Finite Group Databasebeta

← #27157919

Subgroups — #27157919

Order 12416.

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

normal subgroups only

This group has 260 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
1#20261694C211yesmaximal, characteristic
2#20261681C211yesmaximal, characteristic
3#20261608C214yesmaximal
4#125×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604—22no—
7C2×C4⋊#87604C2212yes—
8#15690865—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—
21C4⋊#87604—44no—
22C2×C4×D97—22no—
23C2×C4×D97D412yes—
24C2×C4×D97C2311yescharacteristic
25C4⋊Dic97—22no—
26C4⋊Dic97C2×C412yes—
27#13095490—22no—
28#13095490—22no—
29#13095490—22no—
30C4×Dic97C2×C411yescharacteristic
31C4×Dic97D412yes—
32C4⋊C4×C97—22no—
33C4⋊C4×C97C2×C412yes—
34C42×C97C2×C411yescharacteristic
35C22×D97C2×D411yescharacteristic
36C2×#87604—42no—
37C2×#87604—44no—
38C2×C4×C97—22no—
39C2×C4×C97C22⋊C412yes—
40C2×C4×C97C22×C411yescharacteristic
41C2×Dic97—22no—
42C2×Dic97C22⋊C412yes—
43C2×Dic97C2×D411yescharacteristic
44C4×D97—22no—
45C4×D97—22no—
46C4×D97—42no—
47C4×D97—42no—
48C4×D97C2×Q812yes—
49C4×D97#4112yes—
50#11715897—24no—
51C22×C97C2×C22⋊C411yescharacteristic
52C2×D97#12411yescharacteristic
53C2×D97C4⋊Q811yescharacteristic
54C2×D97AGL(1,ℤ/8)12yes—
55C2×D97#13712yes—
56#87604—82no—
57C4×C97—22no—
58C4×C97—42no—
59C4×C97C4×Q812yes—
60Dic97—22no—
61Dic97—42no—
62Dic97C22⋊Q812yes—
63C2×C97#39111yescharacteristic
64C2×C97#41712yes—
65D97#49011yescharacteristic
66D97#49111yescharacteristic
67D97#50112yes—
68#1992—971nomaximal, Sylow
69C97#199211yescharacteristic, complemented, Sylow
70C2×#125—971no—
71C2×#97—971no—
72#384—971no—
73#340—974no—
74C22×C8—972no—
75#125—972no—
76#125—971no—
77#125—1942no—
78#125#8760411yescharacteristic, complemented
79C2×C4⋊C4—972no—
80C2×C4⋊C4—972no—
81C2×C4⋊C4—1942no—
82C2×C4⋊C4—1942no—
83C2×C42—971no—
84#97—1942no—
85C22×C4—971no—
86C22×C4—972no—
87C22×C4—1942no—
88C22×C4—1942no—
89C2×C8—974no—
90C2×C8—1942no—
91C2×C8—1942no—
92C4⋊C4—22no—
93C4⋊C4—972no—
94C4⋊C4—3884no—
95C4⋊C4—1942no—
96C4⋊C4—1942no—
97C4⋊C4—1942no—
98C4⋊C4—1944no—
99C4⋊C4—1942no—
100C4⋊C4C2×#8760412yes—
101C42—972no—
102C42—971no—
103C42C2×#8760411yescharacteristic
104C23—971no—
105C2×C4—22no—
106C2×C4—971no—
107C2×C4—972no—
108C2×C4—972no—
109C2×C4—972no—
110C2×C4—3882no—
111C2×C4—3882no—
112C2×C4—3884no—
113C2×C4—3882no—
114C2×C4—1942no—
115C2×C4—1942no—
116C2×C4—1942no—
117C2×C4C22⋊#8760412yes—
118C2×C4C22×#8760411yescharacteristic
119C8—1944no—
120C22—971no—
121C22—971no—
122C22—972no—
123C22—972no—
124C22C2×C22⋊#8760411yescharacteristic
125C4—22no—
126C4—42no—
127C4—7762no—
128C4—972no—
129C4—3882no—
130C4—1942no—
131C4Q8×#8760412yes—
132C2—971no—
133C2—971no—
134C2—972no—
135C2(C2×#87604)⋊Q811yescharacteristic
136C2#2026172612yes—
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#1992
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
69. C97#1992semidirect
78. #125#87604semidirect

Maximal subgroups

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

subgroupindex
#202616942
#202616812
#202616082
#125×D972
#199297