Finite Group Databasebeta

← #34102139

Subgroups — #34102139

Order 20096.

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

normal subgroups only

This group has 270 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#23757461C212yesmaximal, complemented
3#23757450C211yesmaximal, characteristic, complemented
4#23757059C212yesmaximal, complemented
5#354×C157C211yesmaximal, characteristic, complemented
6#19097816C2211yescharacteristic
7#19097802—21no—
8#19097760—21no—
9#19097759—21no—
10#19097757—22no—
11#19097741—22no—
12#19097741—22no—
13#19097739C2212yes—
14#19097699—21no—
15#19097688C2211yescharacteristic
16#19097685—22no—
17#124×C157C2211yescharacteristic
18#99×C157C2212yes—
19C23×D157D412yescomplemented
20#14578954—21no—
21#14578953—41no—
22C2×Dic314—21no—
23C2×D628—21no—
24C2×D628—21no—
25C2×D628—42no—
26C2×C22⋊D157—22no—
27C2×C22⋊D157—22no—
28D4×D157—41no—
29#14578944—41no—
30#14578941—41no—
31#14578938—21no—
32C22⋊Dic157D412yescomplemented
33#14578931—22no—
34#14578931—22no—
35#14578931—22no—
36#14578931—42no—
37C4×Dic157—21no—
38C2×Q8×C157D411yescharacteristic, complemented
39C2×D4×C157C2311yescharacteristic
40C22⋊C4×C157—22no—
41C22⋊C4×C157—22no—
42C42×C157D411yescharacteristic, complemented
43C22×D157—22no—
44C22×D157—22no—
45C22×D157—42no—
46C22×D157—82no—
47Dic314—21no—
48Dic314—81no—
49D628—21no—
50D628—81no—
51D628—81no—
52D628—84no—
53C22⋊D157—42no—
54C22⋊D157—41no—
55C22⋊D157—82no—
56C2×Dic157—22no—
57C2×Dic157—22no—
58C4×D157—41no—
59#11929434—41no—
60C23×C157C2×D412yes—
61Q8×C157—41no—
62D4×C157—41no—
63C2×C4×C157—21no—
64C2×C4×C157—21no—
65C2×C4×C157—42no—
66C2×C4×C157C2×D411yescharacteristic
67C22×C157—22no—
68C22×C157—42no—
69C22×C157#12011yescharacteristic
70C2×D157—42no—
71C2×D157—42no—
72C2×D157—44no—
73C2×D157—41no—
74C2×D157—42no—
75C2×D157—82no—
76C2×D157—82no—
77C4×C157—21no—
78C4×C157—41no—
79C4×C157—41no—
80C4×C157—82no—
81Dic157—42no—
82Dic157—41no—
83C2×C157—21no—
84C2×C157—42no—
85C2×C157B(4,𝔽2)11yescharacteristic
86D157—82no—
87D157—81no—
88C157#219911yescharacteristic, complemented, Sylow
89#2199—1571nomaximal, Sylow
90H(3,𝔽4)—1571no—
91B(4,𝔽2)—1572no—
92#455—1571no—
93#354D15711yescharacteristic, complemented
94AΣL(1,𝔽16)—1572no—
95H(5,𝔽2)—1571no—
96#137—3141no—
97#124—3142no—
98#124—3141no—
99#124—3141no—
100#124C2×D15711yescharacteristic
101#120—3142no—
102#120—3142no—
103#120—1572no—
104#104—3141no—
105#100—1571no—
106#99—3142no—
107#99C2×D15712yes—
108C24—1572no—
109#41—3141no—
110#41—6281no—
111C2×Q8—3141no—
112C2×Q8D62811yescharacteristic, complemented
113C2×D4—3141no—
114C2×D4—3141no—
115C2×D4—3142no—
116C2×D4—3142no—
117C2×D4—6282no—
118C2×D4—6281no—
119C2×D4C22×D15711yescharacteristic
120Q16—6281no—
121SD16—6281no—
122M16—3141no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—3142no—
126C22⋊C4—3142no—
127C22⋊C4—3142no—
128C22⋊C4—6282no—
129C22⋊C4—1572no—
130C42—3141no—
131C42D62811yescharacteristic, complemented
132C23—3142no—
133C23—3142no—
134C23—6282no—
135C23—12562no—
136C23D4×D15712yes—
137Q8—41no—
138Q8—3141no—
139Q8—12561no—
140D4—41no—
141D4—3141no—
142D4—6282no—
143D4—6281no—
144D4—12561no—
145D4—12561no—
146D4—12564no—
147D4—12562no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—3142no—
152C2×C4—3142no—
153C2×C4—6281no—
154C2×C4C2×D62811yescharacteristic
155C8—6281no—
156C22—22no—
157C22—42no—
158C22—6282no—
159C22—6282no—
160C22—6284no—
161C22—6281no—
162C22—6282no—
163C22—12562no—
164C22—12562no—
165C22#1909774111yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—82no—
170C4—6282no—
171C4—6281no—
172C2—21no—
173C2—42no—
174C2—12562no—
175C2—12561no—
176C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×D4×C157C23
FrattiniC2×D4C22×D157
Fitting#354×C157C2
socleC2×C157B(4,𝔽2)
hypercenter#354D157
hypocenterC157#2199
LCSG ▷ C2×D4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #23758015C2semidirect
2. #23757461C2semidirect
3. #23757450C2semidirect
4. #23757059C2semidirect
5. #354×C157C2semidirect
19. C23×D157D4semidirect
32. C22⋊Dic157D4semidirect
38. C2×Q8×C157D4semidirect
42. C42×C157D4semidirect
88. C157#2199semidirect
93. #354D157semidirect
112. C2×Q8D628semidirect
131. C42D628semidirect

Maximal subgroups

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

subgroupindex
#237580152
#237574612
#237574502
#237570592
#354×C1572
#2199157