Finite Group Databasebeta

← #34100574

Subgroups — #34100574

Order 20096.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#19097745C211yesmaximal, characteristic
2C2×#19097693C212yesmaximal, complemented
3#23757133C211yesmaximal, characteristic, complemented
4#23757036C212yesmaximal, complemented
5C22×M16×C157C211yesmaximal, characteristic, complemented
6C22×D628C2211yescharacteristic
7C22×C22⋊D157—21no—
8#19097745—22no—
9#19097745C414yescomplemented
10C2×C4⋊Dic157C2212yes—
11C2×C4⋊Dic157C2211yescharacteristic
12C2×C22⋊Dic157—21no—
13C2×#14578931—21no—
14#19097693—24no—
15#19097678—21no—
16C23×C4×C157C2211yescharacteristic
17C2×M16×C157—22no—
18C2×M16×C157—24no—
19C22×C8×C157C2212yes—
20C23×D157—21no—
21C2×D628—22no—
22C2×D628—21no—
23C2×D628—22no—
24C2×D628C2×C412yes—
25C2×C22⋊D157—24no—
26C2×C22⋊D157—44no—
27C22×Dic157—21no—
28C22×Dic157—22no—
29C4⋊Dic157—21no—
30C4⋊Dic157—22no—
31C4⋊Dic157—22no—
32C4⋊Dic157C2×C412yes—
33C22⋊Dic157—42no—
34#14578931—24no—
35C24×C157D411yescharacteristic
36C22×C4×C157—22no—
37C22×C4×C157—22no—
38C22×C4×C157C2×C412yes—
39C22×C4×C157D412yes—
40C22×C4×C157D411yescharacteristic
41C22×C4×C157C2311yescharacteristic
42M16×C157—28no—
43C2×C8×C157—22no—
44C2×C8×C157—22no—
45C2×C8×C157D414yescomplemented
46C22×D157—22no—
47C22×D157—42no—
48C22×D157—42no—
49C22×D157—41no—
50C22×D157—41no—
51D628—24no—
52D628—41no—
53D628—41no—
54C22⋊D157—48no—
55C2×Dic157—22no—
56C2×Dic157—42no—
57C2×Dic157—42no—
58C2×Dic157—42no—
59C2×Dic157—42no—
60C23×C157—22no—
61C23×C157—22no—
62C23×C157—21no—
63C23×C157—21no—
64C23×C157C22⋊C412yes—
65C23×C157C2×D411yescharacteristic
66C2×C4×C157—22no—
67C2×C4×C157—24no—
68C2×C4×C157—22no—
69C2×C4×C157—22no—
70C2×C4×C157C22⋊C412yes—
71C2×C4×C157C22×C411yescharacteristic
72C2×C4×C157C2×D412yes—
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157#4112yes—
75C8×C157—24no—
76C22×C157—22no—
77C22×C157—22no—
78C22×C157—22no—
79C22×C157—42no—
80C22×C157—42no—
81C22×C157C2×C22⋊C411yescharacteristic
82C22×C157#12112yes—
83C22×C157AGL(1,ℤ/8)11yescharacteristic
84C22×C157AGL(1,ℤ/8)12yes—
85C22×C157#13711yescharacteristic
86C2×D157—42no—
87C2×D157—42no—
88C2×D157—42no—
89C2×D157—82no—
90C2×D157—82no—
91C4×C157—22no—
92C4×C157C4×D412yes—
93C4×C157#12011yescharacteristic
94C4×C157#12311yescharacteristic
95Dic157—42no—
96Dic157—82no—
97C2×C157—42no—
98C2×C157#38611yescharacteristic
99C2×C157#41711yescharacteristic
100C2×C157#41811yescharacteristic
101C2×C157#46812yes—
102C2×C157#46912yes—
103D157—82no—
104C157#189511yescharacteristic, complemented, Sylow
105#1895—1571nomaximal, Sylow
106C22×M16D15711yescharacteristic, complemented
107C2×#121—1571no—
108C2×#102—1572no—
109#380—1571no—
110#340—1572no—
111C22×D4—3141no—
112C22×D4—1571no—
113C23×C4C2×D15711yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D15712yes—
117#121—3142no—
118#121—1574no—
119C2×C4⋊C4—1572no—
120C2×C4⋊C4—1571no—
121C2×C22⋊C4—3141no—
122C2×C22⋊C4—3141no—
123#102—3144no—
124#95—3141no—
125C24—3141no—
126C24D62811yescharacteristic
127C2×D4—3144no—
128C2×D4—3142no—
129C2×D4—3141no—
130C2×D4—3142no—
131C2×D4—6284no—
132C2×D4—1572no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—3141no—
136C22×C4—3142no—
137C22×C4C4×D15712yes—
138C22×C4C22⋊D15712yes—
139C22×C4D62811yescharacteristic
140C22×C4C22×D15711yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D15714yescomplemented
145C4⋊C4—3141no—
146C4⋊C4—3142no—
147C4⋊C4—3142no—
148C4⋊C4—1572no—
149C22⋊C4—3144no—
150C22⋊C4—6282no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—3142no—
156C23—6282no—
157C23—6282no—
158C23—6281no—
159C23—6281no—
160C23#1457893112yes—
161C23C2×D62811yescharacteristic
162D4—3144no—
163D4—6288no—
164D4—6281no—
165D4—6281no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—3142no—
171C2×C4—6282no—
172C2×C4—6282no—
173C2×C4—6282no—
174C2×C4—6282no—
175C2×C4#1457893112yes—
176C2×C4C2×C4×D15711yescharacteristic
177C2×C4C2×C22⋊D15712yes—
178C2×C4C2×C22⋊D15711yescharacteristic
179C2×C4#1457895412yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—6282no—
187C22—6282no—
188C22—6282no—
189C22—12562no—
190C22—12562no—
191C22C2×#1457893111yescharacteristic
192C22#1909774512yes—
193C22#1909779912yes—
194C22#1909779911yescharacteristic
195C22#1909780411yescharacteristic
196C4—22no—
197C4—6282no—
198C4—12562no—
199C4C4×C22⋊D15712yes—
200C4#1909774011yescharacteristic
201C4#1909775411yescharacteristic
202C2—42no—
203C2—12562no—
204C2#2375715211yescharacteristic
205C2#2375727511yescharacteristic
206C2#2375727911yescharacteristic
207C2#2375751912yes—
208C2#2375752612yes—
subgroupG/H
centerC23C2×D628
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC22×M16×C157C2
socleC23×C157C2×D4
prime radicalC2×#19097745C2
cosocleC22×C4×C157C23
hypercenterC22×M16D157
hypocenterC157#1895
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#19097693C2semidirect
3. #23757133C2semidirect
4. #23757036C2semidirect
5. C22×M16×C157C2semidirect
9. #19097745C4semidirect
45. C2×C8×C157D4semidirect
104. C157#1895semidirect
106. C22×M16D157semidirect
144. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977452
C2×#190976932
#237571332
#237570362
C22×M16×C1572
#1895157