Finite Group Databasebeta

← #15689636

Subgroups — #15689636

Order 3008.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C22×C4×D47C211yesmaximal, characteristic, complemented
3#13094009C212yesmaximal, complemented
4#13094000C211yesmaximal, characteristic, complemented
5#13093999C211yesmaximal, characteristic, complemented
6#13093998C211yesmaximal, characteristic, complemented
7#13093994C212yesmaximal, complemented
8#13093986C212yesmaximal, complemented
9(C22⋊D47)⋊C4C212yesmaximal, complemented
10#13093982C211yesmaximal, characteristic, complemented
11C22⋊Q8×C47C211yesmaximal, characteristic, complemented
12C23×D47C2211yescharacteristic
13C2×Q8×C47C2211yescharacteristic, complemented
14C22×C4×C47C2211yescharacteristic
15C2×C22⋊D47C2212yes—
16C22×Dic47C2211yescharacteristic
17#10625354—24no—
18C2×D188C2212yes—
19C2×D188C2212yes—
20C2×D188C2211yescharacteristic, complemented
21C2×C4×D47—24no—
22C2×C4×D47C2211yescharacteristic
23C2×C4×D47C2211yescharacteristic
24C2×C4×D47C2211yescharacteristic
25C2×C4×D47C2211yescharacteristic
26C2×C4×D47C2212yescomplemented
27C4⋊C4×C47C2211yescharacteristic, complemented
28C4⋊C4×C47C2212yescomplemented
29C22⋊C4×C47C2212yes—
30(C2×D47)⋊C4C2212yes—
31(C2×D47)⋊C4C2212yes—
32(C2×D47)⋊C4C2212yescomplemented
33(C2×D47)⋊C4C2212yescomplemented
34C4⋊Dic47C2211yescharacteristic, complemented
35Dic47⋊C4C2212yescomplemented
36C4×Dic47C2212yescomplemented
37C23×C47C2311yescharacteristic
38C22×D47—22no—
39C22×D47—22no—
40C22×D47—22no—
41C22×D47C2312yes—
42C22×D47C2311yescharacteristic
43C22×D47C2311yescharacteristic
44Q8×C47—22no—
45C2×C4×C47—22no—
46C2×C4×C47C2312yes—
47C2×C4×C47C2312yes—
48C2×C4×C47C2311yescharacteristic
49C2×C4×C47C2311yescharacteristic
50C22⋊D47—24no—
51C2×Dic47—22no—
52C2×Dic47C2312yes—
53C2×Dic47C2311yescharacteristic
54C2×Dic47C2311yescharacteristic
55D188—24no—
56D188—24no—
57D188—22no—
58C4×D47—22no—
59C4×D47—22no—
60C4×D47—22no—
61C4×D47—24no—
62C4×D47D414yescomplemented
63C22×C47—21no—
64C22×C47—21no—
65C22×C47#4112yescomplemented
66C22×C47C2411yescharacteristic
67C2×D47—28no—
68C2×D47—21no—
69C2×D47—21no—
70C2×D47—21no—
71C2×D47—21no—
72C2×D47—22no—
73C2×D47—22no—
74C2×D47—22no—
75C2×D47C2×D412yes—
76C2×D47#4112yescomplemented
77C4×C47—21no—
78C4×C47—22no—
79C4×C47—22no—
80C4×C47C2×D412yes—
81Dic47—21no—
82Dic47—22no—
83Dic47C2×D412yes—
84C2×C47—22no—
85C2×C47C22×D411yescharacteristic
86C2×C47C2×#4111yescharacteristic
87C2×C47C2×#4111yescharacteristic
88D47—22no—
89D47—22no—
90D47—42no—
91#525—471nomaximal, Sylow
92C47#52511yescharacteristic, complemented, Sylow
93C2×#41—471no—
94C23×C4—471no—
95#123—472no—
96C22⋊Q8—471no—
97C22⋊Q8D4711yescharacteristic, complemented
98#121—471no—
99#121—471no—
