Finite Group Databasebeta

← #27143634

Subgroups — #27143634

Order 12416.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261170C211yesmaximal, characteristic
2#20260848C211yesmaximal, characteristic, complemented
3#20260559C211yesmaximal, characteristic, complemented
4#20260547C211yesmaximal, characteristic, complemented
5#20260545C211yesmaximal, characteristic, complemented
6#20260513C211yesmaximal, characteristic, complemented
7#444×C97C211yesmaximal, characteristic, complemented
8C2×D776C2211yescharacteristic
9C4×D388—21no—
10#15690753C412yescomplemented
11#15690751C412yescomplemented
12#15690750C2211yescharacteristic
13#15690750C2211yescharacteristic
14#15690742C2211yescharacteristic
15#15690732—21no—
16#135×C97C2211yescharacteristic
17#131×C97—21no—
18#108×C97C2211yescharacteristic
19#104×C97—21no—
20#97×C97C2211yescharacteristic
21C2×D388—21no—
22C2×C4×D97—41no—
23D776—21no—
24D776C2×C412yes—
25C4⋊Dic97—21no—
26#13095488—41no—
27#13095486—21no—
28#13095486C2×C412yescomplemented
29#41×C97—21no—
30Q16×C97C2×C411yescharacteristic, complemented
31SD16×C97—21no—
32D8×C97C2×C411yescharacteristic, complemented
33M16×C97—21no—
34M16×C97—41no—
35C2×C8×C97—41no—
36C2×C8×C97D411yescharacteristic
37C2×C8×C97C2311yescharacteristic
38C42×C97D411yescharacteristic
39C22×D97—41no—
40Q8×C97—21no—
41D4×C97—21no—
42D4×C97—41no—
43C2×C4×C97—21no—
44C2×C4×C97—41no—
45C2×C4×C97C2×D411yescharacteristic
46C2×Dic97—41no—
47D388—22no—
48D388—41no—
49C4×D97—82no—
50C8×C97—21no—
51C8×C97—41no—
52C8×C97C22×C411yescharacteristic
53C8×C97#4111yescharacteristic
54C22×C97—41no—
55C22×C97AGL(1,ℤ/8)11yescharacteristic
56C2×D97—42no—
57C2×D97—81no—
58C2×D97—81no—
59C4×C97—41no—
60C4×C97—41no—
61C4×C97C4×D411yescharacteristic
62C4×C97AGL(1,ℤ/8)11yescharacteristic
63Dic97—81no—
64C2×C97—21no—
65C2×C97—81no—
66C2×C97#44211yescharacteristic
67D97—82no—
68AGL(1,ℤ/16)—971nomaximal, Sylow
69C97AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—971no—
71#444D9711yescharacteristic, complemented
72#442—971no—
73#365—971no—
74#361—971no—
75#360—971no—
76#347—971no—
77#135C2×D9711yescharacteristic
78C2×D8—971no—
79#131—21no—
80C4×D4—1941no—
81SD32—972no—
82D16—972no—
83M32—971no—
84M32—971no—
85#108C2×D9711yescharacteristic
86#106—971no—
87#104—21no—
88#102—1941no—
89#97C2×D9711yescharacteristic
90#41—21no—
91C2×D4—1941no—
92C22×C4—3881no—
93Q16C4×D9711yescharacteristic, complemented
94SD16—21no—
95D8—972no—
96D8—1941no—
97D8C4×D9711yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8C22⋊D9711yescharacteristic
102C2×C8C22×D9711yescharacteristic
103C4⋊C4—1941no—
104C22⋊C4—3881no—
105C42C22⋊D9711yescharacteristic
106C16—972no—
107C16—1941no—
108C23—3881no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—3881no—
113D4—1942no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—7762no—
117C2×C4—3881no—
118C2×C4C2×C22⋊D9711yescharacteristic
119C8—21no—
120C8—41no—
121C8C2×C4×D9711yescharacteristic
122C8#1309551111yescharacteristic
123C22—41no—
124C22—7761no—
125C22—7761no—
126C22—3882no—
127C22#1569083511yescharacteristic
128C4—41no—
129C4—41no—
130C4—7761no—
131C4C4×C22⋊D9711yescharacteristic
132C4#1569083611yescharacteristic
133C2—21no—
134C2—81no—
135C2—7762no—
136C2#2026084511yescharacteristic
subgroupG/H
centerC2#20260845
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#444×C97C2
socleC2×C97#442
prime radical#20261170C2
cosocleC2×C8×C97C23
hypercenter#444D97
hypocenterC97AGL(1,ℤ/16)
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #444

Complements

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

NH ≅ G/Ntype
2. #20260848C2semidirect
3. #20260559C2semidirect
4. #20260547C2semidirect
5. #20260545C2semidirect
6. #20260513C2semidirect
7. #444×C97C2semidirect
10. #15690753C4semidirect
11. #15690751C4semidirect
28. #13095486C2×C4semidirect
30. Q16×C97C2×C4semidirect
32. D8×C97C2×C4semidirect
69. C97AGL(1,ℤ/16)semidirect
71. #444D97semidirect
93. Q16C4×D97semidirect
97. D8C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202611702
#202608482
#202605592
#202605472
#202605452
#202605132
#444×C972
AGL(1,ℤ/16)97