Finite Group Databasebeta

← #15689688

Subgroups — #15689688

Order 3008.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×#10625352C211yesmaximal, characteristic, complemented
2C22×C22⋊D47C211yesmaximal, characteristic, complemented
3#13094020C212yesmaximal, complemented
4#13094010C212yesmaximal, complemented
5#13093997C212yesmaximal, complemented
6#13093996C212yesmaximal, complemented
7#13093994C211yesmaximal, characteristic, complemented
8#13093993C211yesmaximal, characteristic, complemented
9#13093992C211yesmaximal, characteristic, complemented
10#13093980C211yesmaximal, characteristic, complemented
11#120×C47C211yesmaximal, characteristic, complemented
12C24×C47C2211yescharacteristic
13C23×D47C2211yescharacteristic
14C2×D4×C47C2211yescharacteristic
15C2×D4×C47C2212yescomplemented
16C2×C22⋊D47—24no—
17C2×C22⋊D47C2211yescharacteristic
18C2×C22⋊D47C2212yes—
19C2×C22⋊D47C2212yes—
20C2×C22⋊D47C2211yescharacteristic, complemented
21C2×C22⋊D47C2212yescomplemented
22C2×C22⋊D47C2212yescomplemented
23C22×Dic47C2211yescharacteristic
24C22×Dic47C2211yescharacteristic, complemented
25#10625352—24no—
26C2×D188C2212yescomplemented
27C2×C4×D47C2211yescharacteristic, complemented
28C2×Dic94C2211yescharacteristic, complemented
29C22⋊C4×C47C2211yescharacteristic
30C22⋊C4×C47C2212yescomplemented
31C22⋊Dic47C2211yescharacteristic
32C22⋊Dic47C2212yescomplemented
33(C2×D47)⋊C4C2212yescomplemented
34(C2×D47)⋊C4C2212yescomplemented
35Dic47⋊C4C2212yescomplemented
36C4×Dic47C2212yescomplemented
37C23×C47—22no—
38C23×C47—42no—
39C23×C47C2312yes—
40C23×C47C2311yescharacteristic
41C23×C47C2311yescharacteristic
42C22×D47—22no—
43C22×D47—42no—
44C22×D47C2312yes—
45C22×D47C2311yescharacteristic
46D4×C47—22no—
47D4×C47—42no—
48C2×C4×C47C2312yes—
49C2×C4×C47C2311yescharacteristic
50C22⋊D47—22no—
51C22⋊D47—22no—
52C22⋊D47—24no—
53C22⋊D47—24no—
54C22⋊D47—42no—
55C22⋊D47—42no—
56C2×Dic47—22no—
57C2×Dic47D414yescomplemented
58C2×Dic47C2312yes—
59C2×Dic47C2311yescharacteristic
60C2×Dic47C2311yescharacteristic
61C2×Dic47C2311yescharacteristic
62D188—42no—
63C4×D47—22no—
64Dic94—22no—
65C22×C47—22no—
66C22×C47—21no—
67C22×C47—22no—
68C22×C47—22no—
69C22×C47—22no—
70C22×C47—22no—
71C22×C47—44no—
72C22×C47C2×D412yes—
73C22×C47C2411yescharacteristic
74C2×D47—21no—
75C2×D47—22no—
76C2×D47—22no—
77C2×D47—22no—
78C2×D47—22no—
79C2×D47—44no—
80C4×C47—21no—
81C4×C47—22no—
82Dic47—21no—
83Dic47—22no—
84Dic47C2×D412yes—
85Dic47C2×D412yes—
86C2×C47—22no—
87C2×C47—41no—
88C2×C47—42no—
89C2×C47C22×D411yescharacteristic
90C2×C47H(5,𝔽2)12yes—
91D47—41no—
92D47—42no—
93#535—471nomaximal, Sylow
94C47#53511yescharacteristic, complemented, Sylow
95C2×#41—471no—
96C22×D4—471no—
97#127—472no—
98#124—472no—
99#121—472no—
100#121—472no—
101#120—471no—
102#120—471no—
103#120—471no—
104#120D4711yescharacteristic, complemented
105#117—471no—
106C24—471no—
107C24C2×D4711yescharacteristic
108#41—944no—
109C2×Q8—471no—
110C2×D4—471no—
111C2×D4—472no—
112C2×D4—472no—
113C2×D4—471no—
114C2×D4—472no—
115C2×D4—472no—
116C2×D4—472no—
117C2×D4—944no—
118C2×D4C2×D4711yescharacteristic
119C2×D4C2×D4712yescomplemented
120C22×C4—471no—
121C22×C4—471no—
122C22×C4—471no—
123C4⋊C4—472no—
124C22⋊C4—471no—
125C22⋊C4—472no—
126C22⋊C4—472no—
127C22⋊C4—472no—
128C22⋊C4C2×D4711yescharacteristic
129C22⋊C4C2×D4712yescomplemented
130C42—472no—
131C23—22no—
132C23—42no—
133C23—472no—
134C23—471no—
135C23—942no—
136C23—1882no—
137C23C22×D4712yes—
138C23C22×D4711yescharacteristic
139C23C22×D4711yescharacteristic
140Q8—942no—
141D4—22no—
142D4—42no—
143D4—944no—
144D4—944no—
145D4—942no—
146D4—942no—
147D4—1882no—
148D4—1882no—
149D4—1882no—
150C2×C4—474no—
151C2×C4—471no—
152C2×C4—471no—
153C2×C4—472no—
154C2×C4—471no—
155C2×C4—942no—
156C2×C4—942no—
157C2×C4C22×D4712yes—
158C2×C4C22×D4711yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—941no—
167C22—942no—
168C22—942no—
169C22—942no—
170C22—942no—
171C22—1884no—
172C22D4×D4712yes—
173C22C23×D4711yescharacteristic
174C4—21no—
175C4—22no—
176C4—472no—
177C4—472no—
178C4—941no—
179C4—942no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—1882no—
184C2—1881no—
185C2C2×D4×D4711yescharacteristic
186C2#1309406912yes—
subgroupG/H
center, FrattiniC22C23×D47
commutator, socle, cosocleC22×C47C24
Fitting#120×C47C2
hypercenter#120D47
hypocenterC47#535
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#10625352C2semidirect
2. C22×C22⋊D47C2semidirect
3. #13094020C2semidirect
4. #13094010C2semidirect
5. #13093997C2semidirect
6. #13093996C2semidirect
7. #13093994C2semidirect
8. #13093993C2semidirect
9. #13093992C2semidirect
10. #13093980C2semidirect
11. #120×C47C2semidirect
15. C2×D4×C47C22semidirect
20. C2×C22⋊D47C22semidirect
21. C2×C22⋊D47C22semidirect
22. C2×C22⋊D47C22semidirect
24. C22×Dic47C22semidirect
26. C2×D188C22semidirect
27. C2×C4×D47C22semidirect
28. C2×Dic94C22semidirect
30. C22⋊C4×C47C22semidirect
32. C22⋊Dic47C22semidirect
33. (C2×D47)⋊C4C22semidirect
34. (C2×D47)⋊C4C22semidirect
35. Dic47⋊C4C22semidirect
36. C4×Dic47C22semidirect
57. C2×Dic47D4semidirect
94. C47#535semidirect
104. #120D47semidirect
119. C2×D4C2×D47semidirect
129. C22⋊C4C2×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#106253522
C22×C22⋊D472
#130940202
#130940102
#130939972
#130939962
#130939942
#130939932
#130939922
#130939802
#120×C472
#53547