Finite Group Databasebeta

← #15689761

Subgroups — #15689761

Order 3008.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×D4×D47C211yesmaximal, characteristic, complemented
2#13094013C211yesmaximal, characteristic, complemented
3#13094007C212yesmaximal, complemented
4#13094000C211yesmaximal, characteristic, complemented
5#13093998C211yesmaximal, characteristic, complemented
6#13093996C212yesmaximal, complemented
7#13093992C212yesmaximal, complemented
8D4×Dic47C211yesmaximal, characteristic, complemented
9C4×C22⋊D47C211yesmaximal, characteristic, complemented
10C2×(C2×D47)⋊C4C212yesmaximal, complemented
11C2×#41×C47C211yesmaximal, characteristic, complemented
12C23×D47C2212yes—
13#41×C47—24no—
14C2×Q8×C47C2211yescharacteristic, complemented
15C2×D4×C47C2211yescharacteristic
16C2×D4×C47C2212yescomplemented
17C22×C4×C47C2212yescomplemented
18C22×C4×C47C2211yescharacteristic, complemented
19C2×C22⋊D47C2211yescharacteristic
20C2×C22⋊D47C2212yes—
21C2×C22⋊D47C2212yescomplemented
22C22×Dic47C2212yescomplemented
23D4×D47—24no—
24C2×D188C2211yescharacteristic
25C2×C4×D47C2211yescharacteristic
26C22⋊Dic47C2211yescharacteristic, complemented
27C22⋊Dic47C2212yescomplemented
28(C2×D47)⋊C4C2211yescharacteristic
29(C2×D47)⋊C4C2212yes—
30(C2×D47)⋊C4C2212yescomplemented
31(C2×D47)⋊C4C2212yescomplemented
32(C2×D47)⋊C4C2212yescomplemented
33C4⋊Dic47C2211yescharacteristic, complemented
34Dic47⋊C4C2211yescharacteristic, complemented
35Dic47⋊C4C2212yescomplemented
36C4×Dic47C2211yescharacteristic, complemented
37C23×C47C2311yescharacteristic
38C23×C47C2312yes—
39C22×D47—24no—
40C22×D47—24no—
41C22×D47—42no—
42C22×D47C2312yes—
43C22×D47C2311yescharacteristic
44Q8×C47—22no—
45D4×C47—24no—
46D4×C47D414yescomplemented
47C2×C4×C47—24no—
48C2×C4×C47—22no—
49C2×C4×C47C2311yescharacteristic
50C2×C4×C47C2312yes—
51C2×C4×C47C2311yescharacteristic
52C22⋊D47—24no—
53C22⋊D47—22no—
54C22⋊D47—42no—
55C2×Dic47—42no—
56C2×Dic47C2311yescharacteristic
57C2×Dic47C2311yescharacteristic
58C2×Dic47C2311yescharacteristic
59C2×Dic47C2312yes—
60D188—22no—
61C4×D47—22no—
62C22×C47—22no—
63C22×C47—22no—
64C22×C47—21no—
65C22×C47—21no—
66C22×C47—21no—
67C22×C47C2×D414yes—
68C22×C47C2411yescharacteristic
69C2×D47—21no—
70C2×D47—21no—
71C2×D47—22no—
72C2×D47—22no—
73C2×D47—22no—
74C2×D47—48no—
75C2×D47#4112yescomplemented
76C4×C47—21no—
77C4×C47—22no—
78C4×C47C2×D412yes—
79Dic47—21no—
80Dic47—21no—
81Dic47—21no—
82Dic47—22no—
83C2×C47—24no—
84C2×C47—41no—
85C2×C47C22×D411yescharacteristic
86C2×C47C2×#4111yescharacteristic
87C2×C47H(5,𝔽2)11yescharacteristic
88D47—22no—
89D47—42no—
90#546—471nomaximal, Sylow
91C47#54611yescharacteristic, complemented, Sylow
92C2×#41D4711yescharacteristic, complemented
93C22×D4—471no—
94#124—471no—
95#123—472no—
96C22⋊Q8—471no—
97#121—472no—
