Finite Group Databasebeta

← C7×C22⋊C4×C3⋊D3 #72246716

Subgroups — C7×C22⋊C4×C3⋊D3 #72246716

Order 2016.

792 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C23×C7×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C7×C3⋊D3C212yesmaximal, complemented
3C32×C7×C22⋊C4C211yesmaximal, characteristic, complemented
4C7×#3757C211yesmaximal, characteristic, complemented
5C7×#3752C212yesmaximal, complemented
6D3×C7×C22⋊C4—34nomaximal
7C23×C32×C7C2211yescharacteristic
8C22×C7×C3⋊D3—24no—
9C22×C7×C3⋊D3C414yescomplemented
10C22×C7×C3⋊D3C2211yescharacteristic
11C22×C7×C3⋊D3C2211yescharacteristic
12C2×C32×C4×C7C2212yescomplemented
13C2×C7×C3⋊Dic3C2212yescomplemented
14C4×C7×C3⋊D3—24no—
15C23×D3×C7—34no—
16C2×C4×D3×C7—38no—
17C3×C7×C22⋊C4D314yescomplemented
18C7×#217—34no—
19C7×(C2×D3)⋊C4—38no—
20C22⋊C4×C3⋊D3C711yesmaximal, characteristic, direct product factor, Hall
21C22×C32×C7—22no—
22C22×C32×C7C2×C412yescomplemented
23C22×C32×C7C2311yescharacteristic
24C2×C7×C3⋊D3—22no—
25C2×C7×C3⋊D3—28no—
26C2×C7×C3⋊D3C2×C412yes—
27C2×C7×C3⋊D3C2×C412yescomplemented
28C2×C7×C3⋊D3D414yes—
29C32×C4×C7—22no—
30C7×C3⋊Dic3—22no—
31C2×C7×C22⋊C4—91noHall
32C23×C3×C7C2×D314yes—
33C22×D3×C7—34no—
34C22×D3×C7—316no—
35C22×D3×C7—34no—
36C22×D3×C7—616no—
37C2×C3×C4×C7C2×D318yescomplemented
38C2×C7×Dic3—38no—
39C4×D3×C7—616no—
40C23×C3⋊D3C2×C711yescharacteristic
41C2×C4×C3⋊D3C2×C712yescomplemented
42C32×C22⋊C4C2×C711yescharacteristic, complemented
43#3757C2×C711yescharacteristic, complemented
44#3752C2×C712yescomplemented
45C2×C32×C7—22no—
46C2×C32×C7C22×C411yescharacteristic
47C2×C32×C7C2×D412yes—
48C7×C3⋊D3—22no—
49C7×C3⋊D3C22⋊C414yesdirect product factor
50C24×C7—91no—
51C22×C4×C7—92no—
52C7×C22⋊C4—91no—
53C7×C22⋊C4—92no—
54C7×C22⋊C4C3⋊D311yescharacteristic, direct product factor
55D3×C22⋊C4—34no—
56C22×C3×C7—28no—
57C22×C3×C7C4×D318yescomplemented
58C22×C3×C7C22×D314yes—
59C2×D3×C7—38no—
60C2×D3×C7—316no—
61C2×D3×C7—38no—
62C2×D3×C7—632no—
63C2×D3×C7—68no—
64C3×C4×C7—28no—
65C7×Dic3—68no—
66C23×C32C22×C711yescharacteristic
67C22×C3⋊D3—24no—
68C22×C3⋊D3C4×C714yescomplemented
69C22×C3⋊D3C22×C711yescharacteristic
70C22×C3⋊D3C22×C711yescharacteristic
71C2×C32×C4C22×C712yescomplemented
72C2×C3⋊Dic3C22×C712yescomplemented
73C4×C3⋊D3—24no—
74C32×C7C2×C22⋊C411yescharacteristic, complemented, Hall
75C23×C7—91no—
76C23×C7—94no—
77C23×C7—91no—
78C23×C7—184no—
79C23×C7C2×C3⋊D311yescharacteristic
80C2×C4×C7—92no—
81C2×C4×C7—184no—
82C2×C4×C7C2×C3⋊D312yescomplemented
83C23×D3—34no—
84C2×C4×D3—38no—
85C3×C22⋊C4D3×C714yescomplemented
86#217—34no—
87(C2×D3)⋊C4—38no—
88C2×C3×C7—28no—
89C2×C3×C7C2×C4×D314yes—
90C2×C3×C7D3×D418yes—
91D3×C7—316no—
92D3×C7—68no—
93C22×C32—22no—
94C22×C32C2×C4×C712yescomplemented
95C22×C32C23×C711yescharacteristic
96C2×C3⋊D3—22no—
97C2×C3⋊D3—28no—
98C2×C3⋊D3C2×C4×C712yes—
99C2×C3⋊D3C2×C4×C712yescomplemented
100C2×C3⋊D3C7×D414yes—
101C32×C4—22no—
102C3⋊Dic3—22no—
103C2×C22⋊C4—91noSylow
104C22×C7—22no—
105C22×C7—92no—
106C22×C7—94no—
107C22×C7—92no—
108C22×C7—188no—
109C22×C7—182no—
110C22×C7C4×C3⋊D312yescomplemented
