Finite Group Databasebeta

← #34099366

Subgroups — #34099366

Order 20096.

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

normal subgroups only

This group has 212 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#23757988C211yesmaximal, characteristic
2#23757624C211yesmaximal, characteristic, complemented
3#23757554C211yesmaximal, characteristic, complemented
4#23757478C211yesmaximal, characteristic, complemented
5#23757021C211yesmaximal, characteristic, complemented
6#23757002C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578942—21no—
9#19097771C2211yescharacteristic
10#19097771C2211yescharacteristic
11#19097757—22no—
12#19097745—21no—
13#19097742—21no—
14C4×D628C2211yescharacteristic
15#19097730—21no—
16#19097729—21no—
17#19097705—21no—
18C4⋊#11929434C2211yescharacteristic
19#19097696—22no—
20#19097694—21no—
21#19097693—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D412yes—
27C2×D628D411yescharacteristic
28C2×C22⋊D157—22no—
29C2×C4×D157—21no—
30#14578942—42no—
31C2×#11929434—21no—
32C4⋊Dic157—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157D411yescharacteristic
37C22⋊Dic157—22no—
38#14578931—22no—
39#14578931—21no—
40C4×Dic157—22no—
41C2×D4×C157D411yescharacteristic
42C22×C4×C157—21no—
43C2×C8×C157—21no—
44C2×C8×C157—21no—
45C4⋊C4×C157D411yescharacteristic
46C22⋊C4×C157—21no—
47C42×C157C2311yescharacteristic
48C22×D157—21no—
49Dic314—42no—
50Dic314—41no—
51Dic314—41no—
52D628—21no—
53D628—21no—
54C22⋊D157—82no—
55C2×Dic157—21no—
56C2×Dic157—22no—
57C2×Dic157—41no—
58C4×D157—41no—
59C4×D157—41no—
60#11929434—41no—
61C23×C157—21no—
62D4×C157—21no—
63D4×C157—21no—
64C2×C4×C157—21no—
65C2×C4×C157—41no—
66C2×C4×C157—41no—
67C2×C4×C157C2×D411yescharacteristic
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C8×C157—41no—
71C8×C157—41no—
72C22×C157—41no—
73C22×C157—41no—
74C22×C157—41no—
75C22×C157#12011yescharacteristic
76C2×D157—41no—
77C2×D157—41no—
78C2×D157—41no—
79C4×C157—21no—
80C4×C157—41no—
81C4×C157#13511yescharacteristic
82C4×C157#13511yescharacteristic
83C4×C157AGL(1,ℤ/8)11yescharacteristic
84C4×C157#13711yescharacteristic
85Dic157—41no—
86Dic157—42no—
87Dic157—81no—
88C2×C157—81no—
89C2×C157#44911yescharacteristic
90C2×C157#45211yescharacteristic
91C2×C157#45511yescharacteristic
92D157—81no—
93C157#164611yescharacteristic, complemented, Sylow
94#1646—1571nomaximal, Sylow
95#555—1571no—
96#487—1571no—
97#474—1571no—
98#461—1571no—
99#333—1571no—
100D4⋊C8—1571no—
101D4⋊C8D15711yescharacteristic, complemented
102C2×SD16—3141no—
103C4⋊Q8—1571no—
104C4⋊Q8—1571no—
105#124—3142no—
106#121—3141no—
107#121—3141no—
108C4×D4—1571no—
109C4×D4C2×D15711yescharacteristic
110#117—3141no—
111#117—3141no—
112#107—3141no—
113C4⋊C8—1571no—
114C4⋊C8C2×D15711yescharacteristic
115#103—3142no—
116#103—3141no—
117#102—3141no—
118C4×C8C2×D15711yescharacteristic
119C2×Q8—3141no—
120C2×Q8—1572no—
121C2×D4—3142no—
122C2×D4—1571no—
123C2×D4D62811yescharacteristic
124C22×C4—21no—
125C22×C4—3141no—
126SD16—6282no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—3141no—
130C4⋊C4—3141no—
131C4⋊C4—3141no—
132C4⋊C4—3141no—
133C4⋊C4—3141no—
134C4⋊C4—1571no—
135C4⋊C4D62811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3142no—
138C22⋊C4—3142no—
139C22⋊C4—3141no—
140C42—3142no—
141C42C22×D15711yescharacteristic
142C23—21no—
143C23—3141no—
144Q8—6282no—
145Q8—6281no—
146Q8—6281no—
147D4—21no—
148D4—21no—
149D4—3141no—
150D4—3141no—
151D4—12562no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—3141no—
156C2×C4—3142no—
157C2×C4—6281no—
158C2×C4—6281no—
159C2×C4—6281no—
160C2×C4D4×D15711yescharacteristic
161C2×C4D4×D15711yescharacteristic
162C2×C4C2×D62811yescharacteristic
163C8—41no—
164C8—41no—
165C8—6281no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—6281no—
170C22—6281no—
171C22—6281no—
172C22#1909774111yescharacteristic
173C4—21no—
174C4—41no—
175C4—6281no—
176C4—6282no—
177C4—12561no—
178C4#1909779011yescharacteristic
179C4#1909779311yescharacteristic
180C4#1909779911yescharacteristic
181C4#1909780011yescharacteristic
182C2—81no—
183C2—12561no—
184C2#2375742011yescharacteristic
185C2#2375743611yescharacteristic
186C2#2375745211yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radical#23757988C2
hypercenterD4⋊C8D157
hypocenterC157#1646
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. #23757554C2semidirect
4. #23757478C2semidirect
5. #23757021C2semidirect
6. #23757002C2semidirect
7. D4⋊C8×C157C2semidirect
93. C157#1646semidirect
101. D4⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237579882
#237576242
#237575542
#237574782
#237570212
#237570022
D4⋊C8×C1572
#1646157