Finite Group Databasebeta

← #30640626

Subgroups — #30640626

Order 16256.

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#22550312C211yesmaximal, characteristic
2#22549999C211yesmaximal, characteristic, complemented
3#22549698C211yesmaximal, characteristic, complemented
4#22549689C211yesmaximal, characteristic, complemented
5#22549685C211yesmaximal, characteristic, complemented
6#22549654C211yesmaximal, characteristic, complemented
7#442×C127C211yesmaximal, characteristic, complemented
8#16861231C2211yescharacteristic
9#16861215—21no—
10#16861153C412yescomplemented
11#16861150C412yescomplemented
12#16861149C2211yescharacteristic
13#16861149C2211yescharacteristic
14#16861145C2211yescharacteristic
15#16861137—21no—
16C2×D8×C127C2211yescharacteristic
17C4×D4×C127—21no—
18#106×C127C2211yescharacteristic
19#102×C127—21no—
20#97×C127C2211yescharacteristic
21#14523248—21no—
22Dic508C2×C411yescharacteristic, complemented
23#14523237—21no—
24D1016C2×C411yescharacteristic, complemented
25#14523232—21no—
26#14523231—41no—
27C8×D127—41no—
28C127⋊C16—21no—
29C127⋊C16C2×C412yescomplemented
30C2×D4×C127—21no—
31C22×C4×C127—41no—
32D8×C127—21no—
33D8×C127C2×C412yes—
34C2×C8×C127D411yescharacteristic
35C2×C8×C127C2311yescharacteristic
36C4⋊C4×C127—21no—
37C22⋊C4×C127—41no—
38C42×C127D411yescharacteristic
39Dic254—21no—
40D508—21no—
41C22⋊D127—41no—
42C4×D127—41no—
43C127⋊C8—41no—
44C23×C127—41no—
45D4×C127—22no—
46D4×C127—41no—
47C2×C4×C127—21no—
48C2×C4×C127—41no—
49C2×C4×C127—82no—
50C2×C4×C127C2×D411yescharacteristic
51C8×C127—21no—
52C8×C127C22×C411yescharacteristic
53C8×C127#4111yescharacteristic
54C22×C127—42no—
55C22×C127—81no—
56C22×C127—81no—
57C22×C127AGL(1,ℤ/8)11yescharacteristic
58C2×D127—41no—
59C4×C127—41no—
60C4×C127—81no—
61C4×C127C4×D411yescharacteristic
62C4×C127AGL(1,ℤ/8)11yescharacteristic
63Dic127—41no—
64C2×C127—21no—
65C2×C127—82no—
66C2×C127#44211yescharacteristic
67D127—81no—
68AGL(1,ℤ/16)—1271nomaximal, Sylow
69C127AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
70#509—1271no—
71#444—1271no—
72#442D12711yescharacteristic, complemented
73#365—1271no—
74#361—1271no—
75#360—1271no—
76#347—1271no—
77#135—1271no—
78C2×D8C2×D12711yescharacteristic
79#131—2541no—
80C4×D4—21no—
81SD32—1272no—
82D16—1272no—
83M32—1271no—
84M32—1271no—
85#108—1271no—
86#106C2×D12711yescharacteristic
87#104—2541no—
88#102—21no—
89#97C2×D12711yescharacteristic
90#41—2541no—
91C2×D4—21no—
92C22×C4—41no—
93Q16—1271no—
94SD16—2541no—
95D8—21no—
96D8—1271no—
97D8C4×D12712yes—
98M16—5081no—
99M16—2541no—
100C2×C8—5081no—
101C2×C8C22⋊D12711yescharacteristic
102C2×C8C22×D12711yescharacteristic
103C4⋊C4—21no—
104C22⋊C4—41no—
105C42C22⋊D12711yescharacteristic
106C16—1272no—
107C16—2541no—
108C23—41no—
109Q8—2541no—
110D4—22no—
111D4—41no—
112D4—5081no—
113D4—2541no—
114C2×C4—21no—
115C2×C4—41no—
116C2×C4—82no—
117C2×C4—5081no—
118C2×C4C2×C22⋊D12711yescharacteristic
119C8—21no—
120C8—5081no—
121C8C2×C4×D12711yescharacteristic
122C8#1452324811yescharacteristic
123C22—42no—
124C22—81no—
125C22—81no—
126C22—5081no—
127C22#1686123511yescharacteristic
128C4—41no—
129C4—81no—
130C4—5081no—
131C4C4×C22⋊D12711yescharacteristic
132C4#1686123411yescharacteristic
133C2—21no—
134C2—82no—
135C2—10161no—
136C2#2254998611yescharacteristic
subgroupG/H
centerC2#22549986
commutatorC8×C127C22×C4
FrattiniC2×C8C22×D127
Fitting#442×C127C2
socleC2×C127#442
prime radical#22550312C2
cosocleC2×C8×C127C23
hypercenter#442D127
hypocenterC127AGL(1,ℤ/16)
LCSG ▷ C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #22549999C2semidirect
3. #22549698C2semidirect
4. #22549689C2semidirect
5. #22549685C2semidirect
6. #22549654C2semidirect
7. #442×C127C2semidirect
10. #16861153C4semidirect
11. #16861150C4semidirect
22. Dic508C2×C4semidirect
24. D1016C2×C4semidirect
29. C127⋊C16C2×C4semidirect
69. C127AGL(1,ℤ/16)semidirect
72. #442D127semidirect

Maximal subgroups

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

subgroupindex
#225503122
#225499992
#225496982
#225496892
#225496852
#225496542
#442×C1272
AGL(1,ℤ/16)127