Finite Group Databasebeta

← #27143631

Subgroups — #27143631

Order 12416.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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#20260857C211yesmaximal, characteristic, complemented
3#20260846C211yesmaximal, characteristic, complemented
4#20260557C211yesmaximal, characteristic, complemented
5#20260549C211yesmaximal, characteristic, complemented
6#20260512C211yesmaximal, characteristic, complemented
7#360×C97C211yesmaximal, characteristic, complemented
8C2×D776C2211yescharacteristic
9#15690817—21no—
10#15690772—21no—
11#15690753C412yescomplemented
12#15690751C412yescomplemented
13#15690750C2211yescharacteristic
14#15690746C2211yescharacteristic
15#15690737—21no—
16#15690730—21no—
17#15690720C2211yescharacteristic
18#135×C97C2211yescharacteristic
19M32×C97C2211yescharacteristic
20#106×C97C2211yescharacteristic
21C2×D388—21no—
22C2×C4×D97—41no—
23D776—21no—
24D776C2×C412yes—
25#13095495—21no—
26#13095495—41no—
27C2×#11715895—41no—
28C2×#11715895D411yescharacteristic
29#13095488—41no—
30C4×Dic97D411yescharacteristic
31#13095486C2×C412yescomplemented
32#41×C97—21no—
33Q16×C97C2×C411yescharacteristic, complemented
34SD16×C97—21no—
35D8×C97C2×C411yescharacteristic, complemented
36C2×C8×C97C2311yescharacteristic
37C4⋊C4×C97—21no—
38C16×C97—21no—
39C22×D97—41no—
40Q8×C97—21no—
41D4×C97—21no—
42D4×C97—41no—
43C2×C4×C97—41no—
44C2×C4×C97—41no—
45C2×C4×C97C2×D411yescharacteristic
46C2×Dic97—21no—
47D388—22no—
48D388—41no—
49C4×D97—82no—
50C8×C97C22×C411yescharacteristic
51C8×C97#4111yescharacteristic
52#11715895—21no—
53#11715895—41no—
54C22×C97—41no—
55C22×C97AGL(1,ℤ/8)11yescharacteristic
56C2×D97—42no—
57C2×D97—81no—
58C2×D97—81no—
59C4×C97—41no—
60C4×C97—81no—
61C4×C97C4×D411yescharacteristic
62C4×C97AGL(1,ℤ/8)11yescharacteristic
63Dic97—41no—
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#444—971no—
72#442—971no—
73#365—971no—
74#361—971no—
75#360D9711yescharacteristic, complemented
76#347—971no—
77#135C2×D9711yescharacteristic
78C2×D8—971no—
79#131—1941no—
80C4×D4—1941no—
81SD32—972no—
82D16—972no—
83M32—971no—
84M32C2×D9711yescharacteristic
85#108—971no—
86#106C2×D9711yescharacteristic
87#104—1941no—
88#102—1941no—
89#97—971no—
90#41—21no—
91C2×D4—1941no—
92C22×C4—3881no—
93Q16C4×D9711yescharacteristic, complemented
94SD16—21no—
95D8—972no—
96D8—1941no—
97D8C4×D9711yescharacteristic, complemented
98M16—3881no—
99M16—1941no—
100C2×C8—971no—
101C2×C8—3881no—
102C2×C8C22×D9711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—3881no—
105C42—971no—
106C16—21no—
107C16—972no—
108C23—3881no—
109Q8—21no—
110D4—21no—
111D4—41no—
112D4—3881no—
113D4—1942no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—7762no—
117C2×C4—1941no—
118C2×C4D4×D9711yescharacteristic
119C8—3881no—
120C8—1941no—
121C8C2×C4×D9711yescharacteristic
122C8#1309551011yescharacteristic
123C22—41no—
124C22—7761no—
125C22—7761no—
126C22—3882no—
127C22#1569083411yescharacteristic
128C4—41no—
129C4—81no—
130C4—3881no—
131C4#1569077011yescharacteristic
132C4#1569083711yescharacteristic
133C2—21no—
134C2—81no—
135C2—7762no—
136C2#2026084411yescharacteristic
subgroupG/H
centerC2#20260844
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#360×C97C2
socleC2×C97#442
prime radical#20261170C2
cosocleC2×C8×C97C23
hypercenter#360D97
hypocenterC97AGL(1,ℤ/16)
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
2. #20260857C2semidirect
3. #20260846C2semidirect
4. #20260557C2semidirect
5. #20260549C2semidirect
6. #20260512C2semidirect
7. #360×C97C2semidirect
11. #15690753C4semidirect
12. #15690751C4semidirect
31. #13095486C2×C4semidirect
33. Q16×C97C2×C4semidirect
35. D8×C97C2×C4semidirect
69. C97AGL(1,ℤ/16)semidirect
75. #360D97semidirect
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
#202608572
#202608462
#202605572
#202605492
#202605122
#360×C972
AGL(1,ℤ/16)97