98#121—471no—
99#120—472no—
100C4×D4—471no—
101C4×D4—471no—
102C2×C22⋊C4—472no—
103C24—472no—
104#41—24no—
105C2×Q8C2×D4711yescharacteristic, complemented
106C2×D4—471no—
107C2×D4—472no—
108C2×D4—472no—
109C2×D4—471no—
110C2×D4—944no—
111C2×D4C2×D4711yescharacteristic
112C2×D4C2×D4712yescomplemented
113C22×C4—472no—
114C22×C4—471no—
115C22×C4C2×D4712yescomplemented
116C22×C4C2×D4711yescharacteristic, complemented
117C4⋊C4—471no—
118C4⋊C4—472no—
119C4⋊C4—471no—
120C22⋊C4—472no—
121C22⋊C4—471no—
122C22⋊C4—471no—
123C22⋊C4—472no—
124C22⋊C4—472no—
125C22⋊C4—472no—
126C22⋊C4—472no—
127C42—471no—
128C23—471no—
129C23—472no—
130C23—944no—
131C23—944no—
132C23—1882no—
133C23C22×D4711yescharacteristic
134C23C22×D4712yes—
135Q8—22no—
136D4—24no—
137D4—942no—
138D4—944no—
139D4—942no—
140D4—1882no—
141D4C22⋊D4714yescomplemented
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—471no—
145C2×C4—471no—
146C2×C4—472no—
147C2×C4—471no—
148C2×C4—942no—
149C2×C4—1882no—
150C2×C4C22×D4711yescharacteristic
151C2×C4C22×D4712yes—
152C2×C4C22×D4711yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—472no—
159C22—941no—
160C22—941no—
161C22—942no—
162C22—942no—
163C22—942no—
164C22—1888no—
165C22C2×C22⋊D4714yes—
166C22C23×D4711yescharacteristic
167C4—21no—
168C4—22no—
169C4—941no—
170C4—941no—
171C4—941no—
172C4—942no—
173C4C2×C22⋊D4712yes—
174C2—24no—
175C2—41no—
176C2—942no—
177C2—1882no—
178C2C22×C22⋊D4711yescharacteristic
179C2#41×D4711yescharacteristic
180C2#1309407011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D47
commutator, socle, cosocleC22×C47C24
FittingC2×#41×C47C2
hypercenterC2×#41D47
hypocenterC47#546
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×D4×D47C2semidirect
2. #13094013C2semidirect
3. #13094007C2semidirect
4. #13094000C2semidirect
5. #13093998C2semidirect
6. #13093996C2semidirect
7. #13093992C2semidirect
8. D4×Dic47C2semidirect
9. C4×C22⋊D47C2semidirect
10. C2×(C2×D47)⋊C4C2semidirect
11. C2×#41×C47C2semidirect
14. C2×Q8×C47C22semidirect
16. C2×D4×C47C22semidirect
17. C22×C4×C47C22semidirect
18. C22×C4×C47C22semidirect
21. C2×C22⋊D47C22semidirect
22. C22×Dic47C22semidirect
26. C22⋊Dic47C22semidirect
27. C22⋊Dic47C22semidirect
30. (C2×D47)⋊C4C22semidirect
31. (C2×D47)⋊C4C22semidirect
32. (C2×D47)⋊C4C22semidirect
33. C4⋊Dic47C22semidirect
34. Dic47⋊C4C22semidirect
35. Dic47⋊C4C22semidirect
36. C4×Dic47C22semidirect
46. D4×C47D4semidirect
75. C2×D47#41semidirect
91. C47#546semidirect
92. C2×#41D47semidirect
105. C2×Q8C2×D47semidirect
112. C2×D4C2×D47semidirect
115. C22×C4C2×D47semidirect
116. C22×C4C2×D47semidirect
141. D4C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D472
#130940132
#130940072
#130940002
#130939982
#130939962
#130939922
D4×Dic472
C4×C22⋊D472
C2×(C2×D47)⋊C42
C2×#41×C472
#54647