Finite Group Databasebeta

← #27156720

Subgroups — #27156720

Order 12416.

124 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 30 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#20261615C211yesmaximal, characteristic
2#20261581C211yesmaximal, characteristic
3#20260666C211yesmaximal, characteristic
4C2×#13095520—21no—
5C22⋊#11715897—21no—
6C2×#13095490—21no—
7C2×#13095490C2211yescharacteristic
8#15690717—21no—
9#15690717C411yescharacteristic
10C2×C4⋊C4×C97C411yescharacteristic
11#13095520—22no—
12#13095520—41no—
13C2×#11715897—41no—
14C2×#11715897—41no—
15C22×Dic97—21no—
16C22×Dic97D411yescharacteristic
17C22×Dic97D411yescharacteristic
18#13095490—41no—
19#13095490—41no—
20#13095490—41no—
21C22×C4×C97—21no—
22C22×C4×C97C2×C411yescharacteristic
23C4⋊C4×C97—41no—
24C23×C97C22⋊C411yescharacteristic
25C2×C4×C97—21no—
26C2×C4×C97—41no—
27C2×C4×C97—41no—
28C2×C4×C97—41no—
29C2×C4×C97—41no—
30C2×Dic97—21no—
31C2×Dic97—21no—
32C2×Dic97—22no—
33C2×Dic97—21no—
34C2×Dic97—41no—
35C2×Dic97—41no—
36C2×Dic97—41no—
37C2×Dic97D811yescharacteristic
38C2×Dic97SD1611yescharacteristic
39#11715897—42no—
40#11715897—81no—
41C22×C97—21no—
42C22×C97—21no—
43C22×C97#9911yescharacteristic
44C22×C97#10211yescharacteristic
45C22×C97#10411yescharacteristic
46C4×C97—41no—
47C4×C97—81no—
48Dic97—21no—
49Dic97—21no—
50Dic97—42no—
51Dic97—41no—
52C2×C97—21no—
53C2×C97—21no—
54C2×C97#32711yescharacteristic
55C2×C97#35211yescharacteristic
56C2×C97#35611yescharacteristic
57#1346—971nomaximal, Sylow
58C97#134611yescharacteristic, complemented, Sylow
59#398—971no—
60#343—971no—
61#324—971no—
62C2×M16—1941no—
63C2×C4⋊C4—971no—
64C2×C4⋊C4—1941no—
65C2×C4⋊C4#8760411yescharacteristic
66C22⋊C8—1941no—
67#95—971no—
68#95—1941no—
69C22×C4—21no—
70C22×C4—971no—
71C22×C4—971no—
72C22×C4—1941no—
73C22×C4C2×#8760411yescharacteristic
74M16—3881no—
75M16—1942no—
76C2×C8—3881no—
77C2×C8—3881no—
78C4⋊C4—41no—
79C4⋊C4—3881no—
80C4⋊C4—3881no—
81C4⋊C4—3881no—
82C23C22⋊#8760411yescharacteristic
83C2×C4—21no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—971no—
89C2×C4—971no—
90C2×C4—3881no—
91C2×C4—3881no—
92C2×C4—3881no—
93C2×C4—1942no—
94C2×C4—1941no—
95C2×C4—1941no—
96C2×C4—1941no—
97C8—7761no—
98C8—3882no—
99C22—21no—
100C22—21no—
101C22#1569086811yescharacteristic
102C22#1569087211yescharacteristic
103C22#1569087411yescharacteristic
104C4—41no—
105C4—81no—
106C4—3882no—
107C4—3881no—
108C4—1941no—
109C4—1941no—
110C2—21no—
111C2—21no—
112C2#2026158611yescharacteristic
113C2#2026163711yescharacteristic
114C2#2026164511yescharacteristic
subgroupG/H
centerC22#15690868
commutatorC22×C4×C97C2×C4
FrattiniC22×C4C2×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97#99
prime radicalC23×C97C22⋊C4
cosocleC2×#13095490C22
hypercenterC2×C4⋊C4#87604
hypocenterC97#1346
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
58. C97#1346semidirect

Maximal subgroups

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

subgroupindex
#202616152
#202615812
#202606662
#134697