Finite Group Databasebeta

← #15689766

Subgroups — #15689766

Order 3008.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#10625354C211yesmaximal, characteristic, complemented
2C2×D4×D47C211yesmaximal, characteristic, complemented
3#13094006C212yesmaximal, complemented
4#13094000C211yesmaximal, characteristic, complemented
5#13093997C212yesmaximal, complemented
6#13093995C211yesmaximal, characteristic, complemented
7#13093994C212yesmaximal, complemented
8C4×D188C211yesmaximal, characteristic, complemented
9C4×D188C211yesmaximal, characteristic, complemented
10C22⋊C4×D47C212yesmaximal, complemented
11#124×C47C211yesmaximal, characteristic, complemented
12C23×D47C2212yes—
13C2×Q8×C47C2211yescharacteristic, complemented
14C2×D4×C47C2211yescharacteristic
15C2×C22⋊D47C2212yes—
16C2×C22⋊D47C2212yescomplemented
17#10625354—24no—
18D4×D47—24no—
19C2×D188C2211yescharacteristic
20C2×D188C2211yescharacteristic
21C2×D188C2212yescomplemented
22C2×C4×D47C2211yescharacteristic
23C2×C4×D47C2212yescomplemented
24C2×C4×D47C2211yescharacteristic, complemented
25C2×C4×D47C2212yescomplemented
26C22⋊C4×C47C2212yescomplemented
27C22⋊C4×C47C2212yescomplemented
28C42×C47C2211yescharacteristic, complemented
29C22⋊Dic47C2212yescomplemented
30(C2×D47)⋊C4C2211yescharacteristic
31(C2×D47)⋊C4C2212yes—
32(C2×D47)⋊C4C2212yescomplemented
33(C2×D47)⋊C4C2211yescharacteristic, complemented
34C4⋊Dic47C2211yescharacteristic, complemented
35C4⋊Dic47C2211yescharacteristic, complemented
36Dic47⋊C4C2212yescomplemented
37C23×C47C2312yes—
38C22×D47—24no—
39C22×D47—24no—
40C22×D47—42no—
41C22×D47C2311yescharacteristic
42C22×D47C2312yes—
43C22×D47C2311yescharacteristic
44Q8×C47—22no—
45D4×C47—22no—
46C2×C4×C47C2312yes—
47C2×C4×C47C2311yescharacteristic
48C2×C4×C47C2311yescharacteristic
49C2×C4×C47C2311yescharacteristic
50C22⋊D47—24no—
51C22⋊D47—42no—
52C2×Dic47C2311yescharacteristic
53C2×Dic47C2312yes—
54C2×Dic47C2311yescharacteristic
55D188—24no—
56D188—22no—
57D188D414yescomplemented
58C4×D47—24no—
59C4×D47—22no—
60C4×D47—22no—
61C4×D47—42no—
62C22×C47—22no—
63C22×C47—22no—
64C22×C47—22no—
65C22×C47C2411yescharacteristic
66C2×D47—21no—
67C2×D47—22no—
68C2×D47—22no—
69C2×D47—21no—
70C2×D47—21no—
71C2×D47—21no—
72C2×D47—21no—
73C2×D47—48no—
74C2×D47C2×D414yes—
75C2×D47#4112yescomplemented
76C4×C47—22no—
77C4×C47—21no—
78C4×C47—21no—
79C4×C47C2×D412yes—
80Dic47—21no—
81Dic47—21no—
82Dic47—22no—
83C2×C47—42no—
84C2×C47C22×D411yescharacteristic
85C2×C47C2×#4111yescharacteristic
86C2×C47H(5,𝔽2)11yescharacteristic
87D47—22no—
88D47—24no—
89D47—41no—
90#546—471nomaximal, Sylow
91C47#54611yescharacteristic, complemented, Sylow
92C2×#41—471no—
93C22×D4—471no—
94#124D4711yescharacteristic, complemented
95#123—472no—
