Finite Group Databasebeta

← C127×#15947 #44579620

Subgroups — C127×#15947 #44579620

Order 32512.

6146 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 6146 conjugacy classes of subgroups — too many to draw. Only its 494 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
1C22×#120×C127C216yesmaximal, complemented
2C127×#1438C219yesmaximal, complemented
3C26×C127C2212yescomplemented
4C23×D4×C127C2219yes—
5C2×#120×C127—272no—
6C22×C22⋊C4×C127C22118yescomplemented
7C2×#95×C127C2216yescomplemented
8C25×C127—236no—
9C25×C127—412no—
10C25×C127C2316yes—
11C22×D4×C127—272no—
12C22×D4×C127—418no—
13C22×D4×C127D4136yescomplemented
14C23×C4×C127C2319yes—
15#120×C127—2192no—
16C2×C22⋊C4×C127—236no—
17C2×C22⋊C4×C127—272no—
18#95×C127—46no—
19C24×C127—236no—
20C24×C127—236no—
21C24×C127—248no—
22C24×C127—4144no—
23C24×C127—472no—
24C24×C127—416no—
25C24×C127—424no—
26C24×C127C2×D4136yes—
27C24×C127C2411yescharacteristic
28C2×D4×C127—2144no—
29C2×D4×C127—436no—
30C2×D4×C127—4144no—
31C22×C4×C127—236no—
32C22×C4×C127—218no—
33C22×C4×C127C2×D4118yes—
34C22⋊C4×C127—2144no—
35C23×C127—236no—
36C23×C127—236no—
37C23×C127—2144no—
38C23×C127—4288no—
39C23×C127—4192no—
40C23×C127—412no—
41C23×C127—436no—
42C23×C127—436no—
43C23×C127—412no—
44C23×C127#120124yescomplemented
45C23×C127C22×D419yes—
46C23×C127H(5,𝔽2)16yes—
47D4×C127—4144no—
48C2×C4×C127—236no—
49C2×C4×C127—218no—
50C2×C4×C127—436no—
51C22×C127—272no—
52C22×C127—4128no—
53C22×C127—472no—
54C22×C127—472no—
55C22×C127C2×#12016yes—
56C22×C127#53419yes—
57C22×C127D42118yes—
58C22×C127H(3,𝔽4)12yes—
59C4×C127—418no—
60#15947C12711yesmaximal, characteristic, direct product factor, Sylow
61C2×C127—424no—
62C2×C127#240319yes—
63C2×C127#267916yes—
64C22×#120C2×C12716yescomplemented
65#1438C2×C12719yescomplemented
66C127#1594711yescharacteristic, direct product factor, Sylow
67C26C22×C12712yescomplemented
68C23×D4C22×C12719yes—
69C2×#120—272no—
70C22×C22⋊C4C22×C127118yescomplemented
71C2×#95C22×C12716yescomplemented
72C25—236no—
73C25—412no—
74C25C23×C12716yes—
75C22×D4—272no—
76C22×D4—418no—
77C22×D4D4×C127136yescomplemented
78C23×C4C23×C12719yes—
79#120—2192no—
80C2×C22⋊C4—236no—
81C2×C22⋊C4—272no—
82#95—46no—
83C24—236no—
84C24—236no—
85C24—248no—
86C24—4144no—
87C24—472no—
88C24—416no—
89C24—424no—
90C24C2×D4×C127136yes—
91C24C24×C12711yescharacteristic
92C2×D4—2144no—
93C2×D4—436no—
94C2×D4—4144no—
95C22×C4—236no—
96C22×C4—218no—
97C22×C4C2×D4×C127118yes—
98C22⋊C4—2144no—
99C23—236no—
100C23—236no—
101C23—2144no—
102C23—4288no—
103C23—4192no—
104C23—412no—
105C23—436no—
106C23—436no—
107C23—412no—
108C23#120×C127124yescomplemented
109C23C22×D4×C12719yes—
110C23H(5,𝔽2)×C12716yes—
111D4—4144no—
112C2×C4—236no—
113C2×C4—218no—
114C2×C4—436no—
115C22—272no—
116C22—4128no—
117C22—472no—
118C22—472no—
119C22C2×#120×C12716yes—
120C22#534×C12719yes—
121C22D42×C127118yes—
122C22H(3,𝔽4)×C12712yes—
123C4—418no—
124C2—424no—
125C2C127×#240319yes—
126C2C127×#267916yes—
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×#120×C127C2semidirect
2. C127×#1438C2semidirect
3. C26×C127C22semidirect
6. C22×C22⋊C4×C127C22semidirect
7. C2×#95×C127C22semidirect
13. C22×D4×C127D4semidirect
44. C23×C127#120semidirect
60. #15947C127direct (N × H)
64. C22×#120C2×C127semidirect
65. #1438C2×C127semidirect
66. C127#15947direct (N × H)
67. C26C22×C127semidirect
70. C22×C22⋊C4C22×C127semidirect
71. C2×#95C22×C127semidirect
77. C22×D4D4×C127semidirect
108. C23#120×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#120×C1272
C127×#14382
#15947127