Finite Group Databasebeta

← #34102152

Subgroups — #34102152

Order 20096.

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

normal subgroups only

This group has 248 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#23758015C211yesmaximal, characteristic, complemented
2#23757446C212yesmaximal, complemented
3#23757070C211yesmaximal, characteristic, complemented
4#23757069C212yesmaximal, complemented
5#454×C157C211yesmaximal, characteristic, complemented
6#19097796—22no—
7#19097760C2212yes—
8#19097759—21no—
9#19097757—22no—
10#19097741—22no—
11#19097739—21no—
12#19097698—22no—
13#19097690C2212yescomplemented
142−1+4×C157C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C157—21no—
16#124×C157C2211yescharacteristic
17#104×C157—21no—
18#101×C157C2211yescharacteristic, complemented
19C23×D157—21no—
20C2×Dic314—21no—
21C2×D628—21no—
22C2×D628D412yes—
23C2×C22⋊D157—22no—
24#14578941—42no—
25#14578939—42no—
26#14578938—22no—
27C22⋊Dic157—21no—
28#14578931—22no—
29#14578931—22no—
30#14578931—22no—
31C4×Dic157D412yes—
32#41×C157—21no—
33#41×C157—22no—
34#41×C157—41no—
35C2×Q8×C157—41no—
36C2×Q8×C157C2311yescharacteristic
37C2×D4×C157D411yescharacteristic
38SD16×C157—41no—
39D8×C157—41no—
40M16×C157—21no—
41C22⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43C22×D157—22no—
44C22×D157—82no—
45C22×D157—81no—
46Dic314—81no—
47D628—42no—
48D628—81no—
49C22⋊D157—82no—
50C2×Dic157—22no—
51#11929434—42no—
52C23×C157—21no—
53Q8×C157—21no—
54Q8×C157—22no—
55Q8×C157—41no—
56D4×C157—21no—
57D4×C157—22no—
58D4×C157—41no—
59D4×C157—41no—
60C2×C4×C157—21no—
61C2×C4×C157—41no—
62C2×C4×C157—42no—
63C2×C4×C157C2×D412yes—
64C2×C4×C157C2×D411yescharacteristic
65C8×C157—41no—
66C22×C157—42no—
67C22×C157—41no—
68C22×C157—41no—
69C22×C157#12011yescharacteristic
70C2×D157—42no—
71C2×D157—42no—
72C2×D157—42no—
73C2×D157—84no—
74C4×C157—21no—
75C4×C157—22no—
76C4×C157—41no—
77C4×C157—41no—
78Dic157—42no—
79C2×C157—21no—
80C2×C157—81no—
81C2×C157—81no—
82C2×C157B(4,𝔽2)11yescharacteristic
83D157—82no—
84C157#220211yescharacteristic, complemented, Sylow
85#2202—1571nomaximal, Sylow
86H(3,𝔽4)—1571no—
87#454—1572no—
88#454D15711yescharacteristic, complemented
89#355—1571no—
90#355—1572no—
912−1+4C2×D15711yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—3142no—
94#124—3142no—
95#124—3141no—
96#124—1572no—
97#124C2×D15711yescharacteristic
98#120—3142no—
99#120—3141no—
100#104—21no—
101#104—3142no—
102#101—1572no—
103#101C2×D15711yescharacteristic, complemented
104C24—3141no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—3141no—
110C2×Q8C22×D15711yescharacteristic
111C2×D4—3141no—
112C2×D4—3142no—
113C2×D4—1572no—
114C2×D4C22⋊D15711yescharacteristic
115SD16—41no—
116SD16—6282no—
117D8—41no—
118D8—6282no—
119M16—21no—
120M16—3142no—
121C22⋊C4—22no—
122C22⋊C4—3142no—
123C22⋊C4—3141no—
124C22⋊C4—3142no—
125C22⋊C4—3142no—
126C42—1572no—
127C42C22⋊D15711yescharacteristic
128C23—21no—
129C23—3142no—
130C23—12562no—
131C23—12561no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—12561no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—6282no—
141D4—12561no—
142D4—12562no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—3142no—
147C2×C4D4×D15712yes—
148C2×C4C2×C22⋊D15711yescharacteristic
149C8—41no—
150C8—6282no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—6282no—
155C22—6282no—
156C22—6282no—
157C22—12564no—
158C22#1909773911yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—6282no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—12562no—
168C2#2375745911yescharacteristic
subgroupG/H
centerC2#23757459
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#454×C157C2
socleC2×C157B(4,𝔽2)
hypercenter#454D157
hypocenterC157#2202
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23758015C2semidirect
2. #23757446C2semidirect
3. #23757070C2semidirect
4. #23757069C2semidirect
5. #454×C157C2semidirect
13. #19097690C22semidirect
14. 2−1+4×C157C22semidirect
18. #101×C157C22semidirect
84. C157#2202semidirect
88. #454D157semidirect
91. 2−1+4C2×D157semidirect
103. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580152
#237574462
#237570702
#237570692
#454×C1572
#2202157