Finite Group Databasebeta

← #30640943

Subgroups — #30640943

Order 16256.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550126C211yesmaximal, characteristic
2#22549923C211yesmaximal, characteristic, complemented
3#22549700C211yesmaximal, characteristic, complemented
4#22549699C211yesmaximal, characteristic, complemented
5#22549689C211yesmaximal, characteristic, complemented
6#22549657C211yesmaximal, characteristic, complemented
7#361×C127C211yesmaximal, characteristic, complemented
8C2×#14523231C2211yescharacteristic
9M16×D127—21no—
10#16861180—21no—
11#16861149C2211yescharacteristic
12#16861145C2211yescharacteristic
13#16861144—21no—
14#16861141C2211yescharacteristic
15#16861130—21no—
16C2×D8×C127C2211yescharacteristic
17M32×C127C2211yescharacteristic
18#108×C127C2211yescharacteristic
19C2×C22⋊D127—41no—
20C2×C4×D127D411yescharacteristic
21#14523232—21no—
22#14523231—21no—
23#14523231—21no—
24#14523231—41no—
25C2×C127⋊C8D411yescharacteristic
26C8×D127—41no—
27C4⋊Dic127—21no—
28C22⋊Dic127—41no—
29C127⋊C16—21no—
30C2×D4×C127—21no—
31D8×C127—21no—
32D8×C127—21no—
33M16×C127—21no—
34C2×C8×C127C2311yescharacteristic
35C16×C127—21no—
36C22×D127SD1611yescharacteristic
37C22⋊D127—82no—
38C2×Dic127—41no—
39C2×Dic127SD1611yescharacteristic
40C4×D127—21no—
41C4×D127—21no—
42C127⋊C8—21no—
43C127⋊C8—41no—
44C23×C127—41no—
45D4×C127—41no—
46D4×C127—41no—
47C2×C4×C127C2×D411yescharacteristic
48C8×C127—41no—
49C8×C127#4111yescharacteristic
50C8×C127#4111yescharacteristic
51C22×C127—81no—
52C22×C127—81no—
53C22×C127—81no—
54C22×C127C2×SD1611yescharacteristic
55C2×D127—21no—
56C2×D127—41no—
57C4×C127#12311yescharacteristic
58C4×C127AGL(1,ℤ/8)11yescharacteristic
59Dic127—21no—
60Dic127—81no—
61C2×C127—21no—
62C2×C127—161no—
63C2×C127#48111yescharacteristic
64D127—41no—
65#2238—1271nomaximal, Sylow
66C127#223811yescharacteristic, complemented, Sylow
67#469—1271no—
68#430—1271no—
69#365—1271no—
70#365—1271no—
71#361—1271no—
72#361D12711yescharacteristic, complemented
73#349—1271no—
74C2×D8C2×D12711yescharacteristic
75C2×M16—1271no—
76C2×M16—2541no—
77#121—2541no—
78M32—1271no—
79M32C2×D12711yescharacteristic
80#108—1271no—
81#108—2541no—
82#108C2×D12711yescharacteristic
83#106—1271no—
84#102—2541no—
85C2×D4—21no—
86C2×D4—5081no—
87C22×C4—1271no—
88D8—21no—
89D8—21no—
90M16—21no—
91M16—5081no—
92M16—2541no—
93M16—2541no—
94M16—2541no—
95C2×C8—1271no—
96C2×C8—5081no—
97C2×C8C22×D12711yescharacteristic
98C4⋊C4—2541no—
99C22⋊C4—5081no—
100C16—21no—
101C16—2541no—
102C23—41no—
103C23—1271no—
104D4—41no—
105D4—41no—
106D4—10162no—
107C2×C4—1271no—
108C2×C4—5081no—
109C2×C4—2541no—
110C2×C4—2541no—
111C2×C4D4×D12711yescharacteristic
112C8—41no—
113C8—5081no—
114C8—2541no—
115C8#1452324711yescharacteristic
116C8#1452324811yescharacteristic
117C22—81no—
118C22—81no—
119C22—81no—
120C22—5081no—
121C22—2541no—
122C22SD16×D12711yescharacteristic
123C4—10161no—
124C4—2541no—
125C4#1686119111yescharacteristic
126C4#1686123311yescharacteristic
127C2—21no—
128C2—161no—
129C2—5081no—
130C2#2255019311yescharacteristic
subgroupG/H
centerC2#22550193
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#361×C127C2
socleC2×C127#481
prime radical#22550126C2
hypercenter#361D127
hypocenterC127#2238
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #22549923C2semidirect
3. #22549700C2semidirect
4. #22549699C2semidirect
5. #22549689C2semidirect
6. #22549657C2semidirect
7. #361×C127C2semidirect
66. C127#2238semidirect
72. #361D127semidirect

Maximal subgroups

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

subgroupindex
#225501262
#225499232
#225497002
#225496992
#225496892
#225496572
#361×C1272
#2238127