Finite Group Databasebeta

← #34099401

Subgroups — #34099401

Order 20096.

234 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×Dic314C211yesmaximal, characteristic
2#23757671C211yesmaximal, characteristic, complemented
3#23757559C211yesmaximal, characteristic, complemented
4#23757478C211yesmaximal, characteristic
5#23757020C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic
8C22×Dic314—21no—
9C2×#14578942—21no—
10#19097771C2211yescharacteristic
11#19097771C2211yescharacteristic
12#19097770—21no—
13#19097750—21no—
14#19097748—22no—
15C4×Dic314C2211yescharacteristic
16D4×Dic157—21no—
17#19097703—22no—
18#19097703—21no—
19C4⋊#11929434C2211yescharacteristic
20#19097696—21no—
21#19097694—21no—
22C4×D4×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—22no—
26C2×Dic314—21no—
27C2×Dic314—21no—
28C2×Dic314—42no—
29C2×Dic314—42no—
30C2×Dic314D411yescharacteristic
31C22×Dic157—22no—
32#14578942—42no—
33C2×#11929434—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38C4⋊Dic157D412yes—
39#14578933—22no—
40#14578933—21no—
41C22⋊Dic157—22no—
42C4×Dic157—21no—
43C2×D4×C157D411yescharacteristic
44C22×C4×C157—21no—
45C2×C8×C157—21no—
46C2×C8×C157—21no—
47C4⋊C4×C157D411yescharacteristic
48C22⋊C4×C157—21no—
49C42×C157C2311yescharacteristic
50Dic314—21no—
51Dic314—44no—
52Dic314—41no—
53Dic314—41no—
54Dic314—41no—
55Dic314—41no—
56Dic314SD1612yescomplemented
57C2×Dic157—21no—
58C2×Dic157—22no—
59C2×Dic157—21no—
60C2×Dic157—44no—
61C2×Dic157—41no—
62#11929434—41no—
63C23×C157—21no—
64D4×C157—21no—
65D4×C157SD1612yes—
66C2×C4×C157—21no—
67C2×C4×C157—41no—
68C2×C4×C157—41no—
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157C2×D411yescharacteristic
72C8×C157—22no—
73C8×C157—41no—
74C22×C157—22no—
75C22×C157—41no—
76C22×C157—41no—
77C22×C157#12011yescharacteristic
78C4×C157—21no—
79C4×C157—41no—
80C4×C157C2×SD1611yescharacteristic
81C4×C157C2×SD1611yescharacteristic
82C4×C157AGL(1,ℤ/8)11yescharacteristic
83C4×C157#13711yescharacteristic
84Dic157—22no—
85Dic157—42no—
86Dic157—41no—
87Dic157—81no—
88C2×C157—42no—
89C2×C157#44811yescharacteristic
90C2×C157#45011yescharacteristic
91C2×C157#45611yescharacteristic
92C157#165111yescharacteristic, complemented, Sylow
93#1651—1571nomaximal, Sylow
94D4×Q8—1571no—
95#498—1571no—
96#475—1571no—
97#461—1571no—
98#332—1571no—
99Q8⋊C8—1571no—
100D4⋊C8D15711yescharacteristic
101C22×Q8—3141no—
102C2×SD16—3141no—
103C4⋊Q8—3141no—
104C4⋊Q8—1571no—
105C4⋊Q8—1571no—
106C22⋊Q8—3142no—
107C22⋊Q8—3141no—
108C4×Q8—1571no—
109C4×D4—3141no—
110C4×D4C2×D15711yescharacteristic
111#106—3141no—
112#106—3142no—
113C4⋊C8—1571no—
114C4⋊C8C2×D15711yescharacteristic
115#103—3141no—
116#103—3141no—
117C4×C8C2×D15711yescharacteristic
118C2×Q8—3142no—
119C2×Q8—3141no—
120C2×Q8—3141no—
121C2×Q8—6282no—
122C2×Q8—6282no—
123C2×Q8—1571no—
124C2×D4D62811yescharacteristic
125C22×C4—21no—
126C22×C4—3142no—
127SD16—6282no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3141no—
131C4⋊C4—3142no—
132C4⋊C4—3141no—
133C4⋊C4—3141no—
134C4⋊C4—3141no—
135C4⋊C4—3141no—
136C4⋊C4—1571no—
137C4⋊C4—1572no—
138C4⋊C4D62811yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—3142no—
141C42—3141no—
142C42C22×D15711yescharacteristic
143C23—21no—
144Q8—3141no—
145Q8—6284no—
146Q8—6281no—
147Q8—6281no—
148Q8—6281no—
149Q8—6281no—
150Q8—1572no—
151D4—21no—
152D4#1457894312yes—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—3141no—
157C2×C4—3141no—
158C2×C4—3142no—
159C2×C4—6284no—
160C2×C4—6281no—
161C2×C4D4×D15711yescharacteristic
162C2×C4D4×D15711yescharacteristic
163C2×C4C2×D62811yescharacteristic
164C8—22no—
165C8—41no—
166C8—6281no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1909774111yescharacteristic
171C4—21no—
172C4—41no—
173C4—3142no—
174C4—6282no—
175C4—6281no—
176C4—12561no—
177C4SD16×D15711yescharacteristic
178C4C2×#1457894311yescharacteristic
179C4#1909779511yescharacteristic
180C4#1909780411yescharacteristic
181C2—42no—
182C2#2375741411yescharacteristic
183C2#2375742511yescharacteristic
184C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenterD4⋊C8D157
hypocenterC157#1651
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. #23757671C2semidirect
3. #23757559C2semidirect
5. #23757020C2semidirect
6. #23757005C2semidirect
56. Dic314SD16semidirect
92. C157#1651semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3142
#237576712
#237575592
#237574782
#237570202
#237570052
D4⋊C8×C1572
#1651157