Finite Group Databasebeta

← #27157918

Subgroups — #27157918

Order 12416.

252 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C4×#13095519C211yesmaximal, characteristic
2#20261679C211yesmaximal, characteristic
3#20261608C214yesmaximal
4#125×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7C4×#11715897—22no—
8(C2×#87604)⋊C4—22no—
9#15690802C411yescharacteristic, complemented
10#15690801C412yescomplemented
11#15690800—22no—
12C4⋊C4×D97—22no—
13C4⋊C4×D97C2212yes—
14C42×D97C2211yescharacteristic
15#125×C97C411yescharacteristic, complemented
16C22×#87604—22no—
17#13095519—22no—
18#13095519Q814yes—
19C2×#11715897—22no—
20C4⋊#87604—24no—
21C2×C4×D97—22no—
22C2×C4×D97D412yes—
23C2×C4×D97C2311yescharacteristic
24C4⋊Dic97—22no—
25C4⋊Dic97C2×C412yes—
26#13095490—22no—
27#13095490—22no—
28#13095490—22no—
29C4×Dic97C2×C411yescharacteristic
30C4×Dic97D412yes—
31C4⋊C4×C97—22no—
32C4⋊C4×C97C2×C412yes—
33C42×C97C2×C411yescharacteristic
34C22×D97C2×D411yescharacteristic
35C2×#87604—42no—
36C2×#87604—44no—
37C2×C4×C97—22no—
38C2×C4×C97C22⋊C412yes—
39C2×C4×C97C22×C411yescharacteristic
40C2×Dic97—22no—
41C2×Dic97C22⋊C412yes—
42C2×Dic97C2×D411yescharacteristic
43C4×D97—24no—
44C4×D97—44no—
45C4×D97C2×Q812yes—
46C4×D97#4112yes—
47#11715897—24no—
48C22×C97C2×C22⋊C411yescharacteristic
49C2×D97#12411yescharacteristic
50C2×D97C4⋊Q811yescharacteristic
51C2×D97#13514yes—
52#87604—82no—
53C4×C97—22no—
54C4×C97—42no—
55C4×C97C4×Q812yes—
56Dic97—22no—
57Dic97—42no—
58Dic97C22⋊Q812yes—
59C2×C97#39111yescharacteristic
60C2×C97#41612yes—
61D97#48812yes—
62D97#49912yes—
63#1991—971nomaximal, Sylow
64C97#199111yescharacteristic, complemented, Sylow
65C2×#125—971no—
66C2×C4×C8—971no—
67#382—971no—
68#340—974no—
69C22×C8—972no—
70#125—972no—
71#125—971no—
72#125—1942no—
73#125#8760411yescharacteristic, complemented
74C2×C4⋊C4—972no—
75C2×C4⋊C4—972no—
76C2×C4⋊C4—1942no—
77C2×C42—971no—
78C4×C8—1942no—
79#95—1942no—
80C22×C4—971no—
81C22×C4—972no—
82C22×C4—1942no—
83C22×C4—1942no—
84C2×C8—974no—
85C2×C8—1942no—
86C2×C8—1942no—
87C4⋊C4—22no—
88C4⋊C4—972no—
89C4⋊C4—1942no—
90C4⋊C4—1942no—
91C4⋊C4—1942no—
92C4⋊C4—1944no—
93C4⋊C4—1942no—
94C4⋊C4C2×#8760412yes—
95C42—972no—
96C42—971no—
97C42C2×#8760411yescharacteristic
98C23—971no—
99C2×C4—22no—
100C2×C4—971no—
101C2×C4—972no—
102C2×C4—972no—
103C2×C4—972no—
104C2×C4—3884no—
105C2×C4—3882no—
106C2×C4—3884no—
107C2×C4—1944no—
108C2×C4—1942no—
109C2×C4C22⋊#8760412yes—
110C2×C4C22×#8760411yescharacteristic
111C8—1944no—
112C22—971no—
113C22—971no—
114C22—974no—
115C22C2×C22⋊#8760411yescharacteristic
116C4—22no—
117C4—42no—
118C4—7762no—
119C4—972no—
120C4—3882no—
121C4—1942no—
122C4Q8×#8760412yes—
123C2—972no—
124C2—972no—
125C2(C2×#87604)⋊Q811yescharacteristic
126C2#2026172512yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenter#125#87604
hypocenterC97#1991
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
9. #15690802C4semidirect
10. #15690801C4semidirect
15. #125×C97C4semidirect
64. C97#1991semidirect
73. #125#87604semidirect

Maximal subgroups

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

subgroupindex
C4×#130955192
#202616792
#202616082
#125×D972
#199197