111C22×C7C22×C3⋊D311yescharacteristic
112C4×C7—22no—
113C4×C7—182no—
114C23×C3C2×D3×C714yes—
115C22×D3—34no—
116C22×D3—316no—
117C22×D3—34no—
118C22×D3—616no—
119C2×C3×C4C2×D3×C718yescomplemented
120C2×Dic3—38no—
121C4×D3—616no—
122C3×C7D3×C22⋊C414yescomplemented
123C2×C32—22no—
124C2×C32C22×C4×C711yescharacteristic
125C2×C32C2×C7×D412yes—
126C3⋊D3—22no—
127C3⋊D3C7×C22⋊C414yesdirect product factor
128C24—91no—
129C22×C4—92no—
130C22⋊C4—91no—
131C22⋊C4—92no—
132C22⋊C4C7×C3⋊D311yescharacteristic, direct product factor
133C2×C7—22no—
134C2×C7—94no—
135C2×C7—182no—
136C2×C7C2×C4×C3⋊D311yescharacteristic
137C2×C7D4×C3⋊D312yes—
138C22×C3—28no—
139C22×C3C4×D3×C718yescomplemented
140C22×C3C22×D3×C714yes—
141C2×D3—38no—
142C2×D3—316no—
143C2×D3—38no—
144C2×D3—632no—
145C2×D3—68no—
146C3×C4—28no—
147Dic3—68no—
148C32C2×C7×C22⋊C411yescharacteristic, complemented, Sylow
149C23—91no—
150C23—94no—
151C23—91no—
152C23—184no—
153C23C2×C7×C3⋊D311yescharacteristic
154C2×C4—92no—
155C2×C4—184no—
156C2×C4C2×C7×C3⋊D312yescomplemented
157C7C22⋊C4×C3⋊D311yescharacteristic, direct product factor, Sylow
158C2×C3—28no—
159C2×C3C2×C4×D3×C714yes—
160C2×C3D3×C7×D418yes—
161D3—316no—
162D3—68no—
163C22—22no—
164C22—92no—
165C22—94no—
166C22—92no—
167C22—188no—
168C22—182no—
169C22C4×C7×C3⋊D312yescomplemented
170C22C22×C7×C3⋊D311yescharacteristic
171C4—22no—
172C4—182no—
173C3D3×C7×C22⋊C414yescomplemented
174C2—22no—
175C2—94no—
176C2—182no—
177C2C2×C4×C7×C3⋊D311yescharacteristic
178C2C7×D4×C3⋊D312yes—
subgroupG/H
centerC22×C7C22×C3⋊D3
commutatorC2×C32C22×C4×C7
FrattiniC22C22×C7×C3⋊D3
FittingC32×C7×C22⋊C4C2
socleC22×C32×C7C23
prime radicalC23×C7×C3⋊D3C2
cosocleC22×C32C23×C7
hypercenterC7×C22⋊C4C3⋊D3
hypocenterC32C2×C7×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C7 ◁ C7×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C7×C3⋊D3C2semidirect
3. C32×C7×C22⋊C4C2semidirect
4. C7×#3757C2semidirect
5. C7×#3752C2semidirect
9. C22×C7×C3⋊D3C4semidirect
12. C2×C32×C4×C7C22semidirect
13. C2×C7×C3⋊Dic3C22semidirect
17. C3×C7×C22⋊C4D3semidirect
20. C22⋊C4×C3⋊D3C7direct (N × H)
22. C22×C32×C7C2×C4semidirect
27. C2×C7×C3⋊D3C2×C4semidirect
37. C2×C3×C4×C7C2×D3semidirect
41. C2×C4×C3⋊D3C2×C7semidirect
42. C32×C22⋊C4C2×C7semidirect
43. #3757C2×C7semidirect
44. #3752C2×C7semidirect
49. C7×C3⋊D3C22⋊C4direct (N × H)
54. C7×C22⋊C4C3⋊D3direct (N × H)
57. C22×C3×C7C4×D3semidirect
68. C22×C3⋊D3C4×C7semidirect
71. C2×C32×C4C22×C7semidirect
72. C2×C3⋊Dic3C22×C7semidirect
74. C32×C7C2×C22⋊C4semidirect
82. C2×C4×C7C2×C3⋊D3semidirect
85. C3×C22⋊C4D3×C7semidirect
94. C22×C32C2×C4×C7semidirect
99. C2×C3⋊D3C2×C4×C7semidirect
110. C22×C7C4×C3⋊D3semidirect
119. C2×C3×C4C2×D3×C7semidirect
122. C3×C7D3×C22⋊C4semidirect
127. C3⋊D3C7×C22⋊C4direct (N × H)
132. C22⋊C4C7×C3⋊D3direct (N × H)
139. C22×C3C4×D3×C7semidirect
148. C32C2×C7×C22⋊C4semidirect
156. C2×C4C2×C7×C3⋊D3semidirect
157. C7C22⋊C4×C3⋊D3direct (N × H)
169. C22C4×C7×C3⋊D3semidirect
173. C3D3×C7×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×C7×C3⋊D32
C2×C4×C7×C3⋊D32
C32×C7×C22⋊C42
C7×#37572
C7×#37522
D3×C7×C22⋊C43
C22⋊C4×C3⋊D37