Finite Group Databasebeta

← #27156680

Subgroups — #27156680

Order 12416.

158 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 36 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#20261578C211yesmaximal, characteristic
2#20261578C211yesmaximal, characteristic
3#20260610C211yesmaximal, characteristic
4C22⋊#11715897—21no—
5C22⋊#11715897—21no—
6C23×Dic97C411yescharacteristic
7C2×C4⋊Dic97—21no—
8C2×C22⋊Dic97C2211yescharacteristic
9#15690717—21no—
10C2×C22⋊C4×C97C411yescharacteristic
11C2×#11715897—41no—
12C2×#11715897—41no—
13C22×Dic97—21no—
14C22×Dic97—42no—
15C22×Dic97—41no—
16C22×Dic97D411yescharacteristic
17C22×Dic97D411yescharacteristic
18C4⋊Dic97—42no—
19C22⋊Dic97—41no—
20C24×C97C2×C411yescharacteristic
21C22×C4×C97—21no—
22C22⋊C4×C97—21no—
23C22⋊C4×C97C812yescomplemented
24C23×C97—21no—
25C23×C97—21no—
26C23×C97—41no—
27C23×C97—41no—
28C23×C97C22⋊C411yescharacteristic
29C23×C97C2×C811yescharacteristic
30C23×C97M1611yescharacteristic
31C2×C4×C97—22no—
32C2×C4×C97—41no—
33C2×C4×C97—41no—
34C2×Dic97—22no—
35C2×Dic97—21no—
36C2×Dic97—21no—
37C2×Dic97—41no—
38C2×Dic97—42no—
39C2×Dic97—41no—
40C2×Dic97—41no—
41C2×Dic97—41no—
42C2×Dic97—42no—
43#11715897—81no—
44#11715897—81no—
45C22×C97—21no—
46C22×C97—21no—
47C22×C97—21no—
48C22×C97—21no—
49C22×C97—41no—
50C22×C97—41no—
51C22×C97—42no—
52C22×C97—41no—
53C22×C97—41no—
54C22×C97C22⋊C811yescharacteristic
55C22×C97#9911yescharacteristic
56C22×C97#10111yescharacteristic
57C4×C97—42no—
58Dic97—41no—
59Dic97—42no—
60Dic97—41no—
61C2×C97—21no—
62C2×C97—21no—
63C2×C97—41no—
64C2×C97—41no—
65C2×C97#32411yescharacteristic
66C2×C97AΣL(1,𝔽16)11yescharacteristic
67C2×C97#35211yescharacteristic
68#1326—971nomaximal, Sylow
69C97#132611yescharacteristic, complemented, Sylow
70#385—971no—
71#323—971no—
72#323—971no—
73C23×C4—971no—
74C2×C4⋊C4—1941no—
75C2×C22⋊C4—971no—
76C2×C22⋊C4#8760411yescharacteristic
77C22⋊C8—1941no—
78C22⋊C8—1941no—
79#95—1941no—
80C24C2×#8760411yescharacteristic
81C22×C4—21no—
82C22×C4—971no—
83C22×C4—971no—
84C22×C4—3881no—
85C22×C4—3882no—
86C22×C4—1941no—
87C2×C8—3881no—
88C2×C8—3881no—
89C4⋊C4—3882no—
90C22⋊C4—21no—
91C22⋊C4—3881no—
92C22⋊C4#1171589712yescomplemented
93C23—21no—
94C23—21no—
95C23—41no—
96C23—41no—
97C23C22⋊#8760411yescharacteristic
98C23C2×#1171589711yescharacteristic
99C23#1309552011yescharacteristic
100C2×C4—22no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—3881no—
104C2×C4—3882no—
105C2×C4—3881no—
106C2×C4—3881no—
107C2×C4—3881no—
108C2×C4—3882no—
109C2×C4—1942no—
110C2×C4—1941no—
111C2×C4—1941no—
112C8—7761no—
113C8—7761no—
114C22—21no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—42no—
121C22—41no—
122C22—41no—
123C22C22⋊#1171589711yescharacteristic
124C22#1569086911yescharacteristic
125C22#1569087111yescharacteristic
126C4—42no—
127C4—3881no—
128C4—3882no—
129C4—3881no—
130C2—21no—
131C2—21no—
132C2—41no—
133C2—41no—
134C2#2026158011yescharacteristic
135C2#2026163611yescharacteristic
136C2#2026163811yescharacteristic
subgroupG/H
centerC22#15690869
commutatorC23×C97C2×C8
FrattiniC24C2×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97#99
prime radicalC24×C97C2×C4
cosocleC2×C22⋊Dic97C22
hypercenterC2×C22⋊C4#87604
hypocenterC97#1326
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
23. C22⋊C4×C97C8semidirect
69. C97#1326semidirect
92. C22⋊C4#11715897semidirect

Maximal subgroups

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

subgroupindex
#202615782
#202606102
#132697