Finite Group Databasebeta

← #34099229

Subgroups — #34099229

Order 20096.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#23757998C211yesmaximal, characteristic, complemented
2#23757470C211yesmaximal, characteristic
3#23757470C211yesmaximal, characteristic
4#23757019C211yesmaximal, characteristic, complemented
5#23757005C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7#486×C157C211yesmaximal, characteristic
8C2×#14578942—21no—
9C2×#14578942—21no—
10#19097769—21no—
11#19097762—21no—
12#19097761—22no—
13Q8×Dic157—21no—
14C4×Dic314C2211yescharacteristic
15C4×Dic314C2211yescharacteristic
16#19097736—21no—
17C4⋊#11929434C2211yescharacteristic
18C4⋊#11929434C2211yescharacteristic
19#19097692—21no—
20#19097692—21no—
21C4⋊Q8×C157C2211yescharacteristic
22#127×C157C2211yescharacteristic
23#102×C157—22no—
24C4×C8×C157C2211yescharacteristic
25C2×Dic314D411yescharacteristic
26C2×Dic314D411yescharacteristic
27#14578942—42no—
28#14578942—42no—
29C2×#11929434—21no—
30C2×#11929434—21no—
31C4⋊Dic157—22no—
32C4⋊Dic157D411yescharacteristic
33C4⋊Dic157D411yescharacteristic
34#14578933—22no—
35#14578933—22no—
36#14578933—21no—
37#14578933—21no—
38C4×Dic157—22no—
39C4×Dic157—21no—
40C4×Dic157—21no—
41C2×Q8×C157—21no—
42C2×D4×C157—21no—
43C2×D4×C157—21no—
44C2×D4×C157—21no—
45C2×C8×C157—21no—
46C4⋊C4×C157—21no—
47C4⋊C4×C157D412yes—
48C42×C157C2311yescharacteristic
49Dic314—21no—
50Dic314—21no—
51Dic314—21no—
52Dic314SD1612yescomplemented
53C2×Dic157—21no—
54C2×Dic157—21no—
55C2×Dic157—21no—
56C2×Dic157—21no—
57#11929434—41no—
58#11929434—41no—
59C23×C157—41no—
60Q8×C157—41no—
61Q8×C157—41no—
62D4×C157—42no—
63D4×C157—41no—
64D4×C157—41no—
65D4×C157—41no—
66D4×C157—41no—
67C2×C4×C157—22no—
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C8×C157—41no—
72C22×C157—81no—
73C22×C157—81no—
74C22×C157—81no—
75C22×C157#12011yescharacteristic
76C4×C157—21no—
77C4×C157—42no—
78C4×C157C2×SD1611yescharacteristic
79C4×C157#13511yescharacteristic
80C4×C157AGL(1,ℤ/8)11yescharacteristic
81C4×C157#13711yescharacteristic
82Dic157—22no—
83Dic157—41no—
84Dic157—41no—
85Dic157—41no—
86C2×C157—161no—
87C2×C157#44811yescharacteristic
88C2×C157#44911yescharacteristic
89C2×C157#45411yescharacteristic
90C157#162611yescharacteristic, complemented, Sylow
91#1626—1571nomaximal, Sylow
92#557—1571no—
93#486D15711yescharacteristic
94#460—1571no—
95#460—1571no—
96#332—1571no—
97Q8⋊C8—1571no—
98Q8⋊C8—1571no—
99C2×SD16—3141no—
100C2×SD16—3141no—
101C4⋊Q8—3141no—
102C4⋊Q8C2×D15711yescharacteristic
103#127C2×D15711yescharacteristic
104#125—3141no—
105#125—3142no—
106C4×Q8—3141no—
107C4×Q8—3141no—
108C4×Q8—1571no—
109C4×Q8—1571no—
110C4⋊C8—1571no—
111C4⋊C8—1571no—
112#102—22no—
113#102—3141no—
114#102—3141no—
115C4×C8C2×D15711yescharacteristic
116C2×Q8—21no—
117C2×Q8—1571no—
118C2×Q8—1571no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122SD16—6282no—
123SD16—6282no—
124C2×C8—21no—
125C2×C8—3141no—
126C2×C8—3141no—
127C4⋊C4—21no—
128C4⋊C4—3142no—
129C4⋊C4—3142no—
130C4⋊C4—3142no—
131C4⋊C4—3141no—
132C4⋊C4—3141no—
133C4⋊C4—1571no—
134C4⋊C4—1571no—
135C4⋊C4C22⋊D15712yes—
136C42—3142no—
137C42—3141no—
138C42—3141no—
139C42C22×D15711yescharacteristic
140C23—41no—
141Q8—41no—
142Q8—41no—
143Q8—3141no—
144Q8—3141no—
145Q8—3141no—
146Q8—1572no—
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152C2×C4—22no—
153C2×C4—3141no—
154C2×C4—3141no—
155C2×C4—3141no—
156C2×C4—3141no—
157C2×C4D4×D15711yescharacteristic
158C2×C4D4×D15711yescharacteristic
159C2×C4C2×C22⋊D15711yescharacteristic
160C8—41no—
161C8—6281no—
162C8—6281no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1909773911yescharacteristic
167C4—21no—
168C4—42no—
169C4—3142no—
170C4—6281no—
171C4—6281no—
172C4—6281no—
173C4SD16×D15711yescharacteristic
174C4#1909779011yescharacteristic
175C4#1909779511yescharacteristic
176C4#1909780011yescharacteristic
177C2—161no—
178C2#2375741011yescharacteristic
179C2#2375741511yescharacteristic
180C2#2375744511yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#486×C157C2
socleC22×C157#120
prime radical#127×C157C22
hypercenter#486D157
hypocenterC157#1626
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #23757998C2semidirect
4. #23757019C2semidirect
5. #23757005C2semidirect
6. #23757005C2semidirect
52. Dic314SD16semidirect
90. C157#1626semidirect

Maximal subgroups

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

subgroupindex
#237579982
#237574702
#237570192
#237570052
#486×C1572
#1626157