Finite Group Databasebeta

← #34099266

Subgroups — #34099266

Order 20096.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#23757944C211yesmaximal, characteristic
2#23757646C211yesmaximal, characteristic, complemented
3#23757569C211yesmaximal, characteristic, complemented
4#23757563C211yesmaximal, characteristic, complemented
5#23757016C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578954—21no—
9C2×#14578943—22no—
10#19097771C2211yescharacteristic
11#19097768C2211yescharacteristic
12#19097767—21no—
13#19097745—21no—
14#19097744—22no—
15C4×Dic314C2211yescharacteristic
16#19097733—21no—
17#19097705—21no—
18#19097702—21no—
19C4⋊#11929434C2211yescharacteristic
20#19097696—21no—
21#19097692—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25#14578954—42no—
26#14578954—42no—
27C2×Dic314—21no—
28C2×Dic314D411yescharacteristic
29C2×D628—21no—
30C2×D628—21no—
31C2×D628D412yes—
32C2×C22⋊D157—22no—
33C2×C22⋊D157—22no—
34C2×C4×D157—22no—
35#14578943—44no—
36C2×#11929434—21no—
37C4⋊Dic157—21no—
38C4⋊Dic157—21no—
39C4⋊Dic157D411yescharacteristic
40#14578933—21no—
41#14578931—22no—
42C4×Dic157—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—22no—
51Dic314—21no—
52Dic314—42no—
53Dic314SD1612yescomplemented
54D628—42no—
55D628—41no—
56D628—41no—
57D628—41no—
58D628—41no—
59C22⋊D157—44no—
60C22⋊D157—82no—
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157—41no—
64C4×D157—44no—
65#11929434—41no—
66C23×C157—21no—
67D4×C157—21no—
68D4×C157—21no—
69C2×C4×C157—21no—
70C2×C4×C157—41no—
71C2×C4×C157—41no—
72C2×C4×C157C2×D411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157C2×D411yescharacteristic
75C8×C157—22no—
76C8×C157—41no—
77C22×C157—41no—
78C22×C157—41no—
79C22×C157—41no—
80C22×C157#12011yescharacteristic
81C2×D157—42no—
82C2×D157—42no—
83C2×D157—42no—
84C4×C157—21no—
85C4×C157—41no—
86C4×C157C2×SD1611yescharacteristic
87C4×C157#13511yescharacteristic
88C4×C157AGL(1,ℤ/8)11yescharacteristic
89C4×C157#13711yescharacteristic
90Dic157—22no—
91Dic157—41no—
92Dic157—81no—
93C2×C157—81no—
94C2×C157#45011yescharacteristic
95C2×C157#45211yescharacteristic
96C2×C157#45311yescharacteristic
97D157—82no—
98C157#163111yescharacteristic, complemented, Sylow
99#1631—1571nomaximal, Sylow
100#550—1571no—
101#492—1571no—
102#477—1571no—
103#476—1571no—
104#331—1571no—
105Q8⋊C8—1571no—
106D4⋊C8D15711yescharacteristic, complemented
107C2×#41—3141no—
108C2×SD16—3142no—
109C4⋊Q8—1571no—
110#127—3141no—
111#127—1571no—
112#121—3142no—
113#121—3141no—
114C4×Q8—1571no—
115C4×D4—3141no—
116C4×D4C2×D15711yescharacteristic
117#107—3141no—
118#106—3141no—
119C4⋊C8—1571no—
120C4⋊C8C2×D15711yescharacteristic
121#103—3141no—
122#102—3141no—
123C4×C8C2×D15711yescharacteristic
124#41—6282no—
125#41—6282no—
126C2×Q8—3141no—
127C2×Q8—1571no—
128C2×D4—3142no—
129C2×D4—3142no—
130C2×D4—3141no—
131C2×D4—3141no—
132C2×D4—1572no—
133C2×D4D62811yescharacteristic
134C22×C4—21no—
135C22×C4—3142no—
136SD16—6284no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—3141no—
140C4⋊C4—3141no—
141C4⋊C4—3141no—
142C4⋊C4—3141no—
143C4⋊C4—1571no—
144C4⋊C4D62811yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—3142no—
147C42—3141no—
148C42C22×D15711yescharacteristic
149C23—21no—
150C23—3142no—
151Q8—3141no—
152Q8—6282no—
153Q8—1572no—
154D4—21no—
155D4—21no—
156D4—6282no—
157D4—6284no—
158D4—6281no—
159D4—6281no—
160D4—6281no—
161D4—6281no—
162D4—12562no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—3141no—
167C2×C4—3141no—
168C2×C4—6284no—
169C2×C4—6281no—
170C2×C4D4×D15711yescharacteristic
171C2×C4D4×D15711yescharacteristic
172C2×C4C2×D62811yescharacteristic
173C8—22no—
174C8—41no—
175C8—6281no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—6282no—
180C22—6282no—
181C22—6282no—
182C22#1909774111yescharacteristic
183C4—21no—
184C4—41no—
185C4—3142no—
186C4—6281no—
187C4—12561no—
188C4SD16×D15711yescharacteristic
189C4#1909779311yescharacteristic
190C4#1909779511yescharacteristic
191C4#1909780411yescharacteristic
192C2—81no—
193C2—12562no—
194C2#2375742511yescharacteristic
195C2#2375743711yescharacteristic
196C2#2375744211yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radical#23757944C2
hypercenterD4⋊C8D157
hypocenterC157#1631
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. #23757646C2semidirect
3. #23757569C2semidirect
4. #23757563C2semidirect
5. #23757016C2semidirect
6. #23757005C2semidirect
7. D4⋊C8×C157C2semidirect
53. Dic314SD16semidirect
98. C157#1631semidirect
106. D4⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237579442
#237576462
#237575692
#237575632
#237570162
#237570052
D4⋊C8×C1572
#1631157