Finite Group Databasebeta

← #34099248

Subgroups — #34099248

Order 20096.

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

normal subgroups only

This group has 216 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, complemented
2#23757620C211yesmaximal, characteristic, complemented
3#23757474C211yesmaximal, characteristic, complemented
4#23757468C211yesmaximal, characteristic
5#23757018C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578943—21no—
9C2×#14578939—21no—
10#19097771C2211yescharacteristic
11#19097768C2211yescharacteristic
12#19097764—22no—
13#19097758—21no—
14#19097750—21no—
15C4×Dic314C2211yescharacteristic
16#19097736—21no—
17#19097730—21no—
18C4⋊#11929434C2211yescharacteristic
19#19097693—22no—
20#19097693—21no—
21#19097691—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×D628—21no—
28C2×D628—21no—
29C2×D628—21no—
30#14578943—42no—
31#14578939—42no—
32C2×#11929434—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157D411yescharacteristic
35C4⋊Dic157D412yes—
36#14578933—22no—
37#14578933—21no—
38C22⋊Dic157—22no—
39C22⋊Dic157—22no—
40C4×Dic157—22no—
41C4×Dic157—21no—
42C2×D4×C157D411yescharacteristic
43C22×C4×C157—21no—
44C2×C8×C157—21no—
45C2×C8×C157—21no—
46C4⋊C4×C157D411yescharacteristic
47C22⋊C4×C157—21no—
48C42×C157C2311yescharacteristic
49C22×D157—41no—
50Dic314—21no—
51Dic314—21no—
52Dic314—41no—
53Dic314—41no—
54D628—42no—
55D628—41no—
56D628—41no—
57D628—41no—
58D628—41no—
59C2×Dic157—22no—
60C2×Dic157—21no—
61C2×Dic157—21no—
62#11929434—41no—
63C23×C157—21no—
64D4×C157—21no—
65D4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—41no—
68C2×C4×C157—41no—
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157C2×D411yescharacteristic
72C8×C157—41no—
73C8×C157—41no—
74C22×C157—41no—
75C22×C157—41no—
76C22×C157—41no—
77C22×C157#12011yescharacteristic
78C2×D157—81no—
79C2×D157—81no—
80C2×D157—81no—
81C4×C157—21no—
82C4×C157—41no—
83C4×C157#13511yescharacteristic
84C4×C157#13511yescharacteristic
85C4×C157AGL(1,ℤ/8)11yescharacteristic
86C4×C157AGL(1,ℤ/8)11yescharacteristic
87Dic157—41no—
88Dic157—42no—
89Dic157—41no—
90C2×C157—81no—
91C2×C157#44911yescharacteristic
92C2×C157#44911yescharacteristic
93C2×C157#45411yescharacteristic
94D157—161no—
95C157#162811yescharacteristic, complemented, Sylow
96#1628—1571nomaximal, Sylow
97#556—1571no—
98#486—1571no—
99#460—1571no—
100#459—1571no—
101#332—1571no—
102Q8⋊C8—1571no—
103D4⋊C8D15711yescharacteristic, complemented
104C2×SD16—3141no—
105C2×D8—3141no—
106C4⋊Q8—1571no—
107#127—1571no—
108#126—3142no—
109#124—3141no—
110C22⋊Q8—3141no—
111C4×Q8—3141no—
112C4×Q8—1571no—
113C4×D4C2×D15711yescharacteristic
114#117—3141no—
115C4⋊C8—1571no—
116C4⋊C8C2×D15711yescharacteristic
117#102—3141no—
118#102—3142no—
119#102—3141no—
120C4×C8C2×D15711yescharacteristic
121C2×Q8—3141no—
122C2×Q8—1571no—
123C2×D4—3141no—
124C2×D4—3141no—
125C2×D4—3141no—
126C2×D4D62811yescharacteristic
127C22×C4—21no—
128SD16—6282no—
129D8—6282no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—3141no—
133C4⋊C4—3142no—
134C4⋊C4—3141no—
135C4⋊C4—3141no—
136C4⋊C4—1571no—
137C4⋊C4—1572no—
138C4⋊C4D62811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3142no—
141C22⋊C4—3142no—
142C42—3142no—
143C42—3141no—
144C42C22×D15711yescharacteristic
145C23—21no—
146C23—6281no—
147Q8—3141no—
148Q8—3141no—
149Q8—6281no—
150Q8—6281no—
151D4—21no—
152D4—21no—
153D4—6282no—
154D4—6281no—
155D4—6281no—
156D4—6281no—
157D4—6281no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3141no—
162C2×C4—3141no—
163C2×C4—3142no—
164C2×C4D4×D15711yescharacteristic
165C2×C4D4×D15711yescharacteristic
166C2×C4C2×D62811yescharacteristic
167C8—41no—
168C8—41no—
169C8—6281no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—12561no—
174C22—12561no—
175C22—12561no—
176C22#1909774111yescharacteristic
177C4—21no—
178C4—41no—
179C4—6281no—
180C4—6282no—
181C4—6281no—
182C4#1909779211yescharacteristic
183C4#1909779311yescharacteristic
184C4#1909779511yescharacteristic
185C4#1909779911yescharacteristic
186C2—81no—
187C2—25121no—
188C2#2375741911yescharacteristic
189C2#2375742011yescharacteristic
190C2#2375744711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radical#23757468C2
hypercenterD4⋊C8D157
hypocenterC157#1628
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
1. #23757993C2semidirect
2. #23757620C2semidirect
3. #23757474C2semidirect
5. #23757018C2semidirect
6. #23757005C2semidirect
7. D4⋊C8×C157C2semidirect
95. C157#1628semidirect
103. D4⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237579932
#237576202
#237574742
#237574682
#237570182
#237570052
D4⋊C8×C1572
#1628157