Finite Group Databasebeta

← #34100663

Subgroups — #34100663

Order 20096.

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

normal subgroups only

This group has 344 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#23757837C211yesmaximal, characteristic
2#23757279C212yesmaximal, complemented
3#23757247C211yesmaximal, characteristic, complemented
4#23757030C212yesmaximal, complemented
5#568×C157C211yesmaximal, characteristic, complemented
6C22×D628C2211yescharacteristic
7#19097767C412yescomplemented
8#19097760C412yescomplemented
9#19097745—22no—
10#19097739—22no—
11#19097730C2212yes—
12#19097730C2211yescharacteristic
13#19097693—24no—
14#19097686—22no—
15C2×#41×C157C2211yescharacteristic
16#131×C157—24no—
17C2×M16×C157—22no—
18C2×M16×C157C2212yes—
19C23×D157—21no—
20C2×D628—22no—
21C2×D628—21no—
22C2×D628—22no—
23C2×D628C2×C412yes—
24C2×C22⋊D157—24no—
25C4⋊Dic157—21no—
26C4⋊Dic157—22no—
27C22⋊Dic157—21no—
28C22⋊Dic157—22no—
29#14578931—24no—
30C4×Dic157—22no—
31C4×Dic157C2×C412yes—
32#41×C157—22no—
33#41×C157—22no—
34C2×Q8×C157C2×C411yescharacteristic
35C2×D4×C157C2×C411yescharacteristic
36C2×D4×C157D412yes—
37C22×C4×C157D412yes—
38C22×C4×C157C2311yescharacteristic
39M16×C157—24no—
40M16×C157—22no—
41M16×C157—22no—
42C2×C8×C157—24no—
43C2×C8×C157D414yescomplemented
44C22×D157—22no—
45C22×D157—42no—
46C22×D157—82no—
47D628—24no—
48D628—41no—
49D628—41no—
50C22⋊D157—84no—
51C2×Dic157—22no—
52C2×Dic157—42no—
53C23×C157C22⋊C412yes—
54C23×C157C2×D411yescharacteristic
55Q8×C157—22no—
56D4×C157—24no—
57D4×C157—22no—
58C2×C4×C157—22no—
59C2×C4×C157—22no—
60C2×C4×C157C22⋊C412yes—
61C2×C4×C157C22×C411yescharacteristic
62C2×C4×C157C2×D412yes—
63C2×C4×C157C2×D411yescharacteristic
64C2×C4×C157#4112yes—
65C8×C157—24no—
66C22×C157—22no—
67C22×C157—41no—
68C22×C157—42no—
69C22×C157C2×C22⋊C411yescharacteristic
70C22×C157#12112yes—
71C2×D157—44no—
72C2×D157—42no—
73C2×D157—84no—
74C4×C157—22no—
75C4×C157C4×D412yes—
76C4×C157#12011yescharacteristic
77C4×C157#12311yescharacteristic
78Dic157—42no—
79Dic157—82no—
80C2×C157—21no—
81C2×C157—22no—
82C2×C157—42no—
83C2×C157#38611yescharacteristic
84D157—82no—
85C157#191011yescharacteristic, complemented, Sylow
86#1910—1571nomaximal, Sylow
87#568D15711yescharacteristic, complemented
88#535—1571no—
89#418—1572no—
90#410—1571no—
91H(3,ℤ/4)—1572no—
92C2×#41C2×D15711yescharacteristic
93C22×D4—1571no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D15712yes—
97#127—1572no—
98#124—1572no—
99#121—3142no—
100#120—3142no—
101#117—1572no—
102#117—1571no—
103#102—3144no—
104#99—3142no—
105C24—3141no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D15711yescharacteristic
109C2×D4—3144no—
110C2×D4—3142no—
111C2×D4—3141no—
112C2×D4—3142no—
113C2×D4—1572no—
114C2×D4C4×D15711yescharacteristic
115C2×D4D62812yes—
116C22×C4C22⋊D15712yes—
117C22×C4C22×D15711yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D15714yescomplemented
123C4⋊C4—3141no—
124C4⋊C4—3142no—
125C22⋊C4—3141no—
126C22⋊C4—3144no—
127C22⋊C4—3142no—
128C42—3142no—
129C42—1572no—
130C23—3142no—
131C23—6282no—
132C23—12562no—
133C23#1457893112yes—
134C23C2×D62811yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—3144no—
139D4—6281no—
140D4—6281no—
141D4—12564no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—3142no—
145C2×C4—6282no—
146C2×C4#1457893112yes—
147C2×C4C2×C4×D15711yescharacteristic
148C2×C4C2×C22⋊D15712yes—
149C2×C4C2×C22⋊D15711yescharacteristic
150C2×C4#1457895412yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—6284no—
156C22—6282no—
157C22—12564no—
158C22C2×#1457893111yescharacteristic
159C22#1909774512yes—
160C4—22no—
161C4—6282no—
162C4—12562no—
163C4C4×C22⋊D15712yes—
164C4#1909774011yescharacteristic
165C4#1909775411yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—12562no—
170C2#2375715211yescharacteristic
subgroupG/H
centerC2#23757152
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#568×C157C2
socleC2×C157#386
prime radical#23757837C2
cosocleC22×C4×C157C23
hypercenter#568D157
hypocenterC157#1910
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23757279C2semidirect
3. #23757247C2semidirect
4. #23757030C2semidirect
5. #568×C157C2semidirect
7. #19097767C4semidirect
8. #19097760C4semidirect
43. C2×C8×C157D4semidirect
85. C157#1910semidirect
87. #568D157semidirect
122. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
#237572792
#237572472
#237570302
#568×C1572
#1910157