Finite Group Databasebeta

← #28399034

Subgroups — #28399034

Order 14464.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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×#16813667C211yesmaximal, characteristic
2#20299073C211yesmaximal, characteristic, complemented
3#20298571C214yesmaximal, complemented
4C2×#16813536C218yesmaximal, complemented
5#518×C113C211yesmaximal, characteristic, complemented
6#16813667—24no—
7#16813667—24no—
8C2×#14268647C2212yes—
9C2×#14268646C414yescomplemented
10C22×C22⋊D113C2214yes—
11C2×D4×D113C414yescomplemented
12D4×Dic113—24no—
13#16813581C2212yescomplemented
14C2×C22⋊Dic113C2214yescomplemented
15#16813536C22116yescomplemented
16C22×D4×C113C2211yescharacteristic
17C4×D4×C113—24no—
18#117×C113C2212yescomplemented
19C2×C22⋊C4×C113C2214yescomplemented
20C23×D113C2×C414yescomplemented
21#14268647—24no—
22#14268647—24no—
23#14268646—216no—
24C2×Dic226C2×C412yescomplemented
25C2×D452C2×C412yescomplemented
26C2×C22⋊D113—28no—
27C2×C22⋊D113—28no—
28C2×C22⋊D113C2×C418yescomplemented
29C2×C22⋊D113C2314yes—
30D4×D113—216no—
31C22×Dic113—24no—
32C22×Dic113C2×C414yescomplemented
33C2×C4×D113C2×C414yescomplemented
34C4⋊Dic113—22no—
35C22⋊Dic113—24no—
36C22⋊Dic113C2314yes—
37C4×Dic113—22no—
38C24×C113C2312yes—
39C2×D4×C113C2×C414yes—
40C2×D4×C113D414yes—
41C2×D4×C113D414yes—
42C22×C4×C113—24no—
43C22×C4×C113C2311yescharacteristic
44C4⋊C4×C113—22no—
45C22⋊C4×C113—24no—
46C22⋊C4×C113C2314yes—
47C42×C113—22no—
48C23×C113—82no—
49C23×C113C22×C414yes—
50C23×C113C2×D414yes—
51C23×C113C2×D414yes—
52C23×C113C2411yescharacteristic
53C22×D113—28no—
54C22×D113—84no—
55C22×D113C22×C414yes—
56D4×C113C22⋊C4116yes—
57C2×C4×C113—24no—
58C2×C4×C113—84no—
59C2×C4×C113C22×C412yes—
60C2×C4×C113C2×D412yes—
61C2×C4×C113C2×D412yes—
62C22⋊D113—216no—
63C22⋊D113—24no—
64C22⋊D113—24no—
65C2×Dic113—28no—
66C2×Dic113—24no—
67C2×Dic113—84no—
68C2×Dic113C22×C414yes—
69D452—24no—
70C4×D113—28no—
71Dic226—24no—
72C22×C113—44no—
73C22×C113—44no—
74C22×C113—44no—
75C22×C113—41no—
76C22×C113C2×C22⋊C418yes—
77C22×C113C23×C411yescharacteristic
78C22×C113C22×D411yescharacteristic
79C22×C113C22×D411yescharacteristic
80C2×D113—24no—
81C2×D113—44no—
82C2×D113—88no—
83C4×C113—44no—
84C4×C113C2×C22⋊C414yes—
85Dic113—24no—
86Dic113—44no—
87C2×C113—21no—
88C2×C113—21no—
89C2×C113—21no—
90C2×C113—28no—
91C2×C113C22×C22⋊C411yescharacteristic
92D113—44no—
93#2883—1131nomaximal, Sylow
94C113#288311yescharacteristic, complemented, Sylow
95C2×H(5,𝔽2)—1131no—
96#518—1131no—
97#518D11311yescharacteristic, complemented
98#410—1134no—
99C2×#99—1138no—
100H(5,𝔽2)—2264no—
101H(5,𝔽2)—2264no—
102C2×#41—1134no—
103C2×#41—1132no—
104C22×D4—1134no—
105C22×D4—1134no—
106C22×D4C2×D11311yescharacteristic
107C4×D4—24no—
108C4×D4—2264no—
109#117—1132no—
110#117C2×D11312yescomplemented
111C2×C22⋊C4—1134no—
112C2×C22⋊C4C2×D11314yescomplemented
113#99—11316no—
114C24—1134no—
115C24C22×D11312yes—
116#41—22616no—
117#41—2264no—
118#41—2264no—
119C2×Q8—1132no—
120C2×D4—1138no—
121C2×D4—1134no—
122C2×D4—1132no—
123C2×D4—2268no—
124C2×D4—2268no—
125C2×D4—22616no—
126C2×D4C4×D11314yes—
127C2×D4D45214yes—
128C2×D4C22⋊D11314yes—
129C22×C4—24no—
130C22×C4—1134no—
131C22×C4—1134no—
132C22×C4—2264no—
133C22×C4C22×D11311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—2262no—
136C22⋊C4—24no—
137C22⋊C4—1134no—
138C22⋊C4—2264no—
139C22⋊C4C22×D11314yes—
140C42—22no—
141C42—2262no—
142C23—82no—
143C23—1134no—
144C23—9044no—
145C23—2268no—
146C23C2×C4×D11314yes—
147C23C2×C22⋊D11314yes—
148C23C2×D45214yes—
149C23C23×D11311yescharacteristic
150Q8—2264no—
151D4—22616no—
152D4—2264no—
153D4—2264no—
154D4—2264no—
155D4#14268624116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—1134no—
159C2×C4—9044no—
160C2×C4—2264no—
161C2×C4—2268no—
162C2×C4—2268no—
163C2×C4C2×C4×D11312yes—
164C2×C4C2×C22⋊D11312yes—
165C2×C4C2×D45212yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—9048no—
171C22—4524no—
172C22—2264no—
173C22C2×#1426862418yes—
174C22C22×C4×D11311yescharacteristic
175C22C22×C22⋊D11311yescharacteristic
176C22C22×D45211yescharacteristic
177C4—44no—
178C4—4524no—
179C4—2264no—
180C4C2×#1426862414yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—4524no—
186C2C22×#1426862411yescharacteristic
subgroupG/H
centerC2C22×#14268624
commutatorC22×C113C23×C4
FrattiniC23C23×D113
Fitting#518×C113C2
socleC2×C113C22×C22⋊C4
prime radicalC2×#16813667C2
cosocleC23×C113C24
hypercenter#518D113
hypocenterC113#2883
LCSG ▷ C22×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20299073C2semidirect
3. #20298571C2semidirect
4. C2×#16813536C2semidirect
5. #518×C113C2semidirect
9. C2×#14268646C4semidirect
11. C2×D4×D113C4semidirect
13. #16813581C22semidirect
14. C2×C22⋊Dic113C22semidirect
15. #16813536C22semidirect
18. #117×C113C22semidirect
19. C2×C22⋊C4×C113C22semidirect
20. C23×D113C2×C4semidirect
24. C2×Dic226C2×C4semidirect
25. C2×D452C2×C4semidirect
28. C2×C22⋊D113C2×C4semidirect
32. C22×Dic113C2×C4semidirect
33. C2×C4×D113C2×C4semidirect
94. C113#2883semidirect
97. #518D113semidirect
110. #117C2×D113semidirect
112. C2×C22⋊C4C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136672
#202990732
#202985712
C2×#168135362
#518×C1132
#2883113