Finite Group Databasebeta

← #27143637

Subgroups — #27143637

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#20261171C211yesmaximal, characteristic
2#20260856C211yesmaximal, characteristic, complemented
3#20260847C211yesmaximal, characteristic, complemented
4#20260559C211yesmaximal, characteristic, complemented
5#20260546C211yesmaximal, characteristic, complemented
6#20260512C211yesmaximal, characteristic, complemented
7#361×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9#15690818—21no—
10D4×Dic97—21no—
11#15690754C412yescomplemented
12#15690751C412yescomplemented
13#15690750C2211yescharacteristic
14#15690742C2211yescharacteristic
15#15690736—21no—
16#15690731—21no—
17#15690720C2211yescharacteristic
18C2×D8×C97C2211yescharacteristic
19M32×C97C2211yescharacteristic
20#108×C97C2211yescharacteristic
21#13095511—21no—
22C22×Dic97—41no—
23Dic388C2×C411yescharacteristic, complemented
24#13095500—21no—
25D776C2×C411yescharacteristic, complemented
26#13095494—41no—
27C2×#11715895D411yescharacteristic
28C8×D97—41no—
29C4⋊Dic97—21no—
30C22⋊Dic97—41no—
31C4×Dic97D411yescharacteristic
32#13095486C2×C412yescomplemented
33C2×D4×C97—21no—
34D8×C97—21no—
35D8×C97C2×C412yes—
36M16×C97—21no—
37C2×C8×C97C2311yescharacteristic
38C16×C97—21no—
39C23×C97—41no—
40D4×C97—22no—
41D4×C97—41no—
42C2×C4×C97C2×D411yescharacteristic
43C22⋊D97—41no—
44C2×Dic97—21no—
45C2×Dic97—41no—
46C2×Dic97—82no—
47D388—21no—
48C4×D97—41no—
49Dic194—21no—
50C8×C97—41no—
51C8×C97C22×C411yescharacteristic
52C8×C97#4111yescharacteristic
53#11715895—21no—
54C22×C97—42no—
55C22×C97—81no—
56C22×C97—81no—
57C22×C97AGL(1,ℤ/8)11yescharacteristic
58C2×D97—41no—
59C4×C97C4×D411yescharacteristic
60C4×C97AGL(1,ℤ/8)11yescharacteristic
61Dic97—41no—
62Dic97—41no—
63Dic97—81no—
64C2×C97—21no—
65C2×C97—82no—
66C2×C97#44211yescharacteristic
67D97—81no—
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#361D9711yescharacteristic, complemented
75#360—971no—
76#347—971no—
77#135—971no—
78C2×D8C2×D9711yescharacteristic
79#131—1941no—
80C4×D4—1941no—
81SD32—972no—
82D16—972no—
83M32—971no—
84M32C2×D9711yescharacteristic
85#108C2×D9711yescharacteristic
86#106—971no—
87#104—1941no—
88#102—1941no—
89#97—971no—
90#41—1941no—
91C2×D4—21no—
92C22×C4—3881no—
93Q16—971no—
94SD16—1941no—
95D8—21no—
96D8—971no—
97D8C4×D9712yes—
98M16—21no—
99M16—3881no—
100C2×C8—971no—
101C2×C8—3881no—
102C2×C8C22×D9711yescharacteristic
103C4⋊C4—1941no—
104C22⋊C4—3881no—
105C42—971no—
106C16—21no—
107C16—972no—
108C23—41no—
109Q8—1941no—
110D4—22no—
111D4—41no—
112D4—3881no—
113D4—1941no—
114C2×C4—7762no—
115C2×C4—3881no—
116C2×C4—3881no—
117C2×C4—1941no—
118C2×C4D4×D9711yescharacteristic
119C8—41no—
120C8—1941no—
121C8C2×C4×D9711yescharacteristic
122C8#1309551011yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—3881no—
127C22#1569083711yescharacteristic
128C4—7761no—
129C4—3881no—
130C4—3881no—
131C4#1569077011yescharacteristic
132C4#1569083411yescharacteristic
133C2—21no—
134C2—82no—
135C2—7761no—
136C2#2026084411yescharacteristic
subgroupG/H
centerC2#20260844
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#361×C97C2
socleC2×C97#442
prime radical#20261171C2
cosocleC2×C8×C97C23
hypercenter#361D97
hypocenterC97AGL(1,ℤ/16)
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #20260856C2semidirect
3. #20260847C2semidirect
4. #20260559C2semidirect
5. #20260546C2semidirect
6. #20260512C2semidirect
7. #361×C97C2semidirect
11. #15690754C4semidirect
12. #15690751C4semidirect
23. Dic388C2×C4semidirect
25. D776C2×C4semidirect
32. #13095486C2×C4semidirect
69. C97AGL(1,ℤ/16)semidirect
74. #361D97semidirect

Maximal subgroups

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

subgroupindex
#202611712
#202608562
#202608472
#202605592
#202605462
#202605122
#361×C972
AGL(1,ℤ/16)97