Finite Group Databasebeta

← #27156730

Subgroups — #27156730

Order 12416.

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

normal subgroups only

This group has 162 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#20261613C211yesmaximal, characteristic
2#20261579C211yesmaximal, characteristic
3#20260648C211yesmaximal, characteristic
4C2×C22⋊#87604—21no—
5C22⋊#11715897—21no—
6C22×Dic194C411yescharacteristic, complemented
7C2×#13095488—21no—
8C2×#13095488C2211yescharacteristic
9#15690717—21no—
10#95×C97C411yescharacteristic, complemented
11C22×#87604—41no—
12C2×#11715897—41no—
13C22⋊#87604—22no—
14C22⋊#87604—41no—
15C23×D97D411yescharacteristic
16C2×Dic194—41no—
17C2×Dic194—42no—
18C22×Dic97—21no—
19C22×Dic97D411yescharacteristic
20#13095488—41no—
21#13095488—42no—
22C22×C4×C97—21no—
23C22×C4×C97C2×C411yescharacteristic
24C23×C97C22⋊C411yescharacteristic
25C22×D97—21no—
26C22×D97—21no—
27C22×D97—41no—
28C22×D97—41no—
29C22×D97SD1611yescharacteristic
30C22×D97Q1611yescharacteristic
31C2×#87604—42no—
32C2×#87604—81no—
33C2×#87604—81no—
34C2×C4×C97—21no—
35C2×C4×C97—41no—
36C2×C4×C97—41no—
37C2×C4×C97—41no—
38C2×C4×C97—41no—
39C2×Dic97—22no—
40C2×Dic97—21no—
41C2×Dic97—41no—
42C2×Dic97—41no—
43C2×Dic97—41no—
44Dic194—42no—
45Dic194—42no—
46#11715897—81no—
47C22×C97—21no—
48C22×C97—21no—
49C22×C97#9911yescharacteristic
50C22×C97#10311yescharacteristic
51C22×C97#10411yescharacteristic
52C2×D97—21no—
53C2×D97—21no—
54C2×D97—41no—
55C2×D97—41no—
56C2×D97—41no—
57C2×D97—41no—
58C2×D97—81no—
59#87604—82no—
60C4×C97—41no—
61C4×C97—81no—
62Dic97—42no—
63Dic97—41no—
64C2×C97—21no—
65C2×C97—21no—
66C2×C97#32811yescharacteristic
67C2×C97AΣL(1,𝔽16)11yescharacteristic
68C2×C97#35411yescharacteristic
69D97—41no—
70D97—41no—
71#1351—971nomaximal, Sylow
72C97#135111yescharacteristic, complemented, Sylow
73#393—971no—
74#342—971no—
75#323—971no—
76C22×Q8—971no—
77C2×C22⋊C4—971no—
78C2×C22⋊C4—1941no—
79C2×C22⋊C4—1941no—
80C22⋊C8—1941no—
81#95—1941no—
82#95#8760411yescharacteristic, complemented
83C24—971no—
84C2×Q8—3881no—
85C2×Q8—3882no—
86C22×C4—21no—
87C22×C4—971no—
88C22×C4—3881no—
89C22×C4—1941no—
90C22×C4C2×#8760411yescharacteristic
91C2×C8—3881no—
92C22⋊C4—3881no—
93C22⋊C4—3882no—
94C22⋊C4—3881no—
95C22⋊C4—1942no—
96C23—971no—
97C23—971no—
98C23—3881no—
99C23—3881no—
100C23—1941no—
101C23—1941no—
102C23C22⋊#8760411yescharacteristic
103Q8—3882no—
104Q8—3882no—
105C2×C4—21no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—7761no—
111C2×C4—7761no—
112C2×C4—3882no—
113C2×C4—3881no—
114C2×C4—3881no—
115C2×C4—3881no—
116C2×C4—1942no—
117C2×C4—1941no—
118C8—7761no—
119C22—21no—
120C22—21no—
121C22—7761no—
122C22—3881no—
123C22—3881no—
124C22—3881no—
125C22—3881no—
126C22—1941no—
127C22—1941no—
128C22#1569086911yescharacteristic
129C22#1569087311yescharacteristic
130C22#1569087411yescharacteristic
131C4—41no—
132C4—81no—
133C4—7762no—
134C4—3882no—
135C4—3881no—
136C2—21no—
137C2—21no—
138C2—3881no—
139C2—3881no—
140C2#2026158811yescharacteristic
141C2#2026163611yescharacteristic
142C2#2026164211yescharacteristic
subgroupG/H
centerC22#15690869
commutatorC22×C4×C97C2×C4
FrattiniC22×C4C2×#87604
Fitting#95×C97C4
socleC22×C97#99
prime radicalC23×D97D4
cosocleC2×#13095488C22
hypercenter#95#87604
hypocenterC97#1351
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #95

Complements

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

NH ≅ G/Ntype
6. C22×Dic194C4semidirect
10. #95×C97C4semidirect
72. C97#1351semidirect
82. #95#87604semidirect

Maximal subgroups

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

subgroupindex
#202616132
#202615792
#202606482
#135197