Finite Group Databasebeta

← #34099400

Subgroups — #34099400

Order 20096.

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

normal subgroups only

This group has 234 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#23757938C211yesmaximal, characteristic
2#23757669C211yesmaximal, characteristic, complemented
3#23757559C211yesmaximal, characteristic, complemented
4#23757479C211yesmaximal, characteristic
5#23757003C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic
7#332×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157—21no—
9C2×#14578943—21no—
10#19097771C2211yescharacteristic
11#19097770—21no—
12#19097749—22no—
13#19097747—21no—
14C4×Dic314C2211yescharacteristic
15C4×D628C2211yescharacteristic
16#19097733—21no—
17#19097703—21no—
18#19097702—22no—
19#19097696—21no—
20#19097696—21no—
21C4×#11929434C2211yescharacteristic
22C4⋊Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4⋊C8×C157C2211yescharacteristic
25C2×Dic314—22no—
26C2×Dic314—21no—
27C2×Dic314D411yescharacteristic
28Q8×D157—42no—
29Q8×D157—42no—
30C2×D628D411yescharacteristic
31C2×C4×D157—22no—
32C2×C4×D157—21no—
33#14578943—42no—
34C2×#11929434—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38C4⋊Dic157D411yescharacteristic
39#14578933—22no—
40#14578933—21no—
41#14578931—22no—
42#14578931—21no—
43C4×Dic157—21no—
44C2×Q8×C157—21no—
45C2×C8×C157—21no—
46C2×C8×C157—21no—
47C4⋊C4×C157—21no—
48C4⋊C4×C157D412yes—
49C42×C157C2311yescharacteristic
50C22×D157—21no—
51Dic314—21no—
52Dic314—44no—
53Dic314—41no—
54Dic314—41no—
55Dic314SD1612yescomplemented
56D628—21no—
57D628SD1612yes—
58C2×Dic157—21no—
59C2×Dic157—21no—
60C2×Dic157—21no—
61C2×Dic157—41no—
62C4×D157—44no—
63C4×D157—41no—
64C4×D157—41no—
65#11929434—22no—
66Q8×C157—41no—
67Q8×C157—41no—
68C2×C4×C157—22no—
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157C2×D411yescharacteristic
72C8×C157—41no—
73C8×C157—41no—
74C22×C157#12011yescharacteristic
75C2×D157—22no—
76C2×D157—41no—
77C2×D157—41no—
78C4×C157—21no—
79C4×C157—42no—
80C4×C157C2×SD1611yescharacteristic
81C4×C157C2×SD1611yescharacteristic
82C4×C157AGL(1,ℤ/8)11yescharacteristic
83C4×C157#13711yescharacteristic
84Dic157—22no—
85Dic157—41no—
86Dic157—41no—
87Dic157—81no—
88C2×C157#44811yescharacteristic
89C2×C157#45011yescharacteristic
90C2×C157#45611yescharacteristic
91D157—42no—
92C157#165111yescharacteristic, complemented, Sylow
93#1651—1571nomaximal, Sylow
94D4×Q8—1571no—
95#498—1571no—
96#475—1571no—
97#461—1571no—
98#332D15711yescharacteristic, complemented
99Q8⋊C8—1571no—
100D4⋊C8—1571no—
101C22×Q8—3141no—
102C2×SD16—3141no—
103C4⋊Q8—3141no—
104C4⋊Q8—1571no—
105C4⋊Q8C2×D15711yescharacteristic
106C22⋊Q8—3141no—
107C22⋊Q8—3142no—
108C4×Q8—1571no—
109C4×D4—3141no—
110C4×D4—1571no—
111#106—3141no—
112#106—3142no—
113C4⋊C8C2×D15711yescharacteristic
114C4⋊C8C2×D15711yescharacteristic
115#103—3141no—
116#103—3141no—
117C4×C8—1571no—
118C2×Q8—21no—
119C2×Q8—3142no—
120C2×Q8—3141no—
121C2×Q8—6282no—
122C2×Q8—6282no—
123C2×Q8—1571no—
124C2×D4—1571no—
125C22×C4—3142no—
126C22×C4—3141no—
127SD16—6282no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—3141no—
131C4⋊C4—21no—
132C4⋊C4—3142no—
133C4⋊C4—3141no—
134C4⋊C4—3141no—
135C4⋊C4—3141no—
136C4⋊C4—1571no—
137C4⋊C4—1571no—
138C4⋊C4D62812yes—
139C22⋊C4—3142no—
140C22⋊C4—3141no—
141C42—3141no—
142C42C22×D15711yescharacteristic
143C23—3141no—
144Q8—41no—
145Q8—41no—
146Q8—3141no—
147Q8—6284no—
148Q8—6281no—
149Q8—6281no—
150Q8—1572no—
151D4—3141no—
152D4—1572no—
153C2×C4—22no—
154C2×C4—3141no—
155C2×C4—3141no—
156C2×C4—3141no—
157C2×C4—6284no—
158C2×C4—6281no—
159C2×C4—6281no—
160C2×C4—6281no—
161C2×C4D4×D15711yescharacteristic
162C2×C4D4×D15711yescharacteristic
163C2×C4C2×D62811yescharacteristic
164C8—41no—
165C8—41no—
166C8—3142no—
167C22—3142no—
168C22—6281no—
169C22—6281no—
170C22#1909774111yescharacteristic
171C4—21no—
172C4—42no—
173C4—3142no—
174C4—6281no—
175C4—6281no—
176C4—12561no—
177C4SD16×D15711yescharacteristic
178C4SD16×D15711yescharacteristic
179C4#1909779511yescharacteristic
180C4#1909780311yescharacteristic
181C2—6282no—
182C2#2375741311yescharacteristic
183C2#2375742511yescharacteristic
184C2#2375745811yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#332×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#332D157
hypocenterC157#1651
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #23757669C2semidirect
3. #23757559C2semidirect
5. #23757003C2semidirect
7. #332×C157C2semidirect
55. Dic314SD16semidirect
92. C157#1651semidirect
98. #332D157semidirect

Maximal subgroups

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

subgroupindex
#237579382
#237576692
#237575592
#237574792
#237570032
#237570002
#332×C1572
#1651157