Finite Group Databasebeta

← #30636615

Subgroups — #30636615

Order 16256.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C4×C22⋊Dic127C212yesmaximal
2#407×C127C211yesmaximal, characteristic
3C2×C22⋊Dic127—22no—
4C2×C4×Dic127—22no—
5C127⋊#95C414yescomplemented
6C23×C4×C127C2211yescharacteristic
7C2×M16×C127C412yescomplemented
8C22⋊C8×C127—22no—
9C22×Dic127—24no—
10C22⋊Dic127—28no—
11C4×Dic127—44no—
12C24×C127D411yescharacteristic
13C22×C4×C127—42no—
14C22×C4×C127C2×C412yes—
15C22×C4×C127C2×C411yescharacteristic
16C22×C4×C127D412yes—
17C22×C4×C127Q811yescharacteristic
18M16×C127—42no—
19C2×C8×C127—22no—
20C2×Dic127—28no—
21C2×Dic127—44no—
22C2×Dic127—44no—
23C23×C127—24no—
24C23×C127—41no—
25C23×C127C22⋊C412yes—
26C23×C127C4⋊C411yescharacteristic
27C2×C4×C127—22no—
28C2×C4×C127—24no—
29C2×C4×C127—42no—
30C2×C4×C127—41no—
31C2×C4×C127C4211yescharacteristic
32C2×C4×C127C22⋊C411yescharacteristic
33C2×C4×C127C4⋊C412yes—
34C8×C127—42no—
35C22×C127—22no—
36C22×C127—22no—
37C22×C127—22no—
38C22×C127—21no—
39C22×C127—44no—
40C22×C127#9511yescharacteristic
41C22×C127#10414yes—
42C4×C127—21no—
43C4×C127—22no—
44C4×C127#9912yes—
45Dic127—48no—
46C2×C127—24no—
47C2×C127—41no—
48C2×C127#33912yes—
49C2×C127#34211yescharacteristic
50#1297—1271nomaximal, Sylow
51C127#129711yescharacteristic, complemented, Sylow
52#407D12711yescharacteristic
53C4×C22⋊C4—1272no—
54C23×C4C2×D12711yescharacteristic
55C2×M16Dic12712yescomplemented
56C2×C22⋊C4—2542no—
57C2×C42—2542no—
58C22⋊C8—22no—
59#95—1274no—
60C24D50811yescharacteristic
61C22×C4—42no—
62C22×C4—2544no—
63C22×C4C4×D12712yes—
64C22×C4C2×Dic12711yescharacteristic
65C22×C4C22⋊D12712yes—
66C22×C4Dic25411yescharacteristic
67M16—42no—
68C2×C8—22no—
69C22⋊C4—2548no—
70C42—5084no—
71C23—24no—
72C23—41no—
73C23(C2×D127)⋊C412yes—
74C23C4⋊Dic12711yescharacteristic
75C2×C4—24no—
76C2×C4—22no—
77C2×C4—42no—
78C2×C4—41no—
79C2×C4—5084no—
80C2×C4—5084no—
81C2×C4—2548no—
82C2×C4C4×Dic12711yescharacteristic
83C2×C4C22⋊Dic12711yescharacteristic
84C2×C4Dic127⋊C412yes—
85C8—42no—
86C22—22no—
87C22—22no—
88C22—22no—
89C22—21no—
90C22—44no—
91C22C127⋊#9511yescharacteristic
92C22#1686113514yes—
93C4—21no—
94C4—22no—
95C4—5088no—
96C4#1686112412yes—
97C2—24no—
98C2—41no—
99C2#2254963712yes—
100C2#2254964311yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic127
commutatorC2×C4×C127C42
FrattiniC23×C4C2×D127
Fitting#407×C127C2
socleC22×C127#95
prime radicalC24×C127D4
cosocleC23×C4×C127C22
hypercenter#407D127
hypocenterC127#1297
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
5. C127⋊#95C4semidirect
7. C2×M16×C127C4semidirect
51. C127#1297semidirect
55. C2×M16Dic127semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Dic1272
#407×C1272
#1297127