100#120—472no—
101C4×D4—472no—
102C4×D4—472no—
103#117—471no—
104C24—471no—
105#41—944no—
106C2×Q8C2×D4711yescharacteristic, complemented
107C2×D4—472no—
108C2×D4—472no—
109C2×D4—472no—
110C2×D4—471no—
111C22×C4—471no—
112C22×C4—471no—
113C22×C4—471no—
114C22×C4—471no—
115C22×C4—471no—
116C22×C4—472no—
117C22×C4—944no—
118C22×C4C2×D4711yescharacteristic
119C4⋊C4—472no—
120C4⋊C4—471no—
121C4⋊C4C2×D4711yescharacteristic, complemented
122C4⋊C4C2×D4712yescomplemented
123C22⋊C4—472no—
124C22⋊C4—472no—
125C22⋊C4—472no—
126C22⋊C4—472no—
127C22⋊C4C2×D4712yes—
128C42—472no—
129C23—471no—
130C23—471no—
131C23—472no—
132C23—942no—
133C23—942no—
134C23—942no—
135C23C22×D4711yescharacteristic
136Q8—22no—
137D4—944no—
138D4—944no—
139D4—944no—
140D4—942no—
141C2×C4—22no—
142C2×C4—471no—
143C2×C4—474no—
144C2×C4—472no—
145C2×C4—471no—
146C2×C4—942no—
147C2×C4—944no—
148C2×C4—942no—
149C2×C4—942no—
150C2×C4—942no—
151C2×C4C22×D4712yes—
152C2×C4C22×D4711yescharacteristic
153C2×C4C22×D4712yes—
154C2×C4C22×D4711yescharacteristic
155C22—21no—
156C22—21no—
157C22—472no—
158C22—472no—
159C22—948no—
160C22—941no—
161C22—941no—
162C22—941no—
163C22—941no—
164C22—942no—
165C22—942no—
166C22—942no—
167C22#1062535412yescomplemented
168C22C23×D4711yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—472no—
173C4—941no—
174C4—942no—
175C4D4×D4712yes—
176C2—22no—
177C2—942no—
178C2—942no—
179C2—1882no—
180C2C2×D4×D4711yescharacteristic
181C2#41×D4711yescharacteristic
182C2C2×#1062535411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D47
commutator, socle, cosocleC22×C47C24
FittingC22⋊Q8×C47C2
hypercenterC22⋊Q8D47
hypocenterC47#525
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10625354C2semidirect
2. C22×C4×D47C2semidirect
3. #13094009C2semidirect
4. #13094000C2semidirect
5. #13093999C2semidirect
6. #13093998C2semidirect
7. #13093994C2semidirect
8. #13093986C2semidirect
9. (C22⋊D47)⋊C4C2semidirect
10. #13093982C2semidirect
11. C22⋊Q8×C47C2semidirect
13. C2×Q8×C47C22semidirect
20. C2×D188C22semidirect
26. C2×C4×D47C22semidirect
27. C4⋊C4×C47C22semidirect
28. C4⋊C4×C47C22semidirect
32. (C2×D47)⋊C4C22semidirect
33. (C2×D47)⋊C4C22semidirect
34. C4⋊Dic47C22semidirect
35. Dic47⋊C4C22semidirect
36. C4×Dic47C22semidirect
62. C4×D47D4semidirect
65. C22×C47#41semidirect
76. C2×D47#41semidirect
92. C47#525semidirect
97. C22⋊Q8D47semidirect
106. C2×Q8C2×D47semidirect
121. C4⋊C4C2×D47semidirect
122. C4⋊C4C2×D47semidirect
167. C22#10625354semidirect

Maximal subgroups

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

subgroupindex
C2×#106253542
C22×C4×D472
#130940092
#130940002
#130939992
#130939982
#130939942
#130939862
(C22⋊D47)⋊C42
#130939822
C22⋊Q8×C472
#52547