Finite Group Databasebeta

← #27157219

Subgroups — #27157219

Order 12416.

196 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261757C212yesmaximal
2#20261592C214yesmaximal, complemented
3#125×D97C211yesmaximal, characteristic
4C4⋊#11715897—22no—
5#15690867—22no—
6#15690865C2214yes—
7#15690802C411yescharacteristic
8#15690801C2212yes—
9#15690800—22no—
10C4⋊C4×D97—22no—
11C4⋊C4×D97C412yes—
12C42×D97C2211yescharacteristic
13#125×C97C411yescharacteristic
14C2×#11715897—24no—
15C2×C4×D97—22no—
16C2×C4×D97C2×C411yescharacteristic
17C2×C4×D97D412yes—
18C4⋊Dic97—22no—
19C4⋊Dic97C2×C412yes—
20#13095490—22no—
21#13095490—22no—
22#13095490—22no—
23C4×Dic97D412yes—
24C4×Dic97C2311yescharacteristic
25C4⋊C4×C97—22no—
26C4⋊C4×C97C2×C412yes—
27C42×C97C2×C411yescharacteristic
28C22×D97C22⋊C411yescharacteristic
29C2×C4×C97—22no—
30C2×C4×C97C22⋊C412yes—
31C2×C4×C97C22×C411yescharacteristic
32C2×Dic97—22no—
33C2×Dic97C22⋊C411yescharacteristic
34C2×Dic97C2×D412yes—
35C4×D97—22no—
36C4×D97—42no—
37C4×D97—44no—
38#11715897—44no—
39C22×C97C2×C22⋊C411yescharacteristic
40C2×D97—22no—
41C2×D97#10112yes—
42C4×C97—21no—
43C4×C97—22no—
44C4×C97—42no—
45Dic97—21no—
46Dic97—22no—
47Dic97—42no—
48C2×C97C2×#10111yescharacteristic
49C2×C97#42112yes—
50D97—22no—
51#1534—971nomaximal, Sylow
52C97#153411yescharacteristic, complemented, Sylow
53C2×#125—971no—
54#432—972no—
55#330—974no—
56#125—972no—
57#125—971no—
58#125—1942no—
59#125#8760411yescharacteristic
60C2×C4⋊C4—972no—
61C2×C4⋊C4—1942no—
62C2×C42—971no—
63C4⋊C8—1942no—
64C22⋊C8—1942no—
65#97—974no—
66C22×C4—972no—
67C22×C4—971no—
68C22×C4—1942no—
69C2×C8—1944no—
70C4⋊C4—22no—
71C4⋊C4—972no—
72C4⋊C4—1942no—
73C4⋊C4—1942no—
74C4⋊C4—1942no—
75C4⋊C4—1942no—
76C4⋊C4C2×#8760412yes—
77C42—972no—
78C42—971no—
79C42C2×#8760411yescharacteristic
80C23—971no—
81C2×C4—22no—
82C2×C4—972no—
83C2×C4—971no—
84C2×C4—3882no—
85C2×C4—3884no—
86C2×C4—1942no—
87C2×C4—1942no—
88C2×C4C22⋊#8760412yes—
89C2×C4C22×#8760411yescharacteristic
90C8—3884no—
91C22—972no—
92C22—1942no—
93C22C2×C22⋊#8760411yescharacteristic
94C4—21no—
95C4—22no—
96C4—42no—
97C4—3882no—
98C4—1941no—
99C4—1942no—
100C2—1942no—
101C2D97⋊#10111yescharacteristic
102C2#2026173212yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenter#125#87604
hypocenterC97#1534
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
2. #20261592C2semidirect
52. C97#1534semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202615922
#125×D972
#153497