Finite Group Databasebeta

← #28393767

Subgroups — #28393767

Order 14464.

456 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16813616C211yesmaximal, characteristic
2#20298493C213yesmaximal, complemented
3#20298488C213yesmaximal
4#20298479C213yesmaximal, complemented
5#20298469C213yesmaximal
6#20298466C211yesmaximal, characteristic, complemented
7#384×C113C211yesmaximal, characteristic, complemented
8C22×C4×D113C2211yescharacteristic
9#16813616C418yescomplemented
10C2×#14268626C2213yes—
11C2×#14268626C2213yes—
12C2×#14268624C2213yes—
13C2×#14268624C2213yes—
14C2×C4×Dic113C2213yes—
15#16813529C2213yes—
16#16813529C2213yes—
17#16813529C2213yes—
18#16813529C2213yes—
19C2×C4⋊C4×C113C2213yes—
20C2×C4⋊C4×C113C2213yes—
21C2×C42×C113C2211yescharacteristic
22C23×D113C2311yescharacteristic
23C22×Dic113C2313yes—
24C22×Dic113C2311yescharacteristic
25C22×Dic113C2313yes—
26C2×C4×D113—43no—
27#14268626—43no—
28#14268626C2×C4112yescomplemented
29#14268624—43no—
30#14268624C2×C4112yes—
31C4×Dic113—43no—
32C22×C4×C113C2313yes—
33C22×C4×C113C2311yescharacteristic
34C22×C4×C113C2313yes—
35C4⋊C4×C113—43no—
36C4⋊C4×C113—43no—
37C42×C113C2×C414yescomplemented
38C23×C113C2411yescharacteristic
39C22×D113—23no—
40C22×D113—23no—
41C22×D113C22×C412yes—
42C2×C4×C113—23no—
43C2×C4×C113—23no—
44C2×C4×C113—23no—
45C2×C4×C113—23no—
46C2×C4×C113—23no—
47C2×C4×C113—23no—
48C2×C4×C113C22×C416yes—
49C2×Dic113—23no—
50C2×Dic113—23no—
51C2×Dic113—23no—
52C2×Dic113—23no—
53C2×Dic113—26no—
54C2×Dic113C22×C416yes—
55C4×D113—44no—
56C22×C113C23×C411yescharacteristic
57C22×C113H(5,𝔽2)13yes—
58C22×C1132−1+413yes—
59C2×D113—26no—
60C2×D113—41no—
61C2×D113—43no—
62C4×C113—26no—
63C4×C113—41no—
64C4×C113—43no—
65Dic113—26no—
66Dic113—41no—
67Dic113—43no—
68C2×C113#52013yes—
69C2×C113#56011yescharacteristic
70C2×C113#56313yes—
71D113—42no—
72#2383—1131nomaximal, Sylow
73C113#238311yescharacteristic, complemented, Sylow
74C2×#126—1131no—
75#389—1133no—
76#388—1133no—
77#387—1133no—
78#385—1133no—
79#384D11311yescharacteristic, complemented
80#383—1131no—
81C23×C4—1131no—
82#126—1138no—
83C2×C4⋊C4—1133no—
84C2×C4⋊C4—1133no—
85C2×C4⋊C4C2×D11313yes—
86C2×C4⋊C4C2×D11313yes—
87C2×C22⋊C4—1133no—
88C2×C22⋊C4—1133no—
89C2×C42—1133no—
90C2×C42C2×D11311yescharacteristic
91#95—1133no—
92#95—1133no—
93#95—1133no—
94#95—1133no—
95C24—1131no—
96C22×C4—1133no—
97C22×C4—1133no—
98C22×C4—1131no—
99C22×C4—4523no—
100C22×C4C22×D11311yescharacteristic
101C22×C4C22×D11313yes—
102C22×C4C22×D11313yes—
103C4⋊C4—43no—
104C4⋊C4—43no—
105C4⋊C4—11312no—
106C4⋊C4—4523no—
107C22⋊C4—11312no—
108C22⋊C4—4523no—
109C42—4523no—
110C42C4×D11314yescomplemented
111C23—1132no—
112C23—2263no—
113C23—2263no—
114C23C23×D11311yescharacteristic
115C2×C4—23no—
116C2×C4—23no—
117C2×C4—23no—
118C2×C4—23no—
119C2×C4—23no—
120C2×C4—23no—
121C2×C4—1136no—
122C2×C4—4524no—
123C2×C4—2263no—
124C2×C4—2263no—
125C2×C4—2266no—
126C2×C4—2263no—
127C2×C4—2263no—
128C2×C4C2×C4×D11316yes—
129C22—4521no—
130C22—4523no—
131C22—2266no—
132C22C22×C4×D11311yescharacteristic
133C22#1681366713yes—
134C22#1681367013yes—
135C4—26no—
136C4—41no—
137C4—43no—
138C4—4521no—
139C4—4523no—
140C4—2266no—
141C2—4522no—
142C2#2029907913yes—
143C2#2029933511yescharacteristic
144C2#2029934713yes—
subgroupG/H
center, FrattiniC23C23×D113
commutatorC22×C113C23×C4
Fitting#384×C113C2
socle, cosocleC23×C113C24
prime radicalC23×D113C23
hypercenter#384D113
hypocenterC113#2383
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #20298493C2semidirect
4. #20298479C2semidirect
6. #20298466C2semidirect
7. #384×C113C2semidirect
9. #16813616C4semidirect
28. #14268626C2×C4semidirect
37. C42×C113C2×C4semidirect
73. C113#2383semidirect
79. #384D113semidirect
110. C42C4×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136162
#202984932
#202984882
#202984792
#202984692
#202984662
#384×C1132
#2383113