Finite Group Databasebeta

← #34100668

Subgroups — #34100668

Order 20096.

362 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 89 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#23757840C211yesmaximal, characteristic
2#23757291C212yesmaximal, complemented
3C2×#19097688C211yesmaximal, characteristic, complemented
4#23757029C212yesmaximal, complemented
5#518×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C2211yescharacteristic
7C2×#14578938C2212yes—
8#19097767C412yescomplemented
9#19097758C412yescomplemented
10#19097742—22no—
11#19097740—22no—
12#19097730C2211yescharacteristic
13#19097699—24no—
14#19097688—22no—
15C22×D4×C157C2211yescharacteristic
16C4×D4×C157—24no—
17#117×C157C2212yes—
18C2×C22⋊C4×C157—22no—
19#14578954—22no—
20#14578954—22no—
21C2×Dic314C2×C411yescharacteristic, complemented
22C2×D628C2×C411yescharacteristic, complemented
23C2×C22⋊D157—24no—
24C2×C22⋊D157—21no—
25C2×C4×D157—21no—
26#14578938—22no—
27#14578938—22no—
28C2×#11929434—22no—
29C4⋊Dic157—21no—
30C22⋊Dic157—24no—
31C22⋊Dic157—21no—
32C4×Dic157C2×C412yescomplemented
33C24×C157D412yes—
34C2×D4×C157—22no—
35C2×D4×C157—22no—
36C2×D4×C157C2×C412yes—
37C2×D4×C157D412yes—
38C22×C4×C157—24no—
39C22×C4×C157C2311yescharacteristic
40C4⋊C4×C157—22no—
41C22⋊C4×C157—24no—
42C22⋊C4×C157—22no—
43C42×C157D414yescomplemented
44C22×D157—21no—
45Dic314—22no—
46D628—22no—
47C22⋊D157—41no—
48C22⋊D157—41no—
49C22⋊D157—84no—
50C2×Dic157—21no—
51C2×Dic157—22no—
52C4×D157—42no—
53#11929434—42no—
54C23×C157—22no—
55C23×C157—22no—
56C23×C157—82no—
57C23×C157C22⋊C414yes—
58C23×C157C2×D411yescharacteristic
59D4×C157—24no—
60D4×C157—24no—
61C2×C4×C157—24no—
62C2×C4×C157—84no—
63C2×C4×C157C22×C411yescharacteristic
64C2×C4×C157C2×D412yes—
65C2×C4×C157C2×D411yescharacteristic
66C2×C4×C157#4112yes—
67C22×C157—24no—
68C22×C157—41no—
69C22×C157—44no—
70C22×C157—44no—
71C22×C157—82no—
72C22×C157C2×C22⋊C411yescharacteristic
73C22×C157#12011yescharacteristic
74C22×C157#12311yescharacteristic
75C2×D157—42no—
76C2×D157—41no—
77C4×C157—44no—
78C4×C157C4×D412yes—
79C4×C157#12112yes—
80Dic157—42no—
81Dic157—41no—
82C2×C157—21no—
83C2×C157—21no—
84C2×C157—21no—
85C2×C157—44no—
86C2×C157#38611yescharacteristic
87D157—81no—
88C157#191111yescharacteristic, complemented, Sylow
89#1911—1571nomaximal, Sylow
90#535—1571no—
91#518D15711yescharacteristic, complemented
92#421—1572no—
93C2×#100—1571no—
94H(3,ℤ/4)—1572no—
95C2×#41—1571no—
96C22×D4C2×D15711yescharacteristic
97C2×M16—1572no—
98#127—1572no—
99#124—1572no—
100#121—3142no—
101#120—3142no—
102C4×D4—24no—
103#117—1571no—
104#117C2×D15712yes—
105C2×C22⋊C4—22no—
106#104—3144no—
107#100—3142no—
108C24C22⋊D15712yes—
109#41—3142no—
110#41—3142no—
111C2×Q8—1571no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—3141no—
115C2×D4—3144no—
116C2×D4—1571no—
117C2×D4C4×D15712yes—
118C2×D4D62812yes—
119C22×C4—24no—
120C22×C4—3141no—
121C22×C4C22×D15711yescharacteristic
122M16—3142no—
123M16—3142no—
124C2×C8—3142no—
125C4⋊C4—22no—
126C4⋊C4—3141no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—3141no—
130C22⋊C4—3144no—
131C42—1572no—
132C42C22⋊D15714yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—3141no—
137C23#1457893114yes—
138C23C2×C22⋊D15711yescharacteristic
139Q8—3142no—
140D4—24no—
141D4—24no—
142D4—3142no—
143D4—6281no—
144D4—6281no—
145D4—12564no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—3142no—
149C2×C4—3141no—
150C2×C4—6282no—
151C2×C4C2×C4×D15711yescharacteristic
152C2×C4C2×C22⋊D15712yes—
153C2×C4C2×D62811yescharacteristic
154C2×C4#1457895412yes—
155C8—6282no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—6282no—
162C22—6281no—
163C22C2×#1457893111yescharacteristic
164C22#1909774011yescharacteristic
165C22#1909775411yescharacteristic
166C4—44no—
167C4—6282no—
168C4—6281no—
169C4C4×C22⋊D15712yes—
170C4#1909774512yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—12561no—
176C2#2375715211yescharacteristic
subgroupG/H
centerC2#23757152
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#518×C157C2
socleC2×C157#386
prime radical#23757840C2
cosocleC22×C4×C157C23
hypercenter#518D157
hypocenterC157#1911
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #23757291C2semidirect
3. C2×#19097688C2semidirect
4. #23757029C2semidirect
5. #518×C157C2semidirect
8. #19097767C4semidirect
9. #19097758C4semidirect
21. C2×Dic314C2×C4semidirect
22. C2×D628C2×C4semidirect
32. C4×Dic157C2×C4semidirect
43. C42×C157D4semidirect
88. C157#1911semidirect
91. #518D157semidirect
132. C42C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578402
#237572912
C2×#190976882
#237570292
#518×C1572
#1911157