Finite Group Databasebeta

← #34099304

Subgroups — #34099304

Order 20096.

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

normal subgroups only

This group has 228 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#23757624C211yesmaximal, characteristic, complemented
3#23757569C211yesmaximal, characteristic, complemented
4#23757563C211yesmaximal, characteristic
5#23757020C211yesmaximal, characteristic, complemented
6#23757005C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic
8C22×Dic314—21no—
9#19097771C2211yescharacteristic
10#19097771C2211yescharacteristic
11#19097770—21no—
12#19097750—21no—
13#19097748—22no—
14C4×Dic314C2211yescharacteristic
15D4×Dic157—21no—
16#19097705—21no—
17#19097702—21no—
18C4⋊#11929434C2211yescharacteristic
19#19097696—22no—
20#19097696—21no—
21#19097692—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—
32C2×#11929434—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157D412yes—
37C4⋊Dic157D411yescharacteristic
38#14578933—22no—
39#14578933—21no—
40C22⋊Dic157—22no—
41C4×Dic157—21no—
42C2×D4×C157D411yescharacteristic
43C22×C4×C157—21no—
44C2×C8×C157—21no—
45C2×C8×C157—21no—
46C4⋊C4×C157D411yescharacteristic
47C22⋊C4×C157—21no—
48C42×C157C2311yescharacteristic
49Dic314—21no—
50Dic314—44no—
51Dic314—41no—
52Dic314—41no—
53Dic314—42no—
54Dic314SD1612yes—
55C2×Dic157—21no—
56C2×Dic157—22no—
57C2×Dic157—21no—
58C2×Dic157—44no—
59C2×Dic157—41no—
60#11929434—41no—
61C23×C157—21no—
62D4×C157—21no—
63D4×C157Q1612yes—
64C2×C4×C157—21no—
65C2×C4×C157—41no—
66C2×C4×C157—41no—
67C2×C4×C157C2×D411yescharacteristic
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C8×C157—41no—
71C8×C157—41no—
72C22×C157—22no—
73C22×C157—41no—
74C22×C157—41no—
75C22×C157#12011yescharacteristic
76C4×C157—21no—
77C4×C157—41no—
78C4×C157C2×SD1611yescharacteristic
79C4×C157C2×Q1611yescharacteristic
80C4×C157AGL(1,ℤ/8)11yescharacteristic
81C4×C157#13711yescharacteristic
82Dic157—22no—
83Dic157—42no—
84Dic157—41no—
85Dic157—81no—
86C2×C157—42no—
87C2×C157#45011yescharacteristic
88C2×C157#45111yescharacteristic
89C2×C157#45411yescharacteristic
90C157#163711yescharacteristic, complemented, Sylow
91#1637—1571nomaximal, Sylow
92D4×Q8—1571no—
93#487—1571no—
94#477—1571no—
95#476—1571no—
96#332—1571no—
97Q8⋊C8—1571no—
98D4⋊C8D15711yescharacteristic
99C22×Q8—3141no—
100C4⋊Q8—3141no—
101C4⋊Q8—1571no—
102C4⋊Q8—1571no—
103C22⋊Q8—3142no—
104C22⋊Q8—3141no—
105C4×Q8—1571no—
106C4×D4—3141no—
107C4×D4C2×D15711yescharacteristic
108#107—3141no—
109#106—3141no—
110C4⋊C8—1571no—
111C4⋊C8C2×D15711yescharacteristic
112#103—3141no—
113#103—3142no—
114#102—3141no—
115C4×C8C2×D15711yescharacteristic
116C2×Q8—3142no—
117C2×Q8—3141no—
118C2×Q8—3141no—
119C2×Q8—6282no—
120C2×Q8—6282no—
121C2×Q8—1571no—
122C2×D4D62811yescharacteristic
123C22×C4—21no—
124C22×C4—3142no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—3141no—
128C4⋊C4—3142no—
129C4⋊C4—3141no—
130C4⋊C4—3141no—
131C4⋊C4—3141no—
132C4⋊C4—3141no—
133C4⋊C4—1572no—
134C4⋊C4—1571no—
135C4⋊C4D62811yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—3142no—
138C42—3141no—
139C42C22×D15711yescharacteristic
140C23—21no—
141Q8—3141no—
142Q8—6282no—
143Q8—6284no—
144Q8—6281no—
145Q8—6281no—
146Q8—1572no—
147D4—21no—
148D4Dic62812yes—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3141no—
153C2×C4—3141no—
154C2×C4—3142no—
155C2×C4—6284no—
156C2×C4—6281no—
157C2×C4D4×D15711yescharacteristic
158C2×C4D4×D15711yescharacteristic
159C2×C4C2×D62811yescharacteristic
160C8—41no—
161C8—41no—
162C8—6281no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1909774111yescharacteristic
167C4—21no—
168C4—41no—
169C4—3142no—
170C4—6282no—
171C4—6281no—
172C4—12561no—
173C4SD16×D15711yescharacteristic
174C4C2×Dic62811yescharacteristic
175C4#1909779511yescharacteristic
176C4#1909780411yescharacteristic
177C2—42no—
178C2#2375742511yescharacteristic
179C2#2375743111yescharacteristic
180C2#2375744711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenterD4⋊C8D157
hypocenterC157#1637
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. #23757624C2semidirect
3. #23757569C2semidirect
5. #23757020C2semidirect
6. #23757005C2semidirect
90. C157#1637semidirect

Maximal subgroups

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

subgroupindex
D4×Dic3142
#237576242
#237575692
#237575632
#237570202
#237570052
D4⋊C8×C1572
#1637157