Finite Group Databasebeta

← #34099107

Subgroups — #34099107

Order 20096.

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

normal subgroups only

This group has 282 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#23757854C211yesmaximal, characteristic
2#23757414C212yesmaximal, complemented
3#23757193C211yesmaximal, characteristic, complemented
4#23757008C212yesmaximal, complemented
5#324×C157C211yesmaximal, characteristic, complemented
6C2×#14578954—21no—
7C22×Dic314C2211yescharacteristic
8C2×#14578943—22no—
9#19097751—22no—
10#19097746—22no—
11#19097745—21no—
12#19097745C2212yes—
13C2×#14578933—21no—
14#19097696—22no—
15#19097678—21no—
16#19097678C2211yescharacteristic
17C2×C4⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19#14578954—42no—
20#14578954—42no—
21C2×Dic314—21no—
22C2×Dic314—22no—
23C2×Dic314—42no—
24C2×D628—21no—
25C2×D628—21no—
26C2×D628D412yes—
27C2×C22⋊D157—22no—
28C22×Dic157—21no—
29C22×Dic157—21no—
30C2×C4×D157—22no—
31#14578943—44no—
32C4⋊Dic157—21no—
33C4⋊Dic157D412yes—
34#14578933—42no—
35#14578931—22no—
36#14578931—22no—
37C22×C4×C157D411yescharacteristic
38C22×C4×C157D411yescharacteristic
39C22×C4×C157C2311yescharacteristic
40C2×C8×C157—22no—
41C4⋊C4×C157—22no—
42C22×D157—22no—
43Dic314—24no—
44Dic314—42no—
45Dic314—42no—
46D628—42no—
47D628—42no—
48D628—81no—
49C22⋊D157—42no—
50C22⋊D157—42no—
51C2×Dic157—22no—
52C2×Dic157—22no—
53C2×Dic157—22no—
54C2×Dic157—41no—
55C2×Dic157—41no—
56C2×Dic157—41no—
57C2×Dic157—41no—
58C4×D157—44no—
59C23×C157C2×D411yescharacteristic
60C2×C4×C157—21no—
61C2×C4×C157—21no—
62C2×C4×C157—21no—
63C2×C4×C157—41no—
64C2×C4×C157—41no—
65C2×C4×C157SD1612yescomplemented
66C2×C4×C157C2×D412yes—
67C8×C157—42no—
68C22×C157—21no—
69C22×C157—21no—
70C22×C157#12011yescharacteristic
71C22×C157C2×SD1611yescharacteristic
72C22×C157AGL(1,ℤ/8)11yescharacteristic
73C2×D157—42no—
74C2×D157—42no—
75C2×D157—42no—
76C4×C157—22no—
77C4×C157—22no—
78C4×C157—41no—
79Dic157—42no—
80Dic157—42no—
81Dic157—42no—
82C2×C157—21no—
83C2×C157—21no—
84C2×C157#45011yescharacteristic
85C2×C157#45611yescharacteristic
86C2×C157B(4,𝔽2)11yescharacteristic
87D157—82no—
88C157#160411yescharacteristic, complemented, Sylow
89#1604—1571nomaximal, Sylow
90#537—1571no—
91#448—1572no—
92#395—1571no—
93#327—1572no—
94#324D15711yescharacteristic, complemented
95C2×#41—3141no—
96C22×Q8—1571no—
97C2×SD16—3142no—
98C22⋊Q8—3142no—
99#121—3142no—
100#121—3141no—
101#121—1572no—
102C2×C4⋊C4—3141no—
103C2×C4⋊C4C2×D15711yescharacteristic
104#103—3142no—
105C22⋊C8C2×D15712yes—
106#95—3141no—
107#95—1571no—
108#41—6282no—
109#41—6282no—
110C2×Q8—3141no—
111C2×Q8—3142no—
112C2×Q8—6282no—
113C2×D4—3141no—
114C2×D4—3141no—
115C2×D4—3142no—
116C2×D4—1572no—
117C22×C4—3142no—
118C22×C4—3141no—
119C22×C4—3141no—
120C22×C4D62811yescharacteristic
121C22×C4D62811yescharacteristic
122C22×C4C22×D15711yescharacteristic
123SD16—6284no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—3141no—
127C4⋊C4—6282no—
128C4⋊C4—1572no—
129C22⋊C4—3142no—
130C22⋊C4—3142no—
131C23—3142no—
132C23C2×D62811yescharacteristic
133Q8—3144no—
134Q8—6282no—
135Q8—6282no—
136D4—6282no—
137D4—6282no—
138D4—6282no—
139D4—6282no—
140D4—12561no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3142no—
147C2×C4—3142no—
148C2×C4—3142no—
149C2×C4—6284no—
150C2×C4—6281no—
151C2×C4—6281no—
152C2×C4—6281no—
153C2×C4—6281no—
154C2×C4#1457894312yescomplemented
155C2×C4D4×D15712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—6282no—
160C22—6282no—
161C22—6282no—
162C22#1909774111yescharacteristic
163C22C2×#1457894311yescharacteristic
164C22#1909779911yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—6282no—
169C4—6282no—
170C4—6282no—
171C2—21no—
172C2—21no—
173C2—12562no—
174C2#2375742611yescharacteristic
175C2#2375745811yescharacteristic
176C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#324×C157C2
socleC22×C157#120
prime radical#23757854C2
hypercenter#324D157
hypocenterC157#1604
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #23757414C2semidirect
3. #23757193C2semidirect
4. #23757008C2semidirect
5. #324×C157C2semidirect
65. C2×C4×C157SD16semidirect
88. C157#1604semidirect
94. #324D157semidirect
154. C2×C4#14578943semidirect

Maximal subgroups

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

subgroupindex
#237578542
#237574142
#237571932
#237570082
#324×C1572
#1604157