Finite Group Databasebeta

← #34102090

Subgroups — #34102090

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#23758075C211yesmaximal, characteristic, complemented
2C2×#19097712C211yesmaximal, characteristic, complemented
3#23757540C211yesmaximal, characteristic, complemented
4#23757087C211yesmaximal, characteristic, complemented
5#23757080C211yesmaximal, characteristic, complemented
6#23757058C211yesmaximal, characteristic, complemented
7#509×C157C211yesmaximal, characteristic, complemented
8#19097817—21no—
9#19097798—22no—
10#19097797—21no—
11#19097796—21no—
12#19097794—21no—
13C2×D1256C2211yescharacteristic
14D8×D157—21no—
15#19097777C2211yescharacteristic
16#19097712—22no—
17C2×#14578929C2211yescharacteristic, complemented
18#19097707C2211yescharacteristic, complemented
19#19097688—21no—
20#135×C157C2211yescharacteristic
21C2×D8×C157C2211yescharacteristic, complemented
22SD32×C157—21no—
23D16×C157—21no—
24M32×C157C2211yescharacteristic, complemented
25#14578955—22no—
26#14578954—41no—
27#14578954D411yescharacteristic
28C2×D628—21no—
29C2×D628—41no—
30D4×D157—22no—
31D4×D157—42no—
32#14578942—41no—
33#14578941—22no—
34D1256—21no—
35D1256D412yes—
36#14578939—22no—
37#14578939—41no—
38#14578938—21no—
39#14578938D411yescharacteristic
40#14578937—21no—
41C8×D157—21no—
42#14578929—21no—
43#41×C157—21no—
44C2×D4×C157—21no—
45Q16×C157D411yescharacteristic
46SD16×C157—21no—
47D8×C157—21no—
48D8×C157—21no—
49D8×C157D411yescharacteristic
50C2×C8×C157C2311yescharacteristic
51C16×C157—21no—
52C22×D157—42no—
53C22×D157—41no—
54Dic314D811yescharacteristic
55D628—22no—
56D628—42no—
57D628—41no—
58D628D811yescharacteristic
59C22⋊D157—21no—
60C22⋊D157—42no—
61C4×D157—21no—
62C4×D157—42no—
63#11929434—21no—
64#11929434—41no—
65C23×C157—41no—
66Q8×C157—21no—
67D4×C157—21no—
68D4×C157—41no—
69D4×C157—41no—
70D4×C157—41no—
71C2×C4×C157—41no—
72C2×C4×C157C2×D411yescharacteristic
73C8×C157C2×D411yescharacteristic
74C8×C157C2×D411yescharacteristic
75C22×C157—41no—
76C22×C157—81no—
77C22×C157—161no—
78C22×C157AGL(1,ℤ/8)11yescharacteristic
79C2×D157—21no—
80C2×D157—42no—
81C2×D157—162no—
82C2×D157—161no—
83C4×C157—41no—
84C4×C157#12011yescharacteristic
85C4×C157C2×D811yescharacteristic
86Dic157—21no—
87C2×C157—21no—
88C2×C157—81no—
89C2×C157—161no—
90C2×C157#44711yescharacteristic
91D157—41no—
92D157—82no—
93C157#219211yescharacteristic, complemented, Sylow
94#2192—1571nomaximal, Sylow
95#576—1571no—
96#509D15711yescharacteristic, complemented
97C2×D16—1571no—
98#472—1571no—
99#361—1571no—
100#359—1571no—
101#350—1571no—
102H(5,𝔽2)—3141no—
103AGL(1,ℤ/8)—3142no—
104AGL(1,ℤ/8)—3141no—
105AGL(1,ℤ/8)—3141no—
106#135—3141no—
107#135C2×D15711yescharacteristic
108C2×D8—3141no—
109C2×D8—1571no—
110C2×D8C2×D15711yescharacteristic, complemented
111#131—1571no—
112SD32—21no—
113D16—21no—
114D16—3142no—
115M32C2×D15711yescharacteristic, complemented
116C2×C16—1571no—
117#108—1571no—
118#100—3141no—
119#41—21no—
120#41—3142no—
121#41—6281no—
122#41—1571no—
123C2×D4—21no—
124C2×D4—3142no—
125C2×D4—3141no—
126C2×D4—6281no—
127C2×D4—6282no—
128Q16C22⋊D15711yescharacteristic
129SD16—21no—
130SD16—3142no—
131SD16—6281no—
132D8—21no—
133D8—21no—
134D8—3142no—
135D8—3141no—
136D8—6281no—
137D8—1572no—
138D8C22⋊D15711yescharacteristic
139M16—3141no—
140M16—3141no—
141M16—1571no—
142C2×C8—3141no—
143C2×C8C22×D15711yescharacteristic
144C16—21no—
145C16—3141no—
146C23—41no—
147C23—6282no—
148C23—6281no—
149Q8—21no—
150Q8—1571no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—3141no—
156D4—3142no—
157D4—6282no—
158D4—6282no—
159D4—6281no—
160D4—1571no—
161C2×C4—41no—
162C2×C4—3141no—
163C2×C4—6282no—
164C2×C4D4×D15711yescharacteristic
165C8—3141no—
166C8—6281no—
167C8D4×D15711yescharacteristic
168C8C2×C22⋊D15711yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—3141no—
173C22—6282no—
174C22—25121no—
175C22—25122no—
176C22#1909779511yescharacteristic
177C4—41no—
178C4—3141no—
179C4#1909773911yescharacteristic
180C4D8×D15711yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—6281no—
185C2—12562no—
186C2#2375740511yescharacteristic
subgroupG/H
centerC2#23757405
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#509×C157C2
socleC2×C157#447
hypercenter#509D157
hypocenterC157#2192
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #23758075C2semidirect
2. C2×#19097712C2semidirect
3. #23757540C2semidirect
4. #23757087C2semidirect
5. #23757080C2semidirect
6. #23757058C2semidirect
7. #509×C157C2semidirect
17. C2×#14578929C22semidirect
18. #19097707C22semidirect
21. C2×D8×C157C22semidirect
24. M32×C157C22semidirect
93. C157#2192semidirect
96. #509D157semidirect
110. C2×D8C2×D157semidirect
115. M32C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580752
C2×#190977122
#237575402
#237570872
#237570802
#237570582
#509×C1572
#2192157