Finite Group Databasebeta

← #34098653

Subgroups — #34098653

Order 20096.

346 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 77 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×#19097745C211yesmaximal, characteristic
2C4×C22⋊Dic157C211yesmaximal, characteristic, complemented
3#23757008C214yesmaximal, complemented
4#407×C157C211yesmaximal, characteristic, complemented
5C22×D628C411yescharacteristic, complemented
6C22×C22⋊D157—21no—
7#19097745—21no—
8#19097745—21no—
9#19097745C412yescomplemented
10#19097745C2212yes—
11C2×C4⋊Dic157C411yescharacteristic, complemented
12C2×C22⋊Dic157—21no—
13C2×#14578931—21no—
14C2×C4×Dic157—21no—
15#19097678C2212yescomplemented
16C23×C4×C157C2211yescharacteristic
17C2×M16×C157—21no—
18C22⋊C8×C157—21no—
19C22⋊C8×C157C2212yescomplemented
20C23×D157—21no—
21C2×D628—21no—
22C2×D628—42no—
23C2×D628C2×C412yes—
24C2×C22⋊D157—22no—
25C2×C22⋊D157—22no—
26C2×C22⋊D157—44no—
27C22×Dic157—21no—
28C22×Dic157—22no—
29C4⋊Dic157—21no—
30C4⋊Dic157C2×C412yes—
31C22⋊Dic157—24no—
32#14578931—22no—
33#14578931—22no—
34C4×Dic157—42no—
35C24×C157D411yescharacteristic
36C22×C4×C157—42no—
37C22×C4×C157C2×C411yescharacteristic
38C22×C4×C157C2×C411yescharacteristic
39C22×C4×C157D411yescharacteristic
40C22×C4×C157D411yescharacteristic
41C22×C4×C157D411yescharacteristic
42C22×C4×C157C2311yescharacteristic
43M16×C157—42no—
44C2×C8×C157—22no—
45C22×D157—22no—
46C22×D157—42no—
47C22×D157—42no—
48C22×D157—42no—
49D628—42no—
50D628—42no—
51C22⋊D157—44no—
52C22⋊D157—44no—
53C2×Dic157—24no—
54C2×Dic157—22no—
55C2×Dic157—42no—
56C2×Dic157—42no—
57C2×Dic157—42no—
58C23×C157—22no—
59C23×C157—22no—
60C23×C157—41no—
61C23×C157C22⋊C411yescharacteristic
62C23×C157C22⋊C411yescharacteristic
63C23×C157C2×D411yescharacteristic
64C2×C4×C157—22no—
65C2×C4×C157—22no—
66C2×C4×C157—22no—
67C2×C4×C157—41no—
68C2×C4×C157—41no—
69C2×C4×C157—41no—
70C2×C4×C157C22⋊C411yescharacteristic
71C2×C4×C157C22⋊C411yescharacteristic
72C2×C4×C157C22×C411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C8×C157—42no—
75C22×C157—21no—
76C22×C157—21no—
77C22×C157—21no—
78C22×C157—21no—
79C22×C157—21no—
80C22×C157—21no—
81C22×C157—21no—
82C22×C157—44no—
83C22×C157#10412yes—
84C22×C157C2×C22⋊C411yescharacteristic
85C22×C157AGL(1,ℤ/8)12yes—
86C2×D157—42no—
87C2×D157—42no—
88C2×D157—42no—
89C2×D157—84no—
90C4×C157—21no—
91C4×C157—21no—
92C4×C157—21no—
93C4×C157#9912yes—
94Dic157—44no—
95Dic157—42no—
96C2×C157—22no—
97C2×C157—22no—
98C2×C157—41no—
99C2×C157C2×#9911yescharacteristic
100C2×C157#41811yescharacteristic
101C2×C157C2×#10411yescharacteristic
102D157—82no—
103C157#150711yescharacteristic, complemented, Sylow
104#1507—1571nomaximal, Sylow
105C2×#121—1571no—
106#407D15711yescharacteristic, complemented
107C4×C22⋊C4—1571no—
108#327—1574no—
109C22×D4—3141no—
110C22×D4—1571no—
111C23×C4C2×D15711yescharacteristic
112C2×M16—21no—
113#121—3141no—
114#121—3141no—
115#121—1572no—
116#121—1572no—
117C2×C4⋊C4—1571no—
118C2×C22⋊C4—3141no—
119C2×C22⋊C4—3141no—
120C2×C42—3141no—
121C22⋊C8—21no—
122C22⋊C8C2×D15712yescomplemented
123#95—1572no—
124C24—3141no—
125C24D62811yescharacteristic
126C2×D4—3142no—
127C2×D4—3142no—
128C2×D4—3141no—
129C2×D4—6282no—
130C2×D4—6284no—
131C2×D4—1572no—
132C22×C4—42no—
133C22×C4—3142no—
134C22×C4—3141no—
135C22×C4C4×D15711yescharacteristic
136C22×C4C4×D15711yescharacteristic
137C22×C4C22⋊D15711yescharacteristic
138C22×C4C22⋊D15711yescharacteristic
139C22×C4D62811yescharacteristic
140C22×C4C22×D15711yescharacteristic
141M16—42no—
142C2×C8—22no—
143C4⋊C4—3141no—
144C4⋊C4—1572no—
145C22⋊C4—3144no—
146C22⋊C4—3142no—
147C22⋊C4—3142no—
148C42—6282no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—3142no—
153C23—6282no—
154C23—6282no—
155C23—6282no—
156C23#1457893111yescharacteristic
157C23#1457893111yescharacteristic
158C23C2×D62811yescharacteristic
159D4—6282no—
160D4—6284no—
161D4—6284no—
162D4—6282no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3144no—
170C2×C4—3142no—
171C2×C4—6282no—
172C2×C4—6282no—
173C2×C4—6282no—
174C2×C4#1457893111yescharacteristic
175C2×C4#1457893111yescharacteristic
176C2×C4C2×C4×D15711yescharacteristic
177C2×C4C2×C22⋊D15711yescharacteristic
178C8—42no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—6282no—
188C22—6282no—
189C22—6282no—
190C22—12564no—
191C22#1909769712yes—
192C22C2×#1457893111yescharacteristic
193C22#1909779912yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—6284no—
198C4—6282no—
199C4#1909768512yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—12562no—
204C2C2×#1909768511yescharacteristic
205C2#2375727911yescharacteristic
206C2C2×#1909769711yescharacteristic
subgroupG/H
centerC22C2×#14578931
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#407×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC2×#19097745C2
cosocleC22×C4×C157C23
hypercenter#407D157
hypocenterC157#1507
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Dic157C2semidirect
3. #23757008C2semidirect
4. #407×C157C2semidirect
5. C22×D628C4semidirect
9. #19097745C4semidirect
11. C2×C4⋊Dic157C4semidirect
15. #19097678C22semidirect
19. C22⋊C8×C157C22semidirect
103. C157#1507semidirect
106. #407D157semidirect
122. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977452
C4×C22⋊Dic1572
#237570082
#407×C1572
#1507157