Finite Group Databasebeta

← #123×D113 #20299098

Subgroups — #123×D113 #20299098

Order 7232.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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×D113C211yesmaximal, characteristic
2C22×C4×D113C211yesmaximal, characteristic, complemented
3#16813607C212yesmaximal, complemented
4#16813606C211yesmaximal, characteristic, complemented
5#16813605C211yesmaximal, characteristic, complemented
6#16813604C212yesmaximal, complemented
7#16813603C211yesmaximal, characteristic, complemented
8C4⋊C4×D113C212yesmaximal, complemented
9C22⋊C4×D113C212yesmaximal, complemented
10C22⋊C4×D113C211yesmaximal, characteristic, complemented
11#123×C113C211yesmaximal, characteristic, complemented
12C23×D113C2211yescharacteristic
13C23×D113C2211yescharacteristic
14C2×D452C2211yescharacteristic
15C2×C22⋊D113C2211yescharacteristic
16C2×C22⋊D113C2211yescharacteristic
17D4×D113—24no—
18C22×Dic113C2211yescharacteristic, complemented
19C2×C4×D113—24no—
20C2×C4×D113C2212yes—
21C2×C4×D113C2212yes—
22C2×C4×D113C2211yescharacteristic
23C2×C4×D113C2212yescomplemented
24C4⋊Dic113C2212yescomplemented
25#14268626C2212yescomplemented
26#14268626C2212yescomplemented
27C22⋊Dic113C2212yescomplemented
28C22⋊Dic113C2211yescharacteristic, complemented
29#14268624C2212yescomplemented
30#14268624C2212yescomplemented
31#14268624C2212yescomplemented
32C2×D4×C113C2211yescharacteristic
33C22×C4×C113C2211yescharacteristic, complemented
34C4⋊C4×C113C2212yescomplemented
35C22⋊C4×C113C2212yescomplemented
36C22⋊C4×C113C2211yescharacteristic, complemented
37C23×C113C2311yescharacteristic
38C23×C113C2311yescharacteristic
39C22×D113—24no—
40C22×D113—22no—
41C22×D113—24no—
42C22×D113D414yes—
43C22×D113C2311yescharacteristic
44C22×D113C2311yescharacteristic
45C22×D113C2311yescharacteristic
46D4×C113—22no—
47C2×C4×C113—22no—
48C2×C4×C113C2312yes—
49C2×C4×C113C2312yes—
50C2×C4×C113C2311yescharacteristic
51C22⋊D113—22no—
52C22⋊D113—22no—
53C2×Dic113—22no—
54C2×Dic113C2312yes—
55C2×Dic113C2312yes—
56C2×Dic113C2311yescharacteristic
57D452—22no—
58C4×D113—24no—
59C4×D113—24no—
60C4×D113—22no—
61C4×D113—24no—
62C22×C113—22no—
63C22×C113—21no—
64C22×C113—22no—
65C22×C113C2×D412yes—
66C22×C113C2411yescharacteristic
67C2×D113—22no—
68C2×D113—28no—
69C2×D113—21no—
70C2×D113—22no—
71C2×D113—44no—
72C2×D113C2×D412yes—
73C2×D113C2×D412yes—
74C2×D113#4114yes—
75C4×C113—22no—
76C4×C113—21no—
77C4×C113—22no—
78Dic113—22no—
79Dic113—21no—
80Dic113—22no—
81C2×C113—22no—
82C2×C113—41no—
83C2×C113C22×D411yescharacteristic
84C2×C113C2×#4112yes—
85D113—22no—
86D113—41no—
87D113#12314yesdirect product factor
88C113C2×#12311yescharacteristic, complemented, Sylow
89C2×#123—1131nomaximal, Sylow
90C22×D4—1131no—
91C23×C4—1131no—
92#123—1131no—
93#123—1131no—
94#123—1132no—
95#123—1132no—
96#123—1131no—
97#123D11311yescharacteristic, direct product factor
98C2×C4⋊C4—1132no—
99C2×C22⋊C4—1131no—
100C2×C22⋊C4—1132no—
101C24—1131no—
102C24—1131no—
103C2×D4—1131no—
104C2×D4—1131no—
105C2×D4—1131no—
106C2×D4—2264no—
107C2×D4C2×D11311yescharacteristic
108C22×C4—1131no—
109C22×C4—1132no—
110C22×C4—1132no—
111C22×C4—1131no—
112C22×C4—1132no—
113C22×C4—2264no—
114C22×C4C2×D11311yescharacteristic, complemented
115C4⋊C4—1132no—
116C4⋊C4—1132no—
117C4⋊C4—1132no—
118C4⋊C4C2×D11312yescomplemented
119C22⋊C4—1131no—
120C22⋊C4—1132no—
121C22⋊C4—1132no—
122C22⋊C4—1132no—
123C22⋊C4—1132no—
124C22⋊C4C2×D11311yescharacteristic, complemented
125C22⋊C4C2×D11312yescomplemented
126C23—1131no—
127C23—1134no—
128C23—1131no—
129C23—1131no—
130C23—2262no—
131C23—2264no—
132C23—2264no—
133C23C22×D11311yescharacteristic
134C23C22×D11311yescharacteristic
135D4—22no—
136D4—2262no—
137D4—2262no—
138D4—2262no—
139C2×C4—22no—
140C2×C4—1132no—
141C2×C4—1132no—
142C2×C4—1131no—
143C2×C4—2264no—
144C2×C4—2264no—
145C2×C4—2262no—
146C2×C4—2264no—
147C2×C4—2262no—
148C2×C4C22×D11312yes—
149C2×C4C22×D11312yes—
150C2×C4C22×D11311yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—1132no—
155C22—1134no—
156C22—1132no—
157C22—4524no—
158C22—2268no—
159C22—2261no—
160C22—2262no—
161C22—2262no—
162C22D4×D11312yes—
163C22C23×D11311yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—2262no—
168C4—2261no—
169C4—2262no—
170C2—22no—
171C2—41no—
172C2—1134no—
173C2—4521no—
174C2—2262no—
175C2C2×D4×D11311yescharacteristic
176C2#41×D11312yes—
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
Fitting#123×C113C2
hypercenter#123D113
hypocenterC113C2×#123
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D113C2semidirect
3. #16813607C2semidirect
4. #16813606C2semidirect
5. #16813605C2semidirect
6. #16813604C2semidirect
7. #16813603C2semidirect
8. C4⋊C4×D113C2semidirect
9. C22⋊C4×D113C2semidirect
10. C22⋊C4×D113C2semidirect
11. #123×C113C2semidirect
18. C22×Dic113C22semidirect
23. C2×C4×D113C22semidirect
24. C4⋊Dic113C22semidirect
25. #14268626C22semidirect
26. #14268626C22semidirect
27. C22⋊Dic113C22semidirect
28. C22⋊Dic113C22semidirect
29. #14268624C22semidirect
30. #14268624C22semidirect
31. #14268624C22semidirect
33. C22×C4×C113C22semidirect
34. C4⋊C4×C113C22semidirect
35. C22⋊C4×C113C22semidirect
36. C22⋊C4×C113C22semidirect
87. D113#123direct (N × H)
88. C113C2×#123semidirect
97. #123D113direct (N × H)
114. C22×C4C2×D113semidirect
118. C4⋊C4C2×D113semidirect
124. C22⋊C4C2×D113semidirect
125. C22⋊C4C2×D113semidirect