Finite Group Databasebeta

← #28389657

Subgroups — #28389657

Order 14464.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#20298972C211yesmaximal, characteristic, complemented
2#20298793C212yesmaximal, complemented
3#20298347C211yesmaximal, characteristic, complemented
4D4⋊#11744751C212yesmaximal, complemented
5#555×C113C211yesmaximal, characteristic, complemented
6C2×#14268634—22no—
7C2×#14268632—22no—
8#16813619C2211yescharacteristic
9C4⋊#11744751C2212yes—
10#16813542—22no—
11C4×#11744751C2211yescharacteristic
12C4⋊Q8×C113C2211yescharacteristic
13#124×C113—22no—
14#121×C113—22no—
15C4×D4×C113C2212yes—
16#117×C113—22no—
17C2×D452—22no—
18C2×D452—21no—
19#14268634—44no—
20#14268632—44no—
21C2×#11744751—21no—
22C2×#11744751—22no—
23C2×Q8×C113D412yes—
24C2×D4×C113—22no—
25C2×D4×C113D412yes—
26C22×C4×C113—22no—
27C4⋊C4×C113—22no—
28C4⋊C4×C113D412yes—
29C22⋊C4×C113—24no—
30C22⋊C4×C113—22no—
31C42×C113—22no—
32C42×C113C2311yescharacteristic
33C23×C113—22no—
34C22×D113—41no—
35Q8×C113—42no—
36D4×C113—22no—
37D4×C113—22no—
38D4×C113—82no—
39C2×C4×C113—22no—
40C2×C4×C113—22no—
41C2×C4×C113—42no—
42C2×C4×C113—42no—
43C2×C4×C113C2×D411yescharacteristic
44C2×C4×C113C2×D412yes—
45D452—42no—
46D452—42no—
47D452—42no—
48#11744751—22no—
49#11744751—42no—
50C22×C113—42no—
51C22×C113—42no—
52C22×C113—42no—
53C22×C113#12011yescharacteristic
54C2×D113—82no—
55C2×D113—81no—
56C4×C113—21no—
57C4×C113—42no—
58C4×C113—42no—
59C4×C113#13512yes—
60C4×C113AGL(1,ℤ/8)12yes—
61C2×C113—82no—
62C2×C113#44912yes—
63C2×C113#45311yescharacteristic
64D113—161no—
65#1624—1131nomaximal, Sylow
66C113#162411yescharacteristic, complemented, Sylow
67#555D11311yescharacteristic, complemented
68#492—1131no—
69#459—1132no—
70#333—1131no—
71D4⋊C8—1132no—
72C2×SD16—2262no—
73C2×D8—2262no—
74C4⋊Q8C2×D11311yescharacteristic
75#127—1131no—
76#124—22no—
77#121—22no—
78C4×D4C2×D11312yes—
79#117—22no—
80C4⋊C8—1132no—
81#102—2262no—
82C4×C8—1131no—
83C2×Q8C22⋊D11312yes—
84C2×D4—22no—
85C2×D4—2262no—
86C2×D4—2261no—
87C2×D4C22⋊D11312yes—
88C22×C4—22no—
89SD16—4524no—
90D8—4524no—
91C2×C8—2262no—
92C2×C8—2261no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D11312yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D11311yescharacteristic
99C23—22no—
100C23—4521no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—4522no—
106D4—4522no—
107D4—4522no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D11311yescharacteristic
113C2×C4C2×C22⋊D11312yes—
114C8—4522no—
115C8—2262no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—9041no—
120C22—9042no—
121C22#1681359111yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1681364212yes—
126C4#1681364712yes—
127C2—82no—
128C2—18081no—
129C2#2029874312yes—
130C2#2029876611yescharacteristic
subgroupG/H
centerC22#16813591
commutator, cosocleC42×C113C23
FrattiniC42C22×D113
Fitting#555×C113C2
socleC22×C113#120
hypercenter#555D113
hypocenterC113#1624
LCSG ▷ C42×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20298972C2semidirect
2. #20298793C2semidirect
3. #20298347C2semidirect
4. D4⋊#11744751C2semidirect
5. #555×C113C2semidirect
66. C113#1624semidirect
67. #555D113semidirect

Maximal subgroups

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

subgroupindex
#202989722
#202987932
#202983472
D4⋊#117447512
#555×C1132
#1624113