96C22⋊Q8—471no—
97#121—471no—
98#121—472no—
99#120—472no—
100C4×D4—471no—
101C4×D4—471no—
102C2×C22⋊C4—472no—
103C24—472no—
104#41—944no—
105C2×Q8C2×D4711yescharacteristic, complemented
106C2×D4—472no—
107C2×D4—472no—
108C2×D4—471no—
109C2×D4—471no—
110C2×D4—472no—
111C2×D4—944no—
112C2×D4C2×D4711yescharacteristic
113C22×C4—471no—
114C22×C4—472no—
115C22×C4—471no—
116C22×C4—472no—
117C4⋊C4—472no—
118C4⋊C4—471no—
119C4⋊C4—471no—
120C22⋊C4—472no—
121C22⋊C4—471no—
122C22⋊C4—472no—
123C22⋊C4—472no—
124C22⋊C4—471no—
125C22⋊C4C2×D4712yescomplemented
126C22⋊C4C2×D4712yescomplemented
127C42C2×D4711yescharacteristic, complemented
128C23—471no—
129C23—472no—
130C23—471no—
131C23—944no—
132C23—944no—
133C23—1882no—
134C23C22×D4712yes—
135Q8—22no—
136D4—22no—
137D4—474no—
138D4—942no—
139D4—944no—
140D4—944no—
141D4—1882no—
142C2×C4—472no—
143C2×C4—471no—
144C2×C4—471no—
145C2×C4—944no—
146C2×C4—942no—
147C2×C4—942no—
148C2×C4—1882no—
149C2×C4C22×D4711yescharacteristic
150C2×C4C22×D4711yescharacteristic
151C2×C4C22×D4712yes—
152C2×C4C22×D4711yescharacteristic
153C22—22no—
154C22—22no—
155C22—22no—
156C22—474no—
157C22—472no—
158C22—941no—
159C22—942no—
160C22—942no—
161C22—941no—
162C22—941no—
163C22—941no—
164C22—941no—
165C22—1888no—
166C22C23×D4711yescharacteristic
167C4—22no—
168C4—21no—
169C4—21no—
170C4—941no—
171C4—941no—
172C4—942no—
173C4D4×D4712yes—
174C2—42no—
175C2—942no—
176C2—944no—
177C2—1881no—
178C2C2×D4×D4711yescharacteristic
179C2#41×D4711yescharacteristic
180C2#1309407011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D47
commutator, socle, cosocleC22×C47C24
Fitting#124×C47C2
hypercenter#124D47
hypocenterC47#546
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#10625354C2semidirect
2. C2×D4×D47C2semidirect
3. #13094006C2semidirect
4. #13094000C2semidirect
5. #13093997C2semidirect
6. #13093995C2semidirect
7. #13093994C2semidirect
8. C4×D188C2semidirect
9. C4×D188C2semidirect
10. C22⋊C4×D47C2semidirect
11. #124×C47C2semidirect
13. C2×Q8×C47C22semidirect
16. C2×C22⋊D47C22semidirect
21. C2×D188C22semidirect
23. C2×C4×D47C22semidirect
24. C2×C4×D47C22semidirect
25. C2×C4×D47C22semidirect
26. C22⋊C4×C47C22semidirect
27. C22⋊C4×C47C22semidirect
28. C42×C47C22semidirect
29. C22⋊Dic47C22semidirect
32. (C2×D47)⋊C4C22semidirect
33. (C2×D47)⋊C4C22semidirect
34. C4⋊Dic47C22semidirect
35. C4⋊Dic47C22semidirect
36. Dic47⋊C4C22semidirect
57. D188D4semidirect
75. C2×D47#41semidirect
91. C47#546semidirect
94. #124D47semidirect
105. C2×Q8C2×D47semidirect
125. C22⋊C4C2×D47semidirect
126. C22⋊C4C2×D47semidirect
127. C42C2×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#106253542
C2×D4×D472
#130940062
#130940002
#130939972
#130939952
#130939942
C4×D1882
C22⋊C4×D472
#124×C472
#54647