Finite Group Databasebeta

← #34099090

Subgroups — #34099090

Order 20096.

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

normal subgroups only

This group has 308 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#23757835C211yesmaximal, characteristic
2#23757414C212yesmaximal, complemented
3#23757185C211yesmaximal, characteristic, complemented
4#23757008C212yesmaximal, complemented
5#323×C157C211yesmaximal, characteristic, complemented
6C22×Dic314C2211yescharacteristic, complemented
7C22×C22⋊D157—21no—
8C2×#14578943—22no—
9#19097755—22no—
10#19097745—21no—
11#19097745C2212yes—
12#19097741—22no—
13C2×C22⋊Dic157—21no—
14#19097696—22no—
15#19097678—21no—
16#19097678C2211yescharacteristic, complemented
17C2×C22⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yescomplemented
19C23×D157—21no—
20C2×Dic314—22no—
21C2×Dic314—42no—
22C2×D628—21no—
23C2×D628D412yes—
24C2×C22⋊D157—22no—
25C2×C22⋊D157—22no—
26C2×C22⋊D157—42no—
27C2×C22⋊D157—42no—
28C22×Dic157—21no—
29C22×Dic157—21no—
30C22×Dic157—21no—
31#14578943—44no—
32C4⋊Dic157—21no—
33C4⋊Dic157D412yes—
34C22⋊Dic157—42no—
35#14578931—22no—
36#14578931—22no—
37C24×C157D411yescharacteristic
38C22×C4×C157D411yescharacteristic
39C22×C4×C157C2311yescharacteristic
40C2×C8×C157—22no—
41C22⋊C4×C157—22no—
42C22×D157—22no—
43C22×D157—42no—
44C22×D157—81no—
45C22×D157—81no—
46Dic314—24no—
47Dic314—42no—
48D628—42no—
49D628—81no—
50C22⋊D157—44no—
51C22⋊D157—42no—
52C22⋊D157—42no—
53C2×Dic157—22no—
54C2×Dic157—22no—
55C2×Dic157—42no—
56C2×Dic157—41no—
57C2×Dic157—41no—
58C2×Dic157—41no—
59C2×Dic157—41no—
60C2×Dic157—41no—
61C23×C157—21no—
62C23×C157—21no—
63C23×C157—41no—
64C23×C157—41no—
65C23×C157SD1612yescomplemented
66C23×C157C2×D411yescharacteristic
67C2×C4×C157—21no—
68C2×C4×C157—41no—
69C2×C4×C157—41no—
70C2×C4×C157C2×D412yes—
71C8×C157—42no—
72C22×C157—21no—
73C22×C157—21no—
74C22×C157—22no—
75C22×C157—42no—
76C22×C157—41no—
77C22×C157—41no—
78C22×C157—81no—
79C22×C157#12011yescharacteristic
80C22×C157C2×SD1611yescharacteristic
81C22×C157AGL(1,ℤ/8)11yescharacteristic
82C2×D157—42no—
83C2×D157—42no—
84C2×D157—42no—
85C2×D157—84no—
86C4×C157—22no—
87C4×C157—41no—
88Dic157—42no—
89Dic157—42no—
90Dic157—81no—
91C2×C157—21no—
92C2×C157—21no—
93C2×C157—42no—
94C2×C157#45011yescharacteristic
95C2×C157#45511yescharacteristic
96C2×C157B(4,𝔽2)11yescharacteristic
97D157—82no—
98C157#160111yescharacteristic, complemented, Sylow
99#1601—1571nomaximal, Sylow
100#534—1571no—
101#448—1572no—
102#393—1571no—
103#327—1572no—
104#323D15711yescharacteristic, complemented
105C22×Q8—1571no—
106C22×D4—3141no—
107C2×SD16—3142no—
108#123—3142no—
109#121—3141no—
110#121—1572no—
111#120—3142no—
112C2×C22⋊C4—3141no—
113C2×C22⋊C4C2×D15711yescharacteristic
114#103—3142no—
115C22⋊C8C2×D15712yescomplemented
116#95—3141no—
117#95—1571no—
118C24—3141no—
119C24D62811yescharacteristic
120C2×Q8—3142no—
121C2×Q8—6282no—
122C2×D4—3141no—
123C2×D4—3142no—
124C2×D4—3142no—
125C2×D4—6282no—
126C2×D4—6282no—
127C2×D4—1572no—
128C22×C4—3141no—
129C22×C4—3141no—
130C22×C4—3141no—
131C22×C4D62811yescharacteristic
132C22×C4C22×D15711yescharacteristic
133SD16—6284no—
134C2×C8—22no—
135C4⋊C4—3141no—
136C4⋊C4—1572no—
137C22⋊C4—22no—
138C22⋊C4—3142no—
139C22⋊C4—3142no—
140C22⋊C4—6282no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—3142no—
146C23—6282no—
147C23—12561no—
148C23—12561no—
149C23#1457894312yescomplemented
150C23C2×D62811yescharacteristic
151Q8—3144no—
152Q8—6282no—
153D4—6282no—
154D4—6284no—
155D4—6282no—
156D4—6282no—
157D4—12561no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3142no—
162C2×C4—3142no—
163C2×C4—6282no—
164C2×C4—6281no—
165C2×C4—6281no—
166C2×C4—6281no—
167C2×C4—6281no—
168C2×C4—6281no—
169C2×C4D4×D15712yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—6282no—
179C22—6282no—
180C22—6282no—
181C22—12564no—
182C22#1909774111yescharacteristic
183C22C2×#1457894311yescharacteristic
184C22#1909779911yescharacteristic
185C4—22no—
186C4—41no—
187C4—6282no—
188C4—6282no—
189C4—12561no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—12562no—
194C2#2375742611yescharacteristic
195C2#2375745311yescharacteristic
196C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#323×C157C2
socleC22×C157#120
prime radical#23757835C2
hypercenter#323D157
hypocenterC157#1601
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #23757414C2semidirect
3. #23757185C2semidirect
4. #23757008C2semidirect
5. #323×C157C2semidirect
6. C22×Dic314C22semidirect
16. #19097678C22semidirect
18. C22⋊C8×C157C22semidirect
65. C23×C157SD16semidirect
98. C157#1601semidirect
104. #323D157semidirect
115. C22⋊C8C2×D157semidirect
149. C23#14578943semidirect

Maximal subgroups

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

subgroupindex
#237578352
#237574142
#237571852
#237570082
#323×C1572
#1601157