Finite Group Databasebeta

← #34102134

Subgroups — #34102134

Order 20096.

246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 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#23758009C211yesmaximal, characteristic, complemented
2#23757464C212yesmaximal, complemented
3#23757440C211yesmaximal, characteristic
4#23757062C212yesmaximal, complemented
5#353×C157C211yesmaximal, characteristic, complemented
6#19097816C2211yescharacteristic
7#19097797—21no—
8#19097758—22no—
9#19097755—22no—
10#19097752—21no—
11#19097752C2212yes—
12C4×Dic314—21no—
13D4×Dic157—22no—
14#19097699—21no—
15#19097688C2211yescharacteristic
16#19097685—22no—
17#127×C157C2211yescharacteristic
18#99×C157C2212yes—
19#14578954—21no—
20#14578953—41no—
21C2×Dic314—21no—
22C2×C22⋊D157—22no—
23D4×D157—41no—
24C22×Dic157—21no—
25C22×Dic157D412yes—
26#14578942—41no—
27#14578939—41no—
28#14578938—21no—
29C4⋊Dic157—21no—
30C4⋊Dic157—42no—
31#14578933—22no—
32C22⋊Dic157—22no—
33C22⋊Dic157—21no—
34C22⋊Dic157—22no—
35C22⋊Dic157D412yes—
36#14578931—42no—
37C4×Dic157—22no—
38C2×D4×C157—22no—
39C2×D4×C157D411yescharacteristic
40C2×D4×C157C2311yescharacteristic
41C22⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43C22×D157—22no—
44Dic314—21no—
45Dic314—42no—
46D628—21no—
47C22⋊D157—42no—
48C22⋊D157—41no—
49C2×Dic157—22no—
50C2×Dic157—22no—
51C2×Dic157—22no—
52C2×Dic157—42no—
53C2×Dic157—81no—
54C4×D157—41no—
55#11929434—41no—
56C23×C157—21no—
57C23×C157C2×D412yes—
58D4×C157—41no—
59D4×C157—41no—
60D4×C157—44no—
61C2×C4×C157—21no—
62C2×C4×C157—42no—
63C2×C4×C157C2×D411yescharacteristic
64C22×C157—22no—
65C22×C157—42no—
66C22×C157—42no—
67C22×C157—41no—
68C22×C157#12011yescharacteristic
69C2×D157—41no—
70C2×D157—82no—
71C4×C157—22no—
72C4×C157—21no—
73C4×C157—82no—
74Dic157—42no—
75Dic157—41no—
76Dic157—42no—
77C2×C157—21no—
78C2×C157—42no—
79C2×C157—81no—
80C2×C157B(4,𝔽2)11yescharacteristic
81D157—81no—
82C157#219811yescharacteristic, complemented, Sylow
83#2198—1571nomaximal, Sylow
84#559—1571no—
85#458—1572no—
86#453—1571no—
87#353D15711yescharacteristic, complemented
88#352—1572no—
89H(5,𝔽2)—1571no—
90AGL(1,ℤ/8)—3141no—
91#127C2×D15711yescharacteristic
92#124—3142no—
93#123—3142no—
94#123—3141no—
95#123—1572no—
96C4×Q8—3141no—
97C4×D4—3142no—
98#104—3141no—
99#100—1571no—
100#99—3142no—
101#99C2×D15712yes—
102#41—3141no—
103#41—6281no—
104C2×Q8—3141no—
105C2×D4—22no—
106C2×D4—3142no—
107C2×D4—6281no—
108C2×D4D62811yescharacteristic
109C2×D4C22×D15711yescharacteristic
110C22×C4—3141no—
111C22×C4—1572no—
112SD16—6281no—
113D8—6281no—
114M16—3141no—
115C4⋊C4—3141no—
116C4⋊C4—3142no—
117C4⋊C4—6282no—
118C22⋊C4—22no—
119C22⋊C4—3142no—
120C22⋊C4—3141no—
121C22⋊C4—3142no—
122C22⋊C4—6282no—
123C22⋊C4—1572no—
124C42—3142no—
125C42D62811yescharacteristic
126C23—21no—
127C23—3142no—
128C23D4×D15712yes—
129Q8—3141no—
130Q8—6282no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3141no—
135D4—6282no—
136D4—6281no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3142no—
140C2×C4—3142no—
141C2×C4—3142no—
142C2×C4—6281no—
143C2×C4—6282no—
144C2×C4—12561no—
145C2×C4C2×D62811yescharacteristic
146C8—6281no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—6281no—
152C22—12562no—
153C22#1909774111yescharacteristic
154C4—22no—
155C4—21no—
156C4—82no—
157C4—6282no—
158C4—6281no—
159C4—6282no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—12561no—
164C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×D4×C157C23
FrattiniC2×D4C22×D157
Fitting#353×C157C2
socleC2×C157B(4,𝔽2)
prime radical#23757440C2
hypercenter#353D157
hypocenterC157#2198
LCSG ▷ C2×D4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #23758009C2semidirect
2. #23757464C2semidirect
4. #23757062C2semidirect
5. #353×C157C2semidirect
82. C157#2198semidirect
87. #353D157semidirect

Maximal subgroups

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

subgroupindex
#237580092
#237574642
#237574402
#237570622
#353×C1572
#2198157