Finite Group Databasebeta

← #34099095

Subgroups — #34099095

Order 20096.

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

normal subgroups only

This group has 256 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#23757899C211yesmaximal, characteristic
2#23757431C212yesmaximal, complemented
3#23757185C211yesmaximal, characteristic
4#23757011C212yesmaximal, complemented
5#323×C157C211yesmaximal, characteristic
6C22×Dic314C2211yescharacteristic
7C2×Dic628—22no—
8#19097750—21no—
9#19097750C2212yes—
10#19097748—22no—
11#19097748—22no—
12C2×C22⋊Dic157—21no—
13C2×C22⋊Dic157—21no—
14#19097696—22no—
15#19097678—21no—
16#19097678C2211yescharacteristic
17C2×C22⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19C2×Dic314—21no—
20C2×Dic314—22no—
21C2×Dic314—42no—
22C2×Dic314D412yes—
23C22×Dic157—21no—
24C22×Dic157—21no—
25C22×Dic157—21no—
26C22×Dic157—21no—
27Dic628—44no—
28C4⋊Dic157—21no—
29C4⋊Dic157D412yes—
30#14578933—22no—
31#14578933—22no—
32C22⋊Dic157—22no—
33C22⋊Dic157—22no—
34C22⋊Dic157—42no—
35C24×C157D411yescharacteristic
36C22×C4×C157D411yescharacteristic
37C22×C4×C157C2311yescharacteristic
38C2×C8×C157—22no—
39C22⋊C4×C157—22no—
40Dic314—24no—
41Dic314—42no—
42Dic314—42no—
43Dic314—81no—
44C2×Dic157—22no—
45C2×Dic157—22no—
46C2×Dic157—22no—
47C2×Dic157—41no—
48C2×Dic157—41no—
49C2×Dic157—41no—
50C2×Dic157—41no—
51C2×Dic157—41no—
52C2×Dic157—81no—
53C2×Dic157—81no—
54C23×C157—21no—
55C23×C157—21no—
56C23×C157—41no—
57C23×C157—41no—
58C23×C157Q1612yescomplemented
59C23×C157C2×D411yescharacteristic
60C2×C4×C157—21no—
61C2×C4×C157—41no—
62C2×C4×C157—41no—
63C2×C4×C157C2×D412yes—
64C8×C157—42no—
65C22×C157—21no—
66C22×C157—21no—
67C22×C157—22no—
68C22×C157—42no—
69C22×C157—41no—
70C22×C157—41no—
71C22×C157—81no—
72C22×C157#12011yescharacteristic
73C22×C157C2×Q1611yescharacteristic
74C22×C157#13711yescharacteristic
75C4×C157—22no—
76C4×C157—41no—
77Dic157—42no—
78Dic157—42no—
79Dic157—42no—
80Dic157—81no—
81C2×C157—21no—
82C2×C157—21no—
83C2×C157—42no—
84C2×C157#45111yescharacteristic
85C2×C157#45511yescharacteristic
86C2×C157B(4,𝔽2)11yescharacteristic
87C157#160211yescharacteristic, complemented, Sylow
88#1602—1571nomaximal, Sylow
89#543—1571no—
90#451—1572no—
91#393—1571no—
92#328—1572no—
93#323D15711yescharacteristic
94C22×Q8—1571no—
95C2×Q16—3142no—
96C22⋊Q8—3142no—
97C22⋊Q8—3142no—
98C22⋊Q8—3141no—
99C22⋊Q8—1572no—
100C2×C22⋊C4—3141no—
101C2×C22⋊C4—3141no—
102C2×C22⋊C4C2×D15711yescharacteristic
103#103—3142no—
104C22⋊C8C2×D15712yes—
105#95—3141no—
106#95—1571no—
107C24D62811yescharacteristic
108C2×Q8—3141no—
109C2×Q8—3142no—
110C2×Q8—6282no—
111C2×Q8—1572no—
112C22×C4—3141no—
113C22×C4—3141no—
114C22×C4—3141no—
115C22×C4—3141no—
116C22×C4D62811yescharacteristic
117C22×C4C22×D15711yescharacteristic
118Q16—6284no—
119C2×C8—22no—
120C4⋊C4—3142no—
121C4⋊C4—3141no—
122C4⋊C4—3142no—
123C4⋊C4—1572no—
124C22⋊C4—22no—
125C22⋊C4—3142no—
126C22⋊C4—3142no—
127C22⋊C4—6282no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic62812yescomplemented
133C23C2×D62811yescharacteristic
134Q8—3144no—
135Q8—6282no—
136Q8—6282no—
137Q8—12561no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—3142no—
142C2×C4—3142no—
143C2×C4—3142no—
144C2×C4—6281no—
145C2×C4—6281no—
146C2×C4—6281no—
147C2×C4—6281no—
148C2×C4—6281no—
149C2×C4—12561no—
150C2×C4—12561no—
151C2×C4D4×D15712yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1909774111yescharacteristic
161C22C2×Dic62811yescharacteristic
162C22#1909780411yescharacteristic
163C4—22no—
164C4—41no—
165C4—6282no—
166C4—6282no—
167C4—6282no—
168C4—12561no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2375743111yescharacteristic
173C2#2375745311yescharacteristic
174C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#323×C157C2
socleC22×C157#120
prime radicalC24×C157D4
hypercenter#323D157
hypocenterC157#1602
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. #23757431C2semidirect
4. #23757011C2semidirect
58. C23×C157Q16semidirect
87. C157#1602semidirect
132. C23Dic628semidirect

Maximal subgroups

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

subgroupindex
#237578992
#237574312
#237571852
#237570112
#323×C1572
#1602157