Finite Group Databasebeta

← #34102154

Subgroups — #34102154

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#23757448C212yesmaximal, complemented
3#23757445C211yesmaximal, characteristic, complemented
4#23757069C212yesmaximal, complemented
5#355×C157C211yesmaximal, characteristic, complemented
6#19097819C2211yescharacteristic, complemented
7#19097799—22no—
8#19097797—21no—
9#19097760C2212yes—
10#19097759—21no—
11#19097757—22no—
12#19097741—22no—
13#19097739—21no—
14#19097699—21no—
15#19097697—22no—
16#19097690C2211yescharacteristic, complemented
17#124×C157C2211yescharacteristic
18#101×C157C2212yescomplemented
19C23×D157—21no—
20#14578955—41no—
21#14578954—22no—
22#14578954—21no—
23C2×Dic314—21no—
24Q8×D157—41no—
25C2×D628—21no—
26C2×D628D412yes—
27C2×C22⋊D157—22no—
28#14578943—42no—
29#14578942—41no—
30D1256—42no—
31#14578939—41no—
32#14578938—21no—
33C22⋊Dic157—21no—
34#14578931—22no—
35#14578931—22no—
36#14578931—22no—
37C4×Dic157D412yes—
38C2×Q8×C157C2311yescharacteristic
39C2×D4×C157D411yescharacteristic
40M16×C157—22no—
41C22⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43C22×D157—22no—
44C22×D157—82no—
45C22×D157—81no—
46Dic314—21no—
47Dic314—22no—
48Dic314—81no—
49D628—21no—
50D628—22no—
51D628—42no—
52D628—81no—
53C22⋊D157—41no—
54C22⋊D157—82no—
55C2×Dic157—22no—
56C4×D157—42no—
57C4×D157—41no—
58#11929434—41no—
59C23×C157—21no—
60Q8×C157—41no—
61D4×C157—41no—
62C2×C4×C157—21no—
63C2×C4×C157C2×D412yes—
64C2×C4×C157C2×D411yescharacteristic
65C8×C157—42no—
66C22×C157—42no—
67C22×C157—41no—
68C22×C157#12011yescharacteristic
69C2×D157—42no—
70C2×D157—42no—
71C2×D157—41no—
72C2×D157—42no—
73C2×D157—84no—
74C4×C157—22no—
75C4×C157—21no—
76C4×C157—41no—
77Dic157—42no—
78Dic157—41no—
79C2×C157—21no—
80C2×C157—81no—
81C2×C157B(4,𝔽2)11yescharacteristic
82D157—81no—
83D157—82no—
84C157#220211yescharacteristic, complemented, Sylow
85#2202—1571nomaximal, Sylow
86H(3,𝔽4)—1571no—
87#454—1571no—
88#454—1572no—
89#355—1572no—
90#355D15711yescharacteristic, complemented
912−1+4—1571no—
92AGL(1,ℤ/8)—3142no—
93AGL(1,ℤ/8)—3141no—
94#124—3142no—
95#124—3141no—
96#124—1572no—
97#124C2×D15711yescharacteristic
98#120—3142no—
99#120—3141no—
100#104—3142no—
101#104—3141no—
102#101—1571no—
103#101C2×D15712yescomplemented
104C24—3141no—
105#41—3142no—
106#41—3141no—
107#41—6281no—
108C2×Q8—3141no—
109C2×Q8—6281no—
110C2×Q8C22×D15711yescharacteristic
111C2×D4—3141no—
112C2×D4—3142no—
113C2×D4—1572no—
114C2×D4D62811yescharacteristic
115SD16—6282no—
116SD16—6281no—
117D8—6282no—
118D8—6281no—
119M16—22no—
120M16—3141no—
121C22⋊C4—22no—
122C22⋊C4—3142no—
123C22⋊C4—3141no—
124C22⋊C4—3142no—
125C22⋊C4—3142no—
126C42—1572no—
127C42D62811yescharacteristic
128C23—21no—
129C23—3142no—
130C23—12562no—
131C23—12561no—
132Q8—41no—
133Q8—3142no—
134Q8—3141no—
135Q8—12561no—
136D4—41no—
137D4—3142no—
138D4—3141no—
139D4—6282no—
140D4—6281no—
141D4—12561no—
142D4—12562no—
143C2×C4—21no—
144C2×C4—3142no—
145C2×C4—6282no—
146C2×C4—6281no—
147C2×C4D4×D15712yes—
148C2×C4C2×D62811yescharacteristic
149C8—42no—
150C8—6281no—
151C22—42no—
152C22—41no—
153C22—6282no—
154C22—6282no—
155C22—6281no—
156C22—6282no—
157C22—12564no—
158C22#1909774111yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—6282no—
163C4—6281no—
164C2—21no—
165C2—81no—
166C2—12562no—
167C2—12561no—
168C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#355×C157C2
socleC2×C157B(4,𝔽2)
hypercenter#355D157
hypocenterC157#2202
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #23758015C2semidirect
2. #23757448C2semidirect
3. #23757445C2semidirect
4. #23757069C2semidirect
5. #355×C157C2semidirect
6. #19097819C22semidirect
16. #19097690C22semidirect
18. #101×C157C22semidirect
84. C157#2202semidirect
90. #355D157semidirect
103. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580152
#237574482
#237574452
#237570692
#355×C1572
#2202157