Finite Group Databasebeta

← #30636875

Subgroups — #30636875

Order 16256.

122 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 25 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#22549669C211yesmaximal, characteristic, complemented
2#22549620C211yesmaximal, characteristic, complemented
3#453×C127C211yesmaximal, characteristic
4C4⋊(C127⋊C8)—21no—
5#16861124—21no—
6H(5,𝔽2)×C127C411yescharacteristic, complemented
7AGL(1,ℤ/8)×C127—21no—
8#127×C127C2211yescharacteristic
9#104×C127—21no—
10#100×C127C411yescharacteristic, complemented
11C2×C127⋊C8—41no—
12C22⋊Dic127—41no—
13#41×C127—21no—
14#41×C127—41no—
15C2×D4×C127—41no—
16C2×D4×C127—41no—
17C2×D4×C127—41no—
18C2×D4×C127C2×C411yescharacteristic
19C2×D4×C127D411yescharacteristic
20SD16×C127—41no—
21D8×C127—41no—
22M16×C127—21no—
23C42×C127D411yescharacteristic
24C2×Dic127—81no—
25C127⋊C8—81no—
26C23×C127—21no—
27C23×C127—21no—
28C23×C127—41no—
29Q8×C127—21no—
30D4×C127—21no—
31D4×C127—41no—
32D4×C127—41no—
33D4×C127—42no—
34D4×C127—41no—
35D4×C127—81no—
36D4×C127—81no—
37C2×C4×C127—21no—
38C2×C4×C127—41no—
39C2×C4×C127—41no—
40C2×C4×C127C22⋊C411yescharacteristic
41C8×C127—41no—
42C22×C127—41no—
43C22×C127—41no—
44C22×C127—41no—
45C22×C127—81no—
46C22×C127—81no—
47C22×C127—161no—
48C22×C127#9911yescharacteristic
49C4×C127—21no—
50C4×C127—21no—
51C4×C127—21no—
52C4×C127—41no—
53Dic127—161no—
54C2×C127—21no—
55C2×C127—81no—
56C2×C127—81no—
57C2×C127—81no—
58C2×C127AΣL(1,𝔽16)11yescharacteristic
59#1402—1271nomaximal, Sylow
60C127#140211yescharacteristic, complemented, Sylow
61#453D12711yescharacteristic
62#353—1271no—
63#331—1271no—
64H(5,𝔽2)Dic12711yescharacteristic, complemented
65AGL(1,ℤ/8)—21no—
66#127C2×D12711yescharacteristic
67C4⋊C8—2541no—
68#104—21no—
69#100Dic12711yescharacteristic, complemented
70#99—2541no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4C2×Dic12711yescharacteristic
77C2×D4C22⋊D12711yescharacteristic
78SD16—41no—
79D8—41no—
80M16—21no—
81C2×C8—5081no—
82C22⋊C4—5081no—
83C42C22⋊D12711yescharacteristic
84C23—21no—
85C23—21no—
86C23—41no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—41no—
91D4—42no—
92D4—41no—
93D4—81no—
94D4—81no—
95C2×C4—21no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—10161no—
99C2×C4C22⋊Dic12711yescharacteristic
100C8—41no—
101C8—10161no—
102C22—41no—
103C22—41no—
104C22—41no—
105C22—81no—
106C22—81no—
107C22—161no—
108C22#1686112411yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—41no—
113C4—20321no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—81no—
118C2#2254966311yescharacteristic
subgroupG/H
centerC2#22549663
commutatorC2×D4×C127C2×C4
Frattini#127C2×D127
Fitting, prime radical#453×C127C2
socleC2×C127AΣL(1,𝔽16)
cosocle#127×C127C22
hypercenter#453D127
hypocenterC127#1402
LCSG ▷ C2×D4×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22549669C2semidirect
2. #22549620C2semidirect
6. H(5,𝔽2)×C127C4semidirect
10. #100×C127C4semidirect
60. C127#1402semidirect
64. H(5,𝔽2)Dic127semidirect
69. #100Dic127semidirect

Maximal subgroups

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

subgroupindex
#225496692
#225496202
#453×C1272
#1402127