D4×D15711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—3141no—
198Q8—3141no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—3141no—
204D4—3141no—
205D4—6281no—
206D4—6281no—
207D4—6284no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—3144no—
214C2×C4—3142no—
215C2×C4—3144no—
216C2×C4—3141no—
217C2×C4—6281no—
218C2×C4—6281no—
219C2×C4—1572no—
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—6281no—
233C22—6282no—
234C22C22⋊C4×D15711yescharacteristic
235C22#1909774311yescharacteristic
236C22#1909774411yescharacteristic
237C4—24no—
238C4—41no—
239C4—3144no—
240C4—3142no—
241C4—6281no—
242C4—1574no—
243C4#1909773111yescharacteristic
244C4C4×C22⋊D15711yescharacteristic
245C4#1909775711yescharacteristic
246C4#1909776711yescharacteristic
247C2—21no—
248C2—21no—
249C2—81no—
250C2—12561no—
251C2#2375717311yescharacteristic
252C2#104×D15712yes—
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757819C2
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1962
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. #23757296C2semidirect
3. C2×#19097699C2semidirect
4. C2×#19097698C2semidirect
5. C2×#19097697C2semidirect
6. C42×Dic157C2semidirect
7. C2×#104×C157C2semidirect
10. #19097769C4semidirect
11. #19097767C4semidirect
12. #19097760C4semidirect
13. #19097758C4semidirect
49. C4×Dic157D4semidirect
64. C42×C157D4semidirect
119. Dic157#104semidirect
126. C157#1962semidirect
133. C2×#104D157semidirect
191. C42C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578192
#237572962
C2×#190976992
C2×#190976982
C2×#190976972
C42×Dic1572
C2×#104×C1572
#1962157