Finite Group Databasebeta

← #34099272

Subgroups — #34099272

Order 20096.

214 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757980C211yesmaximal, characteristic
2#23757653C211yesmaximal, characteristic, complemented
3#23757569C211yesmaximal, characteristic, complemented
4#23757563C211yesmaximal, characteristic
5#23757021C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic
8C2×Dic628—22no—
9#19097771C2211yescharacteristic
10#19097771C2211yescharacteristic
11#19097762—21no—
12#19097750—21no—
13#19097748—22no—
14C4×Dic314C2211yescharacteristic
15D4×Dic157—21no—
16C2×C4⋊Dic157—21no—
17#19097705—21no—
18#19097702—21no—
19C4⋊#11929434C2211yescharacteristic
20#19097696—21no—
21#19097692—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×Dic314D412yes—
28C22×Dic157—22no—
29Dic628—44no—
30C2×#11929434—21no—
31C4⋊Dic157—22no—
32C4⋊Dic157—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157D411yescharacteristic
37#14578933—22no—
38#14578933—21no—
39C22⋊Dic157—22no—
40C4×Dic157—21no—
41C2×D4×C157D411yescharacteristic
42C22×C4×C157—21no—
43C2×C8×C157—21no—
44C2×C8×C157—21no—
45C4⋊C4×C157D411yescharacteristic
46C22⋊C4×C157—21no—
47C42×C157C2311yescharacteristic
48Dic314—21no—
49Dic314—21no—
50Dic314—42no—
51Dic314—42no—
52C2×Dic157—21no—
53C2×Dic157—22no—
54C2×Dic157—21no—
55C2×Dic157—41no—
56C2×Dic157—82no—
57#11929434—41no—
58C23×C157—21no—
59D4×C157—21no—
60D4×C157Q1612yescomplemented
61C2×C4×C157—21no—
62C2×C4×C157—41no—
63C2×C4×C157—41no—
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157C2×D411yescharacteristic
66C2×C4×C157C2×D411yescharacteristic
67C8×C157—22no—
68C8×C157—41no—
69C22×C157—22no—
70C22×C157—41no—
71C22×C157—41no—
72C22×C157#12011yescharacteristic
73C4×C157—21no—
74C4×C157—41no—
75C4×C157C2×Q1611yescharacteristic
76C4×C157#13511yescharacteristic
77C4×C157#13711yescharacteristic
78C4×C157#13711yescharacteristic
79Dic157—41no—
80Dic157—41no—
81Dic157—42no—
82Dic157—81no—
83C2×C157—42no—
84C2×C157#45111yescharacteristic
85C2×C157#45211yescharacteristic
86C2×C157#45311yescharacteristic
87C157#163211yescharacteristic, complemented, Sylow
88#1632—1571nomaximal, Sylow
89#554—1571no—
90#494—1571no—
91#477—1571no—
92#476—1571no—
93#333—1571no—
94Q8⋊C8—1571no—
95D4⋊C8D15711yescharacteristic
96C2×Q16—3142no—
97C4⋊Q8—1571no—
98C4⋊Q8—1571no—
99#125—3141no—
100C22⋊Q8—3142no—
101C22⋊Q8—3141no—
102C4×Q8—1571no—
103C4×D4—3141no—
104C4×D4C2×D15711yescharacteristic
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—1571no—
115C2×Q8—1572no—
116C2×D4D62811yescharacteristic
117C22×C4—21no—
118C22×C4—3142no—
119Q16—6284no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3141no—
123C4⋊C4—3142no—
124C4⋊C4—3142no—
125C4⋊C4—3141no—
126C4⋊C4—3141no—
127C4⋊C4—3141no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4—1571no—
131C4⋊C4D62811yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—3142no—
134C42—3141no—
135C42C22×D15711yescharacteristic
136C23—21no—
137Q8—3141no—
138Q8—3141no—
139Q8—6282no—
140Q8—6282no—
141D4—21no—
142D4Dic62812yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—3141no—
147C2×C4—3141no—
148C2×C4—3142no—
149C2×C4—6281no—
150C2×C4—12562no—
151C2×C4D4×D15711yescharacteristic
152C2×C4D4×D15711yescharacteristic
153C2×C4C2×D62811yescharacteristic
154C8—22no—
155C8—41no—
156C8—6281no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1909774111yescharacteristic
161C4—21no—
162C4—41no—
163C4—6281no—
164C4—6281no—
165C4—6282no—
166C4—12561no—
167C4C2×Dic62811yescharacteristic
168C4#1909779011yescharacteristic
169C4#1909780011yescharacteristic
170C4#1909780411yescharacteristic
171C2—42no—
172C2#2375743111yescharacteristic
173C2#2375743611yescharacteristic
174C2#2375744211yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenterD4⋊C8D157
hypocenterC157#1632
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757653C2semidirect
3. #23757569C2semidirect
5. #23757021C2semidirect
6. #23757005C2semidirect
60. D4×C157Q16semidirect
87. C157#1632semidirect
142. D4Dic628semidirect

Maximal subgroups

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

subgroupindex
#237579802
#237576532
#237575692
#237575632
#237570212
#237570052
D4⋊C8×C1572
#1632157