Finite Group Databasebeta

← #30636666

Subgroups — #30636666

Order 16256.

128 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 37 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#22549658C212yesmaximal, complemented
2#407×C127C211yesmaximal, characteristic
3#16861149—22no—
4C23×C4×C127C411yescharacteristic
5C2×M16×C127C411yescharacteristic
6C2×M16×C127C2211yescharacteristic
7C22⋊C8×C127—22no—
8C127⋊C16—42no—
9C24×C127C811yescharacteristic
10C22×C4×C127—21no—
11C22×C4×C127—21no—
12C22×C4×C127—41no—
13C22×C4×C127—41no—
14C22×C4×C127C811yescharacteristic
15C22×C4×C127C2×C411yescharacteristic
16M16×C127—41no—
17M16×C127—41no—
18C2×C8×C127—21no—
19C2×C8×C127D412yes—
20C23×C127—21no—
21C23×C127—41no—
22C23×C127—42no—
23C23×C127C2×C811yescharacteristic
24C2×C4×C127—21no—
25C2×C4×C127—21no—
26C2×C4×C127—21no—
27C2×C4×C127—42no—
28C2×C4×C127—41no—
29C2×C4×C127—41no—
30C2×C4×C127—41no—
31C2×C4×C127C22⋊C411yescharacteristic
32C2×C4×C127M1611yescharacteristic
33C8×C127—22no—
34C8×C127—41no—
35C22×C127—22no—
36C22×C127—21no—
37C22×C127—41no—
38C22×C127—41no—
39C22×C127—42no—
40C22×C127—42no—
41C22×C127—41no—
42C22×C127C22⋊C811yescharacteristic
43C4×C127—21no—
44C4×C127—41no—
45C4×C127#9911yescharacteristic
46C4×C127#10011yescharacteristic
47C2×C127—21no—
48C2×C127—42no—
49C2×C127—41no—
50C2×C127#32311yescharacteristic
51#1320—1271nomaximal, Sylow
52C127#132011yescharacteristic, complemented, Sylow
53#407D12711yescharacteristic
54#349—1272no—
55C23×C4Dic12711yescharacteristic
56C2×M16Dic12711yescharacteristic
57C2×M16C2×D12711yescharacteristic
58M32—2542no—
59C22⋊C8—22no—
60C24C127⋊C811yescharacteristic
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4C127⋊C811yescharacteristic
66C22×C4C2×Dic12711yescharacteristic
67M16—41no—
68M16—41no—
69C2×C8—21no—
70C2×C8C22⋊D12712yes—
71C16—5082no—
72C23—21no—
73C23—41no—
74C23—42no—
75C23C2×C127⋊C811yescharacteristic
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—42no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4C22⋊Dic12711yescharacteristic
84C2×C4#1452323211yescharacteristic
85C8—22no—
86C8—41no—
87C22—22no—
88C22—21no—
89C22—41no—
90C22—41no—
91C22—42no—
92C22—42no—
93C22—41no—
94C22C22⋊(C127⋊C8)11yescharacteristic
95C4—21no—
96C4—41no—
97C4#1686112411yescharacteristic
98C4#1686112611yescharacteristic
99C2—21no—
100C2—42no—
101C2—41no—
102C2#2254960011yescharacteristic
subgroupG/H
centerC4#16861124
commutatorC23×C127C2×C8
FrattiniC2×M16C2×D127
Fitting#407×C127C2
socleC2×C127#323
prime radicalC24×C127C8
cosocleC2×M16×C127C22
hypercenter#407D127
hypocenterC127#1320
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #22549658C2semidirect
52. C127#1320semidirect

Maximal subgroups

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

subgroupindex
#225496582
#407×C1272
#1320127