Finite Group Databasebeta

← #27158017

Subgroups — #27158017

Order 12416.

270 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261714C212yesmaximal, complemented
2#20261640C214yesmaximal, complemented
3#20261304C211yesmaximal, characteristic
4#15690889—22no—
5#15690869C2214yescomplemented
6#15690868—22no—
7C2×#13095511C2211yescharacteristic
8C22×D388C411yescharacteristic, complemented
9#15690807C2212yes—
10#15690798C412yescomplemented
11#15690785—22no—
12#15690780—22no—
13#117×C97C411yescharacteristic, complemented
14C4⋊#87604—42no—
15C22⋊#87604—24no—
16C4×#87604—42no—
17C23×D97—21no—
18#13095511—42no—
19C2×Dic194C2×C411yescharacteristic
20C2×D388—21no—
21C2×D388—24no—
22C2×D388—42no—
23C2×D388C2×C412yes—
24C2×D388C2311yescharacteristic
25C2×C22⋊D97D412yes—
26C2×C4×D97D412yes—
27#13095488—24no—
28C22×C4×C97C2×C411yescharacteristic
29C4⋊C4×C97—21no—
30C22⋊C4×C97—21no—
31C42×C97C2×C412yes—
32C23×C97C22⋊C411yescharacteristic
33C22×D97—22no—
34C22×D97—42no—
35C22×D97—82no—
36C22×D97C2×D412yes—
37C2×#87604—44no—
38C2×C4×C97—22no—
39C2×C4×C97—22no—
40C2×C4×C97C22⋊C411yescharacteristic
41C2×C4×C97C22×C411yescharacteristic
42C22⋊D97—42no—
43C2×Dic97C22⋊C412yes—
44D388—41no—
45D388—42no—
46D388—42no—
47D388—84no—
48C4×D97—22no—
49C4×D97—22no—
50Dic194—41no—
51C22×C97—21no—
52C22×C97—41no—
53C22×C97C2×C22⋊C411yescharacteristic
54C2×D97—22no—
55C2×D97—42no—
56C2×D97—44no—
57C2×D97—42no—
58C2×D97—84no—
59#87604—84no—
60C4×C97—21no—
61C4×C97—42no—
62C4×C97#9912yes—
63Dic97—22no—
64C2×C97—21no—
65C2×C97—41no—
66C2×C97C2×#9911yescharacteristic
67D97—42no—
68D97—82no—
69#2128—971nomaximal, Sylow
70C97#212811yescharacteristic, complemented, Sylow
71#535—971no—
72#410—972no—
73#353—974no—
74C2×#41—971no—
75C22×D4—971no—
76#127—972no—
77#124—972no—
78#121—1942no—
79#120—1942no—
80#117—1942no—
81#117#8760411yescharacteristic, complemented
82#99—974no—
83#99—1942no—
84C24—1941no—
85#41—3882no—
86C2×Q8—971no—
87C2×D4—972no—
88C2×D4—971no—
89C2×D4—972no—
90C2×D4—3882no—
91C2×D4—1944no—
92C2×D4—1941no—
93C22×C4—972no—
94C22×C4C2×#8760411yescharacteristic
95C4⋊C4—21no—
96C4⋊C4—3882no—
97C22⋊C4—21no—
98C22⋊C4—1944no—
99C22⋊C4—1944no—
100C42—3882no—
101C42C2×#8760412yes—
102C23—7762no—
103C23—972no—
104C23—3882no—
105C23—1942no—
106C23C22⋊#8760411yescharacteristic
107Q8—3881no—
108D4—7764no—
109D4—3882no—
110D4—3881no—
111D4—3882no—
112D4—3882no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—972no—
116C2×C4—3884no—
117C2×C4—1942no—
118C2×C4—1942no—
119C2×C4C22⋊#8760411yescharacteristic
120C2×C4C22×#8760411yescharacteristic
121C22—21no—
122C22—41no—
123C22—7764no—
124C22—3884no—
125C22—3882no—
126C22—3882no—
127C22—1942no—
128C22C2×C22⋊#8760411yescharacteristic
129C4—21no—
130C4—42no—
131C4—7764no—
132C4—1942no—
133C4#1569086912yes—
134C2—21no—
135C2—41no—
136C2—7762no—
137C2—3882no—
138C2C2×#1569086911yescharacteristic
subgroupG/H
centerC2C2×#15690869
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC2×C97C2×#99
prime radical#20261304C2
cosocleC2×D388C23
hypercenter#117#87604
hypocenterC97#2128
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20261714C2semidirect
2. #20261640C2semidirect
5. #15690869C22semidirect
8. C22×D388C4semidirect
10. #15690798C4semidirect
13. #117×C97C4semidirect
70. C97#2128semidirect
81. #117#87604semidirect

Maximal subgroups

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

subgroupindex
#202617142
#202616402
#202613042
#212897