Finite Group Databasebeta

← #34110069

Subgroups — #34110069

Order 20096.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#23757784C211yesmaximal, characteristic
2C2×#19097730C211yesmaximal, characteristic, complemented
3#23757615C214yesmaximal, complemented
4#23757598C214yesmaximal, complemented
5#23757315C212yesmaximal, complemented
6#23757275C212yesmaximal, complemented
7#407×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097754—21no—
10#19097750C2212yes—
11#19097745C2212yes—
12#19097740—21no—
13C4×C22⋊D157—22no—
14#19097730—22no—
15#19097730C2211yescharacteristic
16#19097730C2214yes—
17C2×C4⋊Dic157C2212yes—
18C2×C22⋊Dic157—21no—
19C2×C4×Dic157—21no—
20#19097705C2214yescomplemented
21#19097703C2214yescomplemented
22#19097696C2214yescomplemented
23#19097693C2214yescomplemented
24C23×C4×C157C2211yescharacteristic
25C2×M16×C157C2212yes—
26C22⋊C8×C157C2214yes—
27#14578954—24no—
28C2×Dic314C2311yescharacteristic
29C2×D628C2311yescharacteristic
30C2×C22⋊D157—22no—
31C2×C22⋊D157—21no—
32C22×Dic157—22no—
33C2×C4×D157—21no—
34C4⋊Dic157—22no—
35C4⋊Dic157C2314yes—
36C4⋊Dic157C2312yes—
37#14578933—22no—
38C22⋊Dic157—24no—
39C22⋊Dic157—21no—
40C22⋊Dic157—22no—
41#14578931—22no—
42C4×Dic157—24no—
43C4×Dic157—21no—
44C24×C157D411yescharacteristic
45C22×C4×C157—24no—
46C22×C4×C157D411yescharacteristic
47C22×C4×C157D412yes—
48C22×C4×C157C2312yes—
49C22×C4×C157C2311yescharacteristic
50M16×C157—24no—
51C2×C8×C157C2314yes—
52C22×D157—21no—
53Dic314—22no—
54D628—22no—
55C22⋊D157—44no—
56C22⋊D157—42no—
57C2×Dic157—21no—
58C2×Dic157—24no—
59C2×Dic157—22no—
60C2×Dic157—44no—
61C4×D157—42no—
62C23×C157—22no—
63C23×C157—42no—
64C23×C157C2×D412yes—
65C23×C157C2×D411yescharacteristic
66C2×C4×C157—22no—
67C2×C4×C157—24no—
68C2×C4×C157—24no—
69C2×C4×C157C2×D412yes—
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157#4114yes—
72C2×C4×C157C2411yescharacteristic
73C8×C157—24no—
74C22×C157—22no—
75C22×C157—22no—
76C22×C157—22no—
77C22×C157—22no—
78C22×C157—44no—
79C22×C157#12312yes—
80C22×C157C22×D411yescharacteristic
81C2×D157—41no—
82C2×D157—42no—
83C4×C157—22no—
84C4×C157#12312yes—
85C4×C157C2×#4112yes—
86Dic157—41no—
87Dic157—44no—
88Dic157—42no—
89C2×C157—22no—
90C2×C157—42no—
91C2×C157C2×#12311yescharacteristic
92C2×C157#57512yes—
93D157—81no—
94C157#309311yescharacteristic, complemented, Sylow
95#3093—1571nomaximal, Sylow
96#525—1571no—
97C2×#117—1571no—
98#485—1574no—
99#482—1574no—
100#428—1572no—
101#417—1572no—
102#407D15711yescharacteristic, complemented
103C2×#41—1571no—
104C23×C4C2×D15711yescharacteristic
105C2×M16C2×D15712yes—
106#123—3141no—
107C22⋊Q8—1572no—
108#121—1572no—
109#120—3141no—
110C4×D4—3142no—
111#117—3142no—
112#117—1574no—
113#117—1571no—
114C2×C4⋊C4—1572no—
115C2×C22⋊C4—3141no—
116C2×C42—3141no—
117#107—1574no—
118#106—1574no—
119#103—1574no—
120#102—1574no—
121C22⋊C8C2×D15714yes—
122C24D62811yescharacteristic
123#41—3144no—
124C2×Q8—1571no—
125C2×D4—3142no—
126C2×D4—3141no—
127C2×D4—1571no—
128C22×C4—24no—
129C22×C4—3142no—
130C22×C4—3141no—
131C22×C4D62811yescharacteristic
132C22×C4D62812yes—
133C22×C4C22×D15712yes—
134C22×C4C22×D15711yescharacteristic
135M16—24no—
136C2×C8C22×D15714yes—
137C4⋊C4—3142no—
138C4⋊C4—3142no—
139C4⋊C4—1574no—
140C4⋊C4—1572no—
141C22⋊C4—3144no—
142C22⋊C4—3141no—
143C22⋊C4—3142no—
144C22⋊C4—3142no—
145C42—3144no—
146C42—3141no—
147C23—22no—
148C23—42no—
149C23—3141no—
150C23C2×D62812yes—
151C23C2×D62811yescharacteristic
152Q8—3142no—
153D4—3142no—
154D4—6284no—
155D4—6282no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—3144no—
160C2×C4—3142no—
161C2×C4—3141no—
162C2×C4—6284no—
163C2×C4—6282no—
164C2×C4C2×D62812yes—
165C2×C4C2×D62811yescharacteristic
166C2×C4#1457895314yes—
167C2×C4C23×D15711yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—6281no—
175C22—6282no—
176C22#1909775512yes—
177C22C22×D62811yescharacteristic
178C4—22no—
179C4—6281no—
180C4—6284no—
181C4—6282no—
182C4#1909775512yes—
183C4C2×#1457895312yes—
184C2—22no—
185C2—42no—
186C2—12561no—
187C2C2×#1909775511yescharacteristic
188C2#2375807312yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#407×C157C2
socleC22×C157C22×D4
prime radical#23757784C2
hypercenter#407D157
hypocenterC157#3093
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#19097730C2semidirect
3. #23757615C2semidirect
4. #23757598C2semidirect
5. #23757315C2semidirect
6. #23757275C2semidirect
7. #407×C157C2semidirect
20. #19097705C22semidirect
21. #19097703C22semidirect
22. #19097696C22semidirect
23. #19097693C22semidirect
94. C157#3093semidirect
102. #407D157semidirect

Maximal subgroups

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

subgroupindex
#237577842
C2×#190977302
#237576152
#237575982
#237573152
#237572752
#407×C1572
#3093157