Finite Group Databasebeta

← #34098734

Subgroups — #34098734

Order 20096.

264 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 81 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#23757330C211yesmaximal, characteristic, complemented
2C4×#14578938C211yesmaximal, characteristic, complemented
3#23757013C214yesmaximal, complemented
4C2×#124×C157C211yesmaximal, characteristic
5C2×#14578938—21no—
6C4⋊#11929434—21no—
7C22⋊#11929434—21no—
8#19097682C2212yescomplemented
9#19097682C2212yescomplemented
10C4×#11929434—21no—
11C22×Q8×C157C411yescharacteristic
12C22×D4×C157C411yescharacteristic
13#124×C157—21no—
14#124×C157—21no—
15#124×C157C412yes—
16#124×C157C2212yes—
17C2×C22⋊C4×C157—21no—
18C2×C22⋊C4×C157—21no—
19C2×C42×C157C2211yescharacteristic
20#14578938—24no—
21C2×#11929434—22no—
22C2×#11929434—22no—
23C24×C157—21no—
24C2×Q8×C157—21no—
25C2×Q8×C157—42no—
26C2×Q8×C157C2×C412yes—
27C2×D4×C157—21no—
28C2×D4×C157—42no—
29C2×D4×C157C2×C412yes—
30C22×C4×C157—21no—
31C22×C4×C157C2×C411yescharacteristic
32C22×C4×C157D411yescharacteristic
33C22×C4×C157D411yescharacteristic
34C22⋊C4×C157—22no—
35C22⋊C4×C157—22no—
36C22⋊C4×C157—22no—
37C22⋊C4×C157—22no—
38C42×C157C2×C411yescharacteristic
39C42×C157D411yescharacteristic
40C42×C157D411yescharacteristic
41C42×C157C2311yescharacteristic
42#11929434—24no—
43#11929434—42no—
44C23×C157—22no—
45C23×C157—42no—
46C23×C157—42no—
47C23×C157—42no—
48C23×C157C22⋊C411yescharacteristic
49Q8×C157—42no—
50Q8×C157—42no—
51D4×C157—42no—
52D4×C157—42no—
53C2×C4×C157—22no—
54C2×C4×C157—22no—
55C2×C4×C157—22no—
56C2×C4×C157—41no—
57C2×C4×C157—42no—
58C2×C4×C157C22⋊C411yescharacteristic
59C2×C4×C157C22⋊C411yescharacteristic
60C2×C4×C157C22⋊C411yescharacteristic
61C2×C4×C157C22×C411yescharacteristic
62C2×C4×C157C2×D411yescharacteristic
63C2×C4×C157C2×D411yescharacteristic
64C22×C157—21no—
65C22×C157—21no—
66C22×C157—42no—
67C22×C157—42no—
68C22×C157—42no—
69C22×C157—84no—
70C22×C157#10412yes—
71C22×C157C2×C22⋊C411yescharacteristic
72C4×C157—21no—
73C4×C157—21no—
74C4×C157—21no—
75C4×C157—21no—
76C4×C157—21no—
77C4×C157—42no—
78C4×C157#10012yes—
79C2×C157—22no—
80C2×C157—82no—
81C2×C157C2×#10011yescharacteristic
82C2×C157C2×#10411yescharacteristic
83C2×C157#42111yescharacteristic
84C157#152711yescharacteristic, complemented, Sylow
85#1527—1571nomaximal, Sylow
86C2×#124D15711yescharacteristic
87#432—1571no—
88C4×M16—1571no—
89#329—1574no—
90C22×Q8Dic15711yescharacteristic
91C22×D4Dic15711yescharacteristic
92C2×M16—3141no—
93#124—21no—
94#124—21no—
95#124Dic15712yes—
96#124C2×D15712yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42C2×D15711yescharacteristic
100C4⋊C8—3141no—
101C22⋊C8—3141no—
102#97—1572no—
103#97—1572no—
104C4×C8—3141no—
105C24—21no—
106C2×Q8—21no—
107C2×Q8—42no—
108C2×Q8C2×Dic15712yes—
109C2×D4—21no—
110C2×D4—42no—
111C2×D4C2×Dic15712yes—
112C22×C4—21no—
113C22×C4C2×Dic15711yescharacteristic
114C22×C4C22⋊D15711yescharacteristic
115C22×C4C22⋊D15711yescharacteristic
116M16—3144no—
117C2×C8—3142no—
118C2×C8—3142no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42C2×Dic15711yescharacteristic
124C42C22⋊D15711yescharacteristic
125C42C22⋊D15711yescharacteristic
126C42C22×D15711yescharacteristic
127C23—22no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23C22⋊Dic15711yescharacteristic
132Q8—42no—
133Q8—42no—
134D4—42no—
135D4—42no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4C22⋊Dic15711yescharacteristic
142C2×C4C22⋊Dic15711yescharacteristic
143C2×C4C22⋊Dic15711yescharacteristic
144C2×C4C22×Dic15711yescharacteristic
145C2×C4C2×C22⋊D15711yescharacteristic
146C2×C4C2×C22⋊D15711yescharacteristic
147C8—3144no—
148C8—6282no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—84no—
155C22#1909769812yes—
156C22C2×C22⋊Dic15711yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—21no—
162C4—42no—
163C4#1909768812yes—
164C2—22no—
165C2—82no—
166C2C2×#1909768811yescharacteristic
167C2C2×#1909769811yescharacteristic
168C2#2375729011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
FittingC2×#124×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC22×D4×C157C4
cosocleC42×C157C23
hypercenterC2×#124D157
hypocenterC157#1527
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #23757330C2semidirect
2. C4×#14578938C2semidirect
3. #23757013C2semidirect
8. #19097682C22semidirect
9. #19097682C22semidirect
84. C157#1527semidirect

Maximal subgroups

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

subgroupindex
#237573302
C4×#145789382
#237570132
C2×#124×C1572
#1527157