Finite Group Databasebeta

← #30636876

Subgroups — #30636876

Order 16256.

122 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 23 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#22550043C211yesmaximal, characteristic
2#22549669C211yesmaximal, characteristic, complemented
3#331×C127C211yesmaximal, characteristic, complemented
4#16861254C411yescharacteristic, complemented
5#16861235—21no—
6#16861137—21no—
7#16861126C411yescharacteristic, complemented
8#16861124—21no—
9#127×C127C2211yescharacteristic
10C4⋊C8×C127—21no—
11#14523248—21no—
12#14523247—41no—
13C2×C22⋊D127—41no—
14D4×D127—41no—
15#14523236—41no—
16#14523233—41no—
17#14523232—21no—
18C22⋊Dic127—41no—
19C2×D4×C127—41no—
20C2×D4×C127C2×C411yescharacteristic
21C2×D4×C127D411yescharacteristic
22C2×C8×C127—41no—
23C42×C127D411yescharacteristic
24C22×D127—41no—
25Dic254—21no—
26D508—21no—
27C22⋊D127—42no—
28C22⋊D127—41no—
29C2×Dic127—41no—
30C2×Dic127—81no—
31C4×D127—41no—
32C127⋊C8—41no—
33C23×C127—21no—
34C23×C127—21no—
35D4×C127—41no—
36D4×C127—41no—
37D4×C127—81no—
38D4×C127—81no—
39C2×C4×C127—21no—
40C2×C4×C127C22⋊C411yescharacteristic
41C8×C127—81no—
42C22×C127—41no—
43C22×C127—41no—
44C22×C127—81no—
45C22×C127—81no—
46C22×C127#9911yescharacteristic
47C2×D127—41no—
48C2×D127—161no—
49C4×C127—21no—
50C4×C127—21no—
51C4×C127—21no—
52Dic127—41no—
53Dic127—161no—
54C2×C127—21no—
55C2×C127—81no—
56C2×C127—81no—
57C2×C127AΣL(1,𝔽16)11yescharacteristic
58D127—81no—
59#1402—1271nomaximal, Sylow
60C127#140211yescharacteristic, complemented, Sylow
61#453—1271no—
62#353—1271no—
63#331D12711yescharacteristic, complemented
64H(5,𝔽2)—1271no—
65AGL(1,ℤ/8)—2541no—
66#127C2×D12711yescharacteristic
67C4⋊C8—21no—
68#104—2541no—
69#100—1271no—
70#99—2541no—
71#41—5081no—
72#41—2541no—
73C2×D4—41no—
74C2×D4—5081no—
75C2×D4—5081no—
76C2×D4C4×D12711yescharacteristic
77C2×D4D50811yescharacteristic
78SD16—5081no—
79D8—5081no—
80M16—2541no—
81C2×C8—41no—
82C22⋊C4—5081no—
83C42C22⋊D12711yescharacteristic
84C23—21no—
85C23—21no—
86C23—5081no—
87Q8—2541no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—5082no—
93D4—5081no—
94D4—2541no—
95C2×C4—21no—
96C2×C4—10161no—
97C2×C4—5081no—
98C2×C4—5081no—
99C2×C4(C2×D127)⋊C411yescharacteristic
100C8—81no—
101C8—5081no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—20321no—
107C22—5081no—
108C22#1686112311yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—20321no—
113C4—5081no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—10161no—
118C2#2254966411yescharacteristic
subgroupG/H
centerC2#22549664
commutatorC2×D4×C127C2×C4
Frattini#127C2×D127
Fitting#331×C127C2
socleC2×C127AΣL(1,𝔽16)
prime radical#22550043C2
cosocle#127×C127C22
hypercenter#331D127
hypocenterC127#1402
LCSG ▷ C2×D4×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22549669C2semidirect
3. #331×C127C2semidirect
4. #16861254C4semidirect
7. #16861126C4semidirect
60. C127#1402semidirect
63. #331D127semidirect

Maximal subgroups

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

subgroupindex
#225500432
#225496692
#331×C1272
#1402127