Finite Group Databasebeta

← #34099310

Subgroups — #34099310

Order 20096.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757993C211yesmaximal, characteristic
2#23757624C211yesmaximal, characteristic, complemented
3#23757569C211yesmaximal, characteristic, complemented
4#23757563C211yesmaximal, characteristic
5#23757018C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic
8#19097771C2211yescharacteristic
9#19097771C2211yescharacteristic
10#19097764—22no—
11#19097758—21no—
12#19097750—21no—
13C4×Dic314C2211yescharacteristic
14#19097736—21no—
15#19097730—21no—
16#19097705—21no—
17#19097702—21no—
18C4⋊#11929434C2211yescharacteristic
19#19097696—22no—
20#19097696—21no—
21#19097692—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314—21no—
27C2×Dic314D411yescharacteristic
28C2×#11929434—21no—
29C4⋊Dic157—21no—
30C4⋊Dic157—21no—
31C4⋊Dic157—21no—
32C4⋊Dic157D412yes—
33C4⋊Dic157D411yescharacteristic
34#14578933—22no—
35#14578933—21no—
36C22⋊Dic157—22no—
37C22⋊Dic157—22no—
38C4×Dic157—22no—
39C4×Dic157—21no—
40C2×D4×C157D411yescharacteristic
41C22×C4×C157—21no—
42C2×C8×C157—21no—
43C2×C8×C157—21no—
44C4⋊C4×C157D411yescharacteristic
45C22⋊C4×C157—21no—
46C42×C157C2311yescharacteristic
47Dic314—21no—
48Dic314—21no—
49Dic314—41no—
50Dic314—41no—
51Dic314—42no—
52C2×Dic157—22no—
53C2×Dic157—21no—
54C2×Dic157—21no—
55C2×Dic157—41no—
56#11929434—41no—
57C23×C157—21no—
58D4×C157—21no—
59D4×C157—21no—
60C2×C4×C157—21no—
61C2×C4×C157—41no—
62C2×C4×C157—41no—
63C2×C4×C157C2×D411yescharacteristic
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157C2×D411yescharacteristic
66C8×C157—41no—
67C8×C157—41no—
68C22×C157—41no—
69C22×C157—41no—
70C22×C157—41no—
71C22×C157#12011yescharacteristic
72C4×C157—21no—
73C4×C157—41no—
74C4×C157#13511yescharacteristic
75C4×C157#13511yescharacteristic
76C4×C157#13711yescharacteristic
77C4×C157#13711yescharacteristic
78Dic157—41no—
79Dic157—42no—
80Dic157—41no—
81Dic157—81no—
82C2×C157—81no—
83C2×C157#45211yescharacteristic
84C2×C157#45211yescharacteristic
85C2×C157#45411yescharacteristic
86C157#163811yescharacteristic, complemented, Sylow
87#1638—1571nomaximal, Sylow
88#556—1571no—
89#487—1571no—
90#477—1571no—
91#476—1571no—
92#332—1571no—
93Q8⋊C8—1571no—
94D4⋊C8D15711yescharacteristic
95C4⋊Q8—1571no—
96C4⋊Q8—1571no—
97#126—3142no—
98#124—3141no—
99C22⋊Q8—3141no—
100C4×Q8—3141no—
101C4×Q8—1571no—
102C4×D4C2×D15711yescharacteristic
103#117—3141no—
104#107—3141no—
105#106—3141no—
106C4⋊C8—1571no—
107C4⋊C8C2×D15711yescharacteristic
108#103—3141no—
109#103—3142no—
110#102—3141no—
111C4×C8C2×D15711yescharacteristic
112C2×Q8—3141no—
113C2×Q8—3141no—
114C2×Q8—1571no—
115C2×D4D62811yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—3141no—
120C4⋊C4—3142no—
121C4⋊C4—3141no—
122C4⋊C4—3141no—
123C4⋊C4—3141no—
124C4⋊C4—3141no—
125C4⋊C4—1572no—
126C4⋊C4—1571no—
127C4⋊C4D62811yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—3142no—
130C22⋊C4—3142no—
131C42—3142no—
132C42—3141no—
133C42C22×D15711yescharacteristic
134C23—21no—
135Q8—3141no—
136Q8—3141no—
137Q8—6282no—
138Q8—6281no—
139Q8—6281no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—3141no—
146C2×C4—3141no—
147C2×C4—3142no—
148C2×C4—6281no—
149C2×C4D4×D15711yescharacteristic
150C2×C4D4×D15711yescharacteristic
151C2×C4C2×D62811yescharacteristic
152C8—41no—
153C8—41no—
154C8—6281no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#1909774111yescharacteristic
159C4—21no—
160C4—41no—
161C4—6281no—
162C4—6282no—
163C4—6281no—
164C4—12561no—
165C4#1909779011yescharacteristic
166C4#1909779311yescharacteristic
167C4#1909780011yescharacteristic
168C4#1909780411yescharacteristic
169C2—81no—
170C2#2375743611yescharacteristic
171C2#2375743711yescharacteristic
172C2#2375744711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenterD4⋊C8D157
hypocenterC157#1638
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757624C2semidirect
3. #23757569C2semidirect
5. #23757018C2semidirect
6. #23757005C2semidirect
86. C157#1638semidirect

Maximal subgroups

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

subgroupindex
#237579932
#237576242
#237575692
#237575632
#237570182
#237570052
D4⋊C8×C1572
#1638157