Finite Group Databasebeta

← #34099259

Subgroups — #34099259

Order 20096.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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#23758008C211yesmaximal, characteristic
2#23757646C211yesmaximal, characteristic
3#23757569C211yesmaximal, characteristic, complemented
4#23757568C211yesmaximal, characteristic, complemented
5#23757016C211yesmaximal, characteristic
6#23757005C211yesmaximal, characteristic, complemented
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578943—22no—
9#19097771C2211yescharacteristic
10#19097768C2211yescharacteristic
11#19097760—21no—
12#19097759—21no—
13#19097756—22no—
14C4×Dic314C2211yescharacteristic
15C4×D628—21no—
16#19097733—21no—
17#19097705—21no—
18#19097704—21no—
19C4⋊#11929434C2211yescharacteristic
20#19097696—21no—
21#19097695—21no—
22C4×Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4×C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×D628—21no—
28C2×D628—21no—
29C2×D628D412yes—
30C2×C4×D157—22no—
31#14578943—44no—
32C2×#11929434—21no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157D411yescharacteristic
36#14578933—21no—
37#14578931—22no—
38#14578931—22no—
39#14578931—22no—
40C4×Dic157—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—22no—
49Dic314—21no—
50Dic314—21no—
51Dic314—42no—
52D628—42no—
53D628—41no—
54D628—41no—
55D628—41no—
56D628—41no—
57C2×Dic157—21no—
58C2×Dic157—21no—
59C2×Dic157—41no—
60C4×D157—82no—
61#11929434—41no—
62Q8×C157—21no—
63Q8×C157—21no—
64C2×C4×C157—21no—
65C2×C4×C157—21no—
66C2×C4×C157C2×D411yescharacteristic
67C2×C4×C157C2×D411yescharacteristic
68C2×C4×C157C2×D411yescharacteristic
69C8×C157—22no—
70C8×C157—41no—
71C22×C157#12011yescharacteristic
72C2×D157—42no—
73C2×D157—42no—
74C2×D157—42no—
75C4×C157—21no—
76C4×C157—41no—
77C4×C157—41no—
78C4×C157#13511yescharacteristic
79C4×C157#13511yescharacteristic
80C4×C157#13711yescharacteristic
81C4×C157#13711yescharacteristic
82Dic157—41no—
83Dic157—41no—
84Dic157—81no—
85C2×C157#45211yescharacteristic
86C2×C157#45211yescharacteristic
87C2×C157#45311yescharacteristic
88D157—82no—
89C157#163011yescharacteristic, complemented, Sylow
90#1630—1571nomaximal, Sylow
91#559—1571no—
92#492—1571no—
93#477—1571no—
94#477—1571no—
95#331—1571no—
96Q8⋊C8—1571no—
97Q8⋊C8D15711yescharacteristic, complemented
98C2×SD16—3142no—
99C4⋊Q8—1571no—
100#127—1571no—
101#124—3141no—
102#124—3141no—
103#123—3142no—
104C4×Q8—1571no—
105C4×Q8C2×D15711yescharacteristic
106C4×D4—3141no—
107C4×D4—3141no—
108#107—3141no—
109#107—3141no—
110C4⋊C8—1571no—
111C4⋊C8C2×D15711yescharacteristic
112#103—3141no—
113#103—3141no—
114C4×C8C2×D15711yescharacteristic
115C2×Q8—3141no—
116C2×Q8—1571no—
117C2×Q8D62811yescharacteristic
118C2×D4—3141no—
119C2×D4—3141no—
120C2×D4—1572no—
121C22×C4—3142no—
122SD16—6284no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—3141no—
126C4⋊C4—21no—
127C4⋊C4—3141no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4—1571no—
131C4⋊C4D62811yescharacteristic
132C22⋊C4—3142no—
133C22⋊C4—3142no—
134C22⋊C4—3142no—
135C42—21no—
136C42—3141no—
137C42C22×D15711yescharacteristic
138C23—3142no—
139Q8—21no—
140Q8—21no—
141Q8—3141no—
142Q8—3141no—
143Q8—6282no—
144D4—6282no—
145D4—6281no—
146D4—6281no—
147D4—6281no—
148D4—6281no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—3141no—
152C2×C4—3141no—
153C2×C4—6281no—
154C2×C4—12562no—
155C2×C4D4×D15711yescharacteristic
156C2×C4D4×D15711yescharacteristic
157C2×C4C2×D62811yescharacteristic
158C8—22no—
159C8—41no—
160C8—6281no—
161C22—6282no—
162C22—6282no—
163C22—6282no—
164C22#1909774111yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—6281no—
169C4—6281no—
170C4—12561no—
171C4#1909779211yescharacteristic
172C4#1909779311yescharacteristic
173C4#1909780311yescharacteristic
174C4#1909780411yescharacteristic
175C2—12562no—
176C2#2375743711yescharacteristic
177C2#2375743811yescharacteristic
178C2#2375744211yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingQ8⋊C8×C157C2
socleC22×C157#120
prime radical#19097768C22
hypercenterQ8⋊C8D157
hypocenterC157#1630
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
3. #23757569C2semidirect
4. #23757568C2semidirect
6. #23757005C2semidirect
7. Q8⋊C8×C157C2semidirect
89. C157#1630semidirect
97. Q8⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237580082
#237576462
#237575692
#237575682
#237570162
#237570052
Q8⋊C8×C1572
#1630157