Finite Group Databasebeta

← AGL(1,ℤ/16)×C251 #44546768

Subgroups — AGL(1,ℤ/16)×C251 #44546768

Order 32128.

154 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 72 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#509×C251C211yesmaximal, characteristic
2#444×C251C211yesmaximal, characteristic, complemented
3#442×C251C211yesmaximal, characteristic, complemented
4#365×C251C211yesmaximal, characteristic, complemented
5#361×C251C211yesmaximal, characteristic, complemented
6#360×C251C211yesmaximal, characteristic, complemented
7#347×C251C211yesmaximal, characteristic, complemented
8#135×C251C2211yescharacteristic
9C2×D8×C251C2211yescharacteristic
10#131×C251—21no—
11C4×D4×C251—21no—
12SD32×C251C412yescomplemented
13D16×C251C412yescomplemented
14M32×C251C2211yescharacteristic
15M32×C251C2211yescharacteristic
16#108×C251C2211yescharacteristic
17#106×C251C2211yescharacteristic
18#104×C251—21no—
19#102×C251—21no—
20#97×C251C2211yescharacteristic
21#41×C251—21no—
22C2×D4×C251—21no—
23C22×C4×C251—41no—
24Q16×C251C2×C411yescharacteristic, complemented
25SD16×C251—21no—
26D8×C251—21no—
27D8×C251C2×C412yes—
28D8×C251C2×C411yescharacteristic, complemented
29M16×C251—21no—
30M16×C251—41no—
31C2×C8×C251—41no—
32C2×C8×C251D411yescharacteristic
33C2×C8×C251C2311yescharacteristic
34C4⋊C4×C251—21no—
35C22⋊C4×C251—41no—
36C42×C251D411yescharacteristic
37C16×C251—21no—
38C16×C251C2×C412yescomplemented
39C23×C251—41no—
40Q8×C251—21no—
41D4×C251—22no—
42D4×C251—21no—
43D4×C251—41no—
44D4×C251—41no—
45C2×C4×C251—21no—
46C2×C4×C251—41no—
47C2×C4×C251—41no—
48C2×C4×C251—82no—
49C2×C4×C251C2×D411yescharacteristic
50C8×C251—21no—
51C8×C251—41no—
52C8×C251C22×C411yescharacteristic
53C8×C251#4111yescharacteristic
54C22×C251—41no—
55C22×C251—42no—
56C22×C251—81no—
57C22×C251—81no—
58C22×C251AGL(1,ℤ/8)11yescharacteristic
59C4×C251—41no—
60C4×C251—41no—
61C4×C251—81no—
62C4×C251C4×D411yescharacteristic
63C4×C251AGL(1,ℤ/8)11yescharacteristic
64C2×C251—21no—
65C2×C251—81no—
66C2×C251—82no—
67C2×C251#44211yescharacteristic
68C251AGL(1,ℤ/16)11yescharacteristic, direct product factor, Sylow
69AGL(1,ℤ/16)C25111yesmaximal, characteristic, direct product factor, Sylow
70#509C2×C25111yescharacteristic
71#444C2×C25111yescharacteristic, complemented
72#442C2×C25111yescharacteristic, complemented
73#365C2×C25111yescharacteristic, complemented
74#361C2×C25111yescharacteristic, complemented
75#360C2×C25111yescharacteristic, complemented
76#347C2×C25111yescharacteristic, complemented
77#135C22×C25111yescharacteristic
78C2×D8C22×C25111yescharacteristic
79#131—21no—
80C4×D4—21no—
81SD32C4×C25112yescomplemented
82D16C4×C25112yescomplemented
83M32C22×C25111yescharacteristic
84M32C22×C25111yescharacteristic
85#108C22×C25111yescharacteristic
86#106C22×C25111yescharacteristic
87#104—21no—
88#102—21no—
89#97C22×C25111yescharacteristic
90#41—21no—
91C2×D4—21no—
92C22×C4—41no—
93Q16C2×C4×C25111yescharacteristic, complemented
94SD16—21no—
95D8—21no—
96D8C2×C4×C25112yes—
97D8C2×C4×C25111yescharacteristic, complemented
98M16—21no—
99M16—41no—
100C2×C8—41no—
101C2×C8D4×C25111yescharacteristic
102C2×C8C23×C25111yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—41no—
105C42D4×C25111yescharacteristic
106C16—21no—
107C16C2×C4×C25112yescomplemented
108C23—41no—
109Q8—21no—
110D4—22no—
111D4—21no—
112D4—41no—
113D4—41no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—82no—
118C2×C4C2×D4×C25111yescharacteristic
119C8—21no—
120C8—41no—
121C8C22×C4×C25111yescharacteristic
122C8#41×C25111yescharacteristic
123C22—41no—
124C22—42no—
125C22—81no—
126C22—81no—
127C22AGL(1,ℤ/8)×C25111yescharacteristic
128C4—41no—
129C4—41no—
130C4—81no—
131C4C4×D4×C25111yescharacteristic
132C4AGL(1,ℤ/8)×C25111yescharacteristic
133C2—21no—
134C2—82no—
135C2—81no—
136C2#442×C25111yescharacteristic
subgroupG/H
center, socleC2×C251#442
commutatorC8C22×C4×C251
Frattini, cosocleC2×C8C23×C251
prime radical#509×C251C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C2×C4×C251 ◁ #97×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #444×C251C2semidirect
3. #442×C251C2semidirect
4. #365×C251C2semidirect
5. #361×C251C2semidirect
6. #360×C251C2semidirect
7. #347×C251C2semidirect
12. SD32×C251C4semidirect
13. D16×C251C4semidirect
24. Q16×C251C2×C4semidirect
28. D8×C251C2×C4semidirect
38. C16×C251C2×C4semidirect
68. C251AGL(1,ℤ/16)direct (N × H)
69. AGL(1,ℤ/16)C251direct (N × H)
71. #444C2×C251semidirect
72. #442C2×C251semidirect
73. #365C2×C251semidirect
74. #361C2×C251semidirect
75. #360C2×C251semidirect
76. #347C2×C251semidirect
81. SD32C4×C251semidirect
82. D16C4×C251semidirect
93. Q16C2×C4×C251semidirect
97. D8C2×C4×C251semidirect
107. C16C2×C4×C251semidirect

Maximal subgroups

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

subgroupindex
#509×C2512
#444×C2512
#442×C2512
#365×C2512
#361×C2512
#360×C2512
#347×C2512
AGL(1,ℤ/16)251