Finite Group Databasebeta

← #34102096

Subgroups — #34102096

Order 20096.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#23758074C211yesmaximal, characteristic, complemented
2#23757711C211yesmaximal, characteristic, complemented
3C2×#19097712C211yesmaximal, characteristic, complemented
4#23757544C211yesmaximal, characteristic, complemented
5#23757079C211yesmaximal, characteristic, complemented
6#23757058C211yesmaximal, characteristic, complemented
7#361×C157C211yesmaximal, characteristic, complemented
8#19097816—21no—
9#19097799—21no—
10#19097797—21no—
11#19097795—22no—
12#19097793C2211yescharacteristic
13#19097790—21no—
14C2×D1256C2211yescharacteristic, complemented
15D8×D157—21no—
16#19097777C2211yescharacteristic
17#19097716—21no—
18D2512—21no—
19#19097712—22no—
20C2×#14578929C2211yescharacteristic, complemented
21#19097687—21no—
22C2×D8×C157C2211yescharacteristic
23M32×C157C2211yescharacteristic, complemented
24#108×C157C2211yescharacteristic, complemented
25#14578954—21no—
26#14578954D411yescharacteristic
27#14578953—22no—
28#14578953—41no—
29C2×D628—21no—
30C2×C22⋊D157—42no—
31D4×D157—22no—
32D4×D157—41no—
33Dic628D411yescharacteristic
34#14578943—21no—
35#14578943—41no—
36#14578942—22no—
37D1256—21no—
38D1256—21no—
39D1256—41no—
40D1256D411yescharacteristic
41#14578939—22no—
42#14578938D411yescharacteristic
43#14578937—21no—
44C8×D157—21no—
45#14578929—21no—
46C2×D4×C157—21no—
47D8×C157—21no—
48D8×C157D412yes—
49M16×C157—21no—
50C2×C8×C157C2311yescharacteristic
51C16×C157—21no—
52C22×D157—42no—
53C22×D157—41no—
54Dic314—21no—
55Dic314D811yescharacteristic
56D628—21no—
57D628—41no—
58D628—41no—
59D628D811yescharacteristic
60C22⋊D157—21no—
61C22⋊D157—42no—
62C22⋊D157—42no—
63C22⋊D157—41no—
64C2×Dic157—42no—
65C4×D157—21no—
66C4×D157—41no—
67#11929434—21no—
68C23×C157—41no—
69D4×C157—22no—
70D4×C157—41no—
71C2×C4×C157C2×D411yescharacteristic
72C8×C157—41no—
73C8×C157C2×D411yescharacteristic
74C8×C157C2×D411yescharacteristic
75C22×C157—42no—
76C22×C157—161no—
77C22×C157AGL(1,ℤ/8)11yescharacteristic
78C2×D157—21no—
79C2×D157—41no—
80C2×D157—81no—
81C2×D157—162no—
82C2×D157—161no—
83C4×C157#12011yescharacteristic
84C4×C157C2×D811yescharacteristic
85Dic157—21no—
86Dic157—41no—
87C2×C157—21no—
88C2×C157—82no—
89C2×C157#44711yescharacteristic
90D157—41no—
91D157—81no—
92D157—161no—
93C157#219211yescharacteristic, complemented, Sylow
94#2192—1571nomaximal, Sylow
95#576—1571no—
96#509—1571no—
97C2×D16—1571no—
98#472—1571no—
99#361D15711yescharacteristic, complemented
100#359—1571no—
101#350—1571no—
102H(5,𝔽2)—3141no—
103AGL(1,ℤ/8)—3141no—
104AGL(1,ℤ/8)—3142no—
105AGL(1,ℤ/8)—3141no—
106#135—3141no—
107#135—1571no—
108C2×D8—3141no—
109C2×D8—1571no—
110C2×D8C2×D15711yescharacteristic
111#131—1571no—
112SD32—3141no—
113D16—3141no—
114D16—3142no—
115M32C2×D15711yescharacteristic, complemented
116C2×C16—1571no—
117#108C2×D15711yescharacteristic, complemented
118#100—3141no—
119#41—3142no—
120#41—3141no—
121#41—6281no—
122#41—1571no—
123C2×D4—21no—
124C2×D4—3142no—
125C2×D4—3141no—
126C2×D4—6281no—
127C2×D4—6282no—
128Q16—1571no—
129SD16—3142no—
130SD16—3141no—
131SD16—6281no—
132D8—21no—
133D8—3142no—
134D8—3141no—
135D8—3141no—
136D8—6281no—
137D8—1571no—
138D8D62812yes—
139M16—21no—
140M16—3141no—
141M16—1571no—
142C2×C8—3141no—
143C2×C8C22×D15711yescharacteristic
144C16—21no—
145C16—3141no—
146C23—41no—
147C23—6282no—
148C23—6281no—
149Q8—3141no—
150Q8—1571no—
151D4—22no—
152D4—41no—
153D4—3141no—
154D4—3141no—
155D4—6282no—
156D4—6282no—
157D4—6281no—
158D4—6281no—
159D4—6281no—
160D4—1571no—
161C2×C4—3141no—
162C2×C4—6282no—
163C2×C4—6281no—
164C2×C4D4×D15711yescharacteristic
165C8—41no—
166C8—3141no—
167C8D4×D15711yescharacteristic
168C8C2×D62811yescharacteristic
169C22—42no—
170C22—161no—
171C22—3141no—
172C22—6281no—
173C22—25121no—
174C22—25122no—
175C22—12561no—
176C22#1909779811yescharacteristic
177C4—3141no—
178C4—6281no—
179C4#1909774111yescharacteristic
180C4D8×D15711yescharacteristic
181C2—21no—
182C2—82no—
183C2—6281no—
184C2—25121no—
185C2—12561no—
186C2#2375740811yescharacteristic
subgroupG/H
centerC2#23757408
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#361×C157C2
socleC2×C157#447
hypercenter#361D157
hypocenterC157#2192
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #23758074C2semidirect
2. #23757711C2semidirect
3. C2×#19097712C2semidirect
4. #23757544C2semidirect
5. #23757079C2semidirect
6. #23757058C2semidirect
7. #361×C157C2semidirect
14. C2×D1256C22semidirect
20. C2×#14578929C22semidirect
23. M32×C157C22semidirect
24. #108×C157C22semidirect
93. C157#2192semidirect
99. #361D157semidirect
115. M32C2×D157semidirect
117. #108C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580742
#237577112
C2×#190977122
#237575442
#237570792
#237570582
#361×C1572
#2192157