Finite Group Databasebeta

← #34099423

Subgroups — #34099423

Order 20096.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757927C211yesmaximal, characteristic
2#23757671C211yesmaximal, characteristic, complemented
3#23757553C211yesmaximal, characteristic, complemented
4#23757479C211yesmaximal, characteristic, complemented
5#23757018C211yesmaximal, characteristic, complemented
6#23757002C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578953—21no—
9C2×#14578943—21no—
10#19097771C2211yescharacteristic
11#19097771C2211yescharacteristic
12#19097755—22no—
13#19097751—21no—
14#19097745—21no—
15C4×D628C2211yescharacteristic
16C2×C4⋊Dic157—21no—
17#19097704—21no—
18#19097703—22no—
19C4⋊#11929434C2211yescharacteristic
20#19097696—21no—
21#19097692—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25#14578953—42no—
26#14578953—42no—
27C2×Dic314—21no—
28C2×Dic314—21no—
29C2×D628D411yescharacteristic
30C2×C22⋊D157—22no—
31C22×Dic157—22no—
32C2×C4×D157—21no—
33#14578943—42no—
34C2×#11929434—21no—
35C4⋊Dic157—22no—
36C4⋊Dic157—21no—
37C4⋊Dic157—21no—
38C4⋊Dic157—21no—
39C4⋊Dic157D411yescharacteristic
40C4⋊Dic157D412yes—
41#14578931—22no—
42#14578931—21no—
43C2×D4×C157D411yescharacteristic
44C22×C4×C157—21no—
45C2×C8×C157—21no—
46C2×C8×C157—21no—
47C4⋊C4×C157D411yescharacteristic
48C22⋊C4×C157—21no—
49C42×C157C2311yescharacteristic
50C22×D157—21no—
51Dic314—41no—
52Dic314—41no—
53Dic314—41no—
54Dic314—41no—
55D628—21no—
56D628—21no—
57C22⋊D157—44no—
58C2×Dic157—21no—
59C2×Dic157—22no—
60C2×Dic157—44no—
61C2×Dic157—41no—
62C4×D157—41no—
63C4×D157—41no—
64#11929434—41no—
65C23×C157—21no—
66D4×C157—21no—
67D4×C157SD1612yescomplemented
68C2×C4×C157—21no—
69C2×C4×C157—41no—
70C2×C4×C157—41no—
71C2×C4×C157C2×D411yescharacteristic
72C2×C4×C157C2×D411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C8×C157—22no—
75C8×C157—41no—
76C22×C157—22no—
77C22×C157—41no—
78C22×C157—41no—
79C22×C157#12011yescharacteristic
80C2×D157—41no—
81C2×D157—41no—
82C2×D157—41no—
83C4×C157—21no—
84C4×C157—41no—
85C4×C157C2×SD1611yescharacteristic
86C4×C157#13511yescharacteristic
87C4×C157AGL(1,ℤ/8)11yescharacteristic
88C4×C157AGL(1,ℤ/8)11yescharacteristic
89Dic157—41no—
90Dic157—42no—
91Dic157—81no—
92C2×C157—42no—
93C2×C157#44911yescharacteristic
94C2×C157#45011yescharacteristic
95C2×C157#45611yescharacteristic
96D157—81no—
97C157#165411yescharacteristic, complemented, Sylow
98#1654—1571nomaximal, Sylow
99#547—1571no—
100#498—1571no—
101#474—1571no—
102#461—1571no—
103#332—1571no—
104D4⋊C8—1571no—
105D4⋊C8D15711yescharacteristic, complemented
106C2×#41—3141no—
107C2×SD16—3141no—
108C4⋊Q8—1571no—
109C4⋊Q8—1571no—
110#123—3142no—
111C22⋊Q8—3141no—
112#121—3141no—
113C4×D4—1571no—
114C4×D4C2×D15711yescharacteristic
115C2×C4⋊C4—3141no—
116#107—3141no—
117#106—3142no—
118C4⋊C8—1571no—
119C4⋊C8C2×D15711yescharacteristic
120#103—3141no—
121#102—3141no—
122C4×C8C2×D15711yescharacteristic
123#41—6282no—
124#41—6282no—
125C2×Q8—3141no—
126C2×Q8—3141no—
127C2×D4—3142no—
128C2×D4—1571no—
129C2×D4D62811yescharacteristic
130C22×C4—21no—
131C22×C4—3142no—
132C22×C4—3141no—
133SD16—6282no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—3141no—
137C4⋊C4—3142no—
138C4⋊C4—3141no—
139C4⋊C4—3141no—
140C4⋊C4—3141no—
141C4⋊C4—1571no—
142C4⋊C4—1572no—
143C4⋊C4D62811yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—3142no—
146C22⋊C4—3141no—
147C42C22×D15711yescharacteristic
148C23—21no—
149C23—3141no—
150Q8—6281no—
151Q8—6281no—
152Q8—6281no—
153Q8—6281no—
154D4—21no—
155D4—3141no—
156D4—3141no—
157D4—6284no—
158D4#1457894312yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—3141no—
163C2×C4—3142no—
164C2×C4—6284no—
165C2×C4—6281no—
166C2×C4—6281no—
167C2×C4—6281no—
168C2×C4D4×D15711yescharacteristic
169C2×C4D4×D15711yescharacteristic
170C2×C4C2×D62811yescharacteristic
171C8—22no—
172C8—41no—
173C8—6281no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—6281no—
178C22—6281no—
179C22—6281no—
180C22#1909774111yescharacteristic
181C4—21no—
182C4—41no—
183C4—6281no—
184C4—6282no—
185C4—12561no—
186C4C2×#1457894311yescharacteristic
187C4#1909779211yescharacteristic
188C4#1909779511yescharacteristic
189C4#1909779911yescharacteristic
190C2—42no—
191C2—12561no—
192C2#2375741911yescharacteristic
193C2#2375742611yescharacteristic
194C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radical#23757927C2
hypercenterD4⋊C8D157
hypocenterC157#1654
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. #23757671C2semidirect
3. #23757553C2semidirect
4. #23757479C2semidirect
5. #23757018C2semidirect
6. #23757002C2semidirect
7. D4⋊C8×C157C2semidirect
67. D4×C157SD16semidirect
97. C157#1654semidirect
105. D4⋊C8D157semidirect
158. D4#14578943semidirect

Maximal subgroups

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

subgroupindex
#237579272
#237576712
#237575532
#237574792
#237570182
#237570022
D4⋊C8×C1572
#1654157