Finite Group Databasebeta

← #34099093

Subgroups — #34099093

Order 20096.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757428C212yesmaximal, complemented
2#23757185C211yesmaximal, characteristic
3#23757010C212yesmaximal, complemented
4#23756997C211yesmaximal, characteristic
5#543×C157C211yesmaximal, characteristic
6C22×Dic314C2211yescharacteristic
7C2×#14578944—22no—
8C2×C22⋊Dic157—21no—
9#19097694—22no—
10C22⋊#11929434C2212yes—
11#19097678—21no—
12#19097678C2211yescharacteristic
13C22⋊Q8×C157—22no—
14C22⋊Q8×C157—21no—
15C22⋊Q8×C157—22no—
16C22⋊Q8×C157C2212yes—
17C2×C22⋊C4×C157—21no—
18C2×C22⋊C4×C157C2211yescharacteristic
19C2×Dic314—22no—
20C2×Dic314—42no—
21C22×Dic157—21no—
22C22×Dic157—21no—
23#14578944—44no—
24C2×#11929434—22no—
25C22⋊Dic157—42no—
26C24×C157D411yescharacteristic
27C2×Q8×C157—21no—
28C2×Q8×C157D412yes—
29C22×C4×C157—21no—
30C22×C4×C157—21no—
31C22×C4×C157D411yescharacteristic
32C22×C4×C157C2311yescharacteristic
33C4⋊C4×C157—21no—
34C4⋊C4×C157—22no—
35C4⋊C4×C157—22no—
36C4⋊C4×C157D412yes—
37C22⋊C4×C157—22no—
38C22⋊C4×C157—22no—
39C22⋊C4×C157—22no—
40Dic314—24no—
41Dic314—42no—
42C2×Dic157—22no—
43C2×Dic157—41no—
44C2×Dic157—41no—
45C2×Dic157—41no—
46C2×Dic157—41no—
47C2×Dic157—41no—
48#11929434—42no—
49C23×C157—21no—
50C23×C157—21no—
51C23×C157—41no—
52C23×C157—41no—
53C23×C157Q1612yescomplemented
54C23×C157C2×D411yescharacteristic
55Q8×C157—42no—
56Q8×C157—81no—
57C2×C4×C157—22no—
58C2×C4×C157—21no—
59C2×C4×C157—22no—
60C2×C4×C157—41no—
61C2×C4×C157—41no—
62C2×C4×C157—81no—
63C2×C4×C157—81no—
64C2×C4×C157C2×D412yes—
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—
77C4×C157—42no—
78C4×C157—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#543D15711yescharacteristic
90#451—1572no—
91#393—1571no—
92#328—1572no—
93#323—1571no—
94C22×Q8—1571no—
95C2×Q16—3142no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C2×D15712yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—3141no—
102C2×C22⋊C4C2×D15711yescharacteristic
103#103—3142no—
104C22⋊C8—1572no—
105#95—3141no—
106#95—1571no—
107C24C22⋊D15711yescharacteristic
108C2×Q8—21no—
109C2×Q8—3142no—
110C2×Q8—6282no—
111C2×Q8C22⋊D15712yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—3141no—
115C22×C4—3141no—
116C22×C4C22⋊D15711yescharacteristic
117C22×C4C22×D15711yescharacteristic
118Q16—6284no—
119C2×C8—3142no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4C22⋊D15712yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—6282no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23#1457894412yescomplemented
133C23C2×C22⋊D15711yescharacteristic
134Q8—42no—
135Q8—81no—
136Q8—3144no—
137Q8—6282no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—3142no—
146C2×C4—6281no—
147C2×C4—6281no—
148C2×C4—6281no—
149C2×C4—6281no—
150C2×C4—6281no—
151C2×C4C2×C22⋊D15712yes—
152C8—6282no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1909774011yescharacteristic
161C22C2×#1457894411yescharacteristic
162C22#1909780211yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—6282no—
168C4—12561no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#2375742911yescharacteristic
173C2#2375745111yescharacteristic
174C2#2375746011yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#543×C157C2
socleC22×C157#120
prime radicalC24×C157D4
hypercenter#543D157
hypocenterC157#1602
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #23757428C2semidirect
3. #23757010C2semidirect
53. C23×C157Q16semidirect
87. C157#1602semidirect
132. C23#14578944semidirect

Maximal subgroups

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

subgroupindex
#237574282
#237571852
#237570102
#237569972
#543×C1572
#1602157