Finite Group Databasebeta

← #27158148

Subgroups — #27158148

Order 12416.

154 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 44 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#20261781C211yesmaximal, characteristic, complemented
2#20261776C211yesmaximal, characteristic, complemented
3#20261659C211yesmaximal, characteristic, complemented
4#20261653C211yesmaximal, characteristic, complemented
5#20261652C211yesmaximal, characteristic, complemented
6#20261626C211yesmaximal, characteristic, complemented
7#20261172C211yesmaximal, characteristic
8#15690898—21no—
9D4×#87604—21no—
10#15690885C2211yescharacteristic
11#15690880C2211yescharacteristic
12#15690878C2211yescharacteristic
13#15690875—21no—
14#15690872—21no—
15#15690864C2211yescharacteristic
16#15690833C2211yescharacteristic
17D8×D97C2211yescharacteristic
18#15690753C411yescharacteristic, complemented
19D1552C411yescharacteristic, complemented
20#15690751C411yescharacteristic, complemented
21#15690749C2211yescharacteristic
22SD32×C97C411yescharacteristic, complemented
23C22×#87604—41no—
24#13095521—21no—
25#13095521—41no—
26#13095519—41no—
27#13095519D411yescharacteristic
28C4⋊#87604—21no—
29C22⋊#87604—41no—
30C4×#87604D411yescharacteristic
31#13095514—21no—
32#13095512—21no—
33D4×D97—21no—
34#13095498—21no—
35D776C2×C411yescharacteristic
36D776C2×C411yescharacteristic, complemented
37#13095496—21no—
38C8×D97C2311yescharacteristic
39#13095486C2×C411yescharacteristic, complemented
40Q16×C97C2×C411yescharacteristic, complemented
41D8×C97C2×C411yescharacteristic
42C16×C97C2×C411yescharacteristic, complemented
43C22×D97—41no—
44C2×#87604—21no—
45C2×#87604—41no—
46C2×#87604—82no—
47Q8×C97—21no—
48D4×C97—21no—
49C22⋊D97—41no—
50D388—21no—
51D388—21no—
52D388—41no—
53C4×D97—41no—
54C4×D97C2×D411yescharacteristic
55#11715897—21no—
56#11715897—41no—
57C8×C97C22×C411yescharacteristic
58#11715895#4111yescharacteristic
59C22×C97—41no—
60C2×D97—41no—
61C2×D97—41no—
62C2×D97—81no—
63C2×D97—81no—
64C2×D97AGL(1,ℤ/8)11yescharacteristic
65#87604—41no—
66#87604—81no—
67C4×C97—41no—
68C4×C97C4×D411yescharacteristic
69Dic97AGL(1,ℤ/8)11yescharacteristic
70C2×C97—81no—
71C2×C97#44211yescharacteristic
72D97—21no—
73D97—81no—
74D97—81no—
75AGL(1,ℤ/16)—971nomaximal, Sylow
76C97AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
77#509—971no—
78#444—971no—
79#442—971no—
80#365—971no—
81#361—971no—
82#360—971no—
83#347—971no—
84#135—971no—
85C2×D8—971no—
86#131—1941no—
87C4×D4—1941no—
88SD32—971no—
89SD32#8760411yescharacteristic, complemented
90D16—971no—
91D16—971no—
92M32—971no—
93M32—971no—
94#108—971no—
95#106—971no—
96#104—1941no—
97#102—1941no—
98#97—971no—
99#41—1941no—
100C2×D4—1941no—
101C22×C4—3881no—
102Q16C2×#8760411yescharacteristic, complemented
103SD16—1941no—
104D8—971no—
105D8—971no—
106D8—1941no—
107D8C2×#8760411yescharacteristic
108M16—3881no—
109M16—1941no—
110C2×C8—971no—
111C2×C8—971no—
112C2×C8—3881no—
113C4⋊C4—1941no—
114C22⋊C4—3881no—
115C42—971no—
116C16—971no—
117C16—1941no—
118C16C2×#8760411yescharacteristic, complemented
119C23—3881no—
120Q8—21no—
121D4—21no—
122D4—3881no—
123D4—3881no—
124D4—1941no—
125D4—1941no—
126C2×C4—7762no—
127C2×C4—971no—
128C2×C4—3881no—
129C2×C4—3881no—
130C2×C4—1941no—
131C8—971no—
132C8—3881no—
133C8—1941no—
134C8C22×#8760411yescharacteristic
135C22—41no—
136C22—7761no—
137C22—7761no—
138C22—971no—
139C22—3881no—
140C22—3881no—
141C4—41no—
142C4—7761no—
143C4—971no—
144C4—3881no—
145C4D4×#8760411yescharacteristic
146C2—81no—
147C2—7761no—
148C2—7761no—
149C2—1941no—
150C2#2026177611yescharacteristic
subgroupG/H
centerC2#20261776
commutatorC8×C97C22×C4
FrattiniC8C22×#87604
FittingSD32×C97C4
socleC2×C97#442
prime radical#20261172C2
cosocleC8×D97C23
hypercenterSD32#87604
hypocenterC97AGL(1,ℤ/16)
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32

Complements

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

NH ≅ G/Ntype
1. #20261781C2semidirect
2. #20261776C2semidirect
3. #20261659C2semidirect
4. #20261653C2semidirect
5. #20261652C2semidirect
6. #20261626C2semidirect
18. #15690753C4semidirect
19. D1552C4semidirect
20. #15690751C4semidirect
22. SD32×C97C4semidirect
36. D776C2×C4semidirect
39. #13095486C2×C4semidirect
40. Q16×C97C2×C4semidirect
42. C16×C97C2×C4semidirect
76. C97AGL(1,ℤ/16)semidirect
89. SD32#87604semidirect
102. Q16C2×#87604semidirect
118. C16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202617812
#202617762
#202616592
#202616532
#202616522
#202616262
#202611722
AGL(1,ℤ/16)97