Finite Group Databasebeta

← #34099273

Subgroups — #34099273

Order 20096.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#23757983C211yesmaximal, characteristic
2#23757653C211yesmaximal, characteristic, complemented
3#23757568C211yesmaximal, characteristic, complemented
4#23757564C211yesmaximal, characteristic
5#23757021C211yesmaximal, characteristic, complemented
6#23757002C211yesmaximal, characteristic
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×Dic628—22no—
9#19097771C2211yescharacteristic
10#19097771C2211yescharacteristic
11#19097763—21no—
12#19097749—22no—
13#19097747—21no—
14C4×D628—21no—
15C4×D628C2211yescharacteristic
16C4⋊C4×D157—21no—
17#19097704—21no—
18#19097703—21no—
19C4⋊#11929434C2211yescharacteristic
20#19097695—21no—
21#19097693—21no—
22C4×Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D412yes—
27C2×D628D411yescharacteristic
28C2×C4×D157—22no—
29C2×C4×D157—21no—
30Dic628—44no—
31C2×#11929434—21no—
32C4⋊Dic157—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157D411yescharacteristic
37#14578933—22no—
38#14578933—22no—
39#14578931—22no—
40#14578931—21no—
41C2×Q8×C157D411yescharacteristic
42C2×C8×C157—21no—
43C2×C8×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157D411yescharacteristic
46C42×C157—21no—
47C42×C157C2311yescharacteristic
48C22×D157—21no—
49Dic314—42no—
50Dic314—42no—
51D628—21no—
52D628Q1612yescomplemented
53C2×Dic157—22no—
54C2×Dic157—21no—
55C2×Dic157—41no—
56C4×D157—41no—
57C4×D157—41no—
58C4×D157—82no—
59#11929434—41no—
60Q8×C157—21no—
61Q8×C157—21no—
62C2×C4×C157—21no—
63C2×C4×C157—21no—
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157C2×D411yescharacteristic
66C2×C4×C157C2×D411yescharacteristic
67C8×C157—22no—
68C8×C157—41no—
69C22×C157#12011yescharacteristic
70C2×D157—22no—
71C2×D157—41no—
72C2×D157—41no—
73C4×C157—21no—
74C4×C157—41no—
75C4×C157—41no—
76C4×C157C2×Q1611yescharacteristic
77C4×C157#13511yescharacteristic
78C4×C157#13711yescharacteristic
79C4×C157#13711yescharacteristic
80Dic157—41no—
81Dic157—42no—
82Dic157—81no—
83C2×C157#45111yescharacteristic
84C2×C157#45211yescharacteristic
85C2×C157#45311yescharacteristic
86D157—42no—
87C157#163211yescharacteristic, complemented, Sylow
88#1632—1571nomaximal, Sylow
89#554—1571no—
90#494—1571no—
91#477—1571no—
92#476—1571no—
93#333—1571no—
94Q8⋊C8D15711yescharacteristic, complemented
95D4⋊C8—1571no—
96C2×Q16—3142no—
97C4⋊Q8—1571no—
98C4⋊Q8—1571no—
99#125—3141no—
100C22⋊Q8—3142no—
101C22⋊Q8—3141no—
102C4×Q8C2×D15711yescharacteristic
103C4×D4—3141no—
104C4×D4—1571no—
105C2×C4⋊C4—3141no—
106#107—3141no—
107#106—3141no—
108C4⋊C8—1571no—
109C4⋊C8C2×D15711yescharacteristic
110#103—3141no—
111#102—3141no—
112C4×C8C2×D15711yescharacteristic
113C2×Q8—3141no—
114C2×Q8—1572no—
115C2×Q8D62811yescharacteristic
116C2×D4—1571no—
117C22×C4—3142no—
118C22×C4—3141no—
119Q16—6284no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3141no—
123C4⋊C4—21no—
124C4⋊C4—3142no—
125C4⋊C4—3142no—
126C4⋊C4—3141no—
127C4⋊C4—3141no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4—1571no—
131C4⋊C4D62811yescharacteristic
132C22⋊C4—3142no—
133C22⋊C4—3141no—
134C42—21no—
135C42C22×D15711yescharacteristic
136C23—3141no—
137Q8—21no—
138Q8—21no—
139Q8—6282no—
140Q8—6282no—
141D4—3141no—
142D4—1572no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—3141no—
146C2×C4—3142no—
147C2×C4—6281no—
148C2×C4—6281no—
149C2×C4—6281no—
150C2×C4—12562no—
151C2×C4D4×D15711yescharacteristic
152C2×C4D4×D15711yescharacteristic
153C2×C4C2×D62811yescharacteristic
154C8—22no—
155C8—41no—
156C8—6281no—
157C22—3142no—
158C22—6281no—
159C22—6281no—
160C22#1909774111yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—6281no—
165C4—6282no—
166C4—12561no—
167C4Q16×D15711yescharacteristic
168C4#1909779311yescharacteristic
169C4#1909780011yescharacteristic
170C4#1909780411yescharacteristic
171C2—6282no—
172C2#2375743011yescharacteristic
173C2#2375743711yescharacteristic
174C2#2375744211yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingQ8⋊C8×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenterQ8⋊C8D157
hypocenterC157#1632
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757653C2semidirect
3. #23757568C2semidirect
5. #23757021C2semidirect
7. Q8⋊C8×C157C2semidirect
52. D628Q16semidirect
87. C157#1632semidirect
94. Q8⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237579832
#237576532
#237575682
#237575642
#237570212
#237570022
Q8⋊C8×C1572
#1632157