Finite Group Databasebeta

← C127×#1687 #30634639

Subgroups — C127×#1687 #30634639

Order 16256.

198 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 68 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#493×C127C211yesmaximal, characteristic, complemented
2#459×C127C212yesmaximal, complemented
3#436×C127C211yesmaximal, characteristic, complemented
4#334×C127C211yesmaximal, characteristic, complemented
5#331×C127C212yesmaximal, complemented
6C2×D8×C127—22no—
7C2×D8×C127—21no—
8C2×D8×C127—21no—
9C2×M16×C127—21no—
10#127×C127C2212yes—
11C4×D4×C127C2211yescharacteristic
12C4⋊C8×C127C2211yescharacteristic
13C4⋊C8×C127C2212yes—
14#102×C127—22no—
15C22⋊C8×C127—21no—
16C4×C8×C127C2211yescharacteristic
17C2×D4×C127—22no—
18C2×D4×C127—22no—
19C2×D4×C127—22no—
20C2×D4×C127D411yescharacteristic
21C22×C4×C127—21no—
22D8×C127—42no—
23D8×C127—44no—
24D8×C127—42no—
25M16×C127—42no—
26C2×C8×C127—21no—
27C2×C8×C127—22no—
28C2×C8×C127D411yescharacteristic
29C2×C8×C127D411yescharacteristic
30C4⋊C4×C127D411yescharacteristic
31C22⋊C4×C127—21no—
32C42×C127C2311yescharacteristic
33C23×C127—21no—
34C23×C127—42no—
35D4×C127—22no—
36D4×C127—42no—
37D4×C127—42no—
38D4×C127—42no—
39D4×C127—42no—
40D4×C127—82no—
41C2×C4×C127—21no—
42C2×C4×C127—41no—
43C2×C4×C127—41no—
44C2×C4×C127C2×D411yescharacteristic
45C2×C4×C127C2×D411yescharacteristic
46C2×C4×C127#4111yescharacteristic
47C8×C127—21no—
48C8×C127—41no—
49C8×C127—42no—
50C8×C127D812yescomplemented
51C22×C127—41no—
52C22×C127—41no—
53C22×C127—41no—
54C22×C127—82no—
55C22×C127—82no—
56C22×C127—82no—
57C22×C127#12111yescharacteristic
58C4×C127—21no—
59C4×C127—41no—
60C4×C127C2×D811yescharacteristic
61C4×C127AGL(1,ℤ/8)11yescharacteristic
62C4×C127AGL(1,ℤ/8)12yes—
63C2×C127—81no—
64C2×C127—162no—
65C2×C127#45911yescharacteristic
66C2×C127#46911yescharacteristic
67C2×C127#47211yescharacteristic
68#1687C12711yesmaximal, characteristic, direct product factor, Sylow
69C127#168711yescharacteristic, direct product factor, Sylow
70#493C2×C12711yescharacteristic, complemented
71#459C2×C12712yescomplemented
72#436C2×C12711yescharacteristic, complemented
73#334C2×C12711yescharacteristic, complemented
74#331C2×C12712yescomplemented
75C2×D8—22no—
76C2×D8—21no—
77C2×D8—21no—
78C2×M16—21no—
79#127C22×C12712yes—
80C4×D4C22×C12711yescharacteristic
81C4⋊C8C22×C12711yescharacteristic
82C4⋊C8C22×C12712yes—
83#102—22no—
84C22⋊C8—21no—
85C4×C8C22×C12711yescharacteristic
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4D4×C12711yescharacteristic
90C22×C4—21no—
91D8—42no—
92D8—44no—
93D8—42no—
94M16—42no—
95C2×C8—21no—
96C2×C8—22no—
97C2×C8D4×C12711yescharacteristic
98C2×C8D4×C12711yescharacteristic
99C4⋊C4D4×C12711yescharacteristic
100C22⋊C4—21no—
101C42C23×C12711yescharacteristic
102C23—21no—
103C23—42no—
104D4—22no—
105D4—42no—
106D4—42no—
107D4—42no—
108D4—42no—
109D4—82no—
110C2×C4—21no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4C2×D4×C12711yescharacteristic
114C2×C4C2×D4×C12711yescharacteristic
115C2×C4#41×C12711yescharacteristic
116C8—21no—
117C8—41no—
118C8—42no—
119C8D8×C12712yescomplemented
120C22—41no—
121C22—41no—
122C22—41no—
123C22—82no—
124C22—82no—
125C22—82no—
126C22#121×C12711yescharacteristic
127C4—21no—
128C4—41no—
129C4C2×D8×C12711yescharacteristic
130C4AGL(1,ℤ/8)×C12711yescharacteristic
131C4AGL(1,ℤ/8)×C12712yes—
132C2—81no—
133C2—162no—
134C2#459×C12711yescharacteristic
135C2#469×C12711yescharacteristic
136C2#472×C12711yescharacteristic
subgroupG/H
center, socleC22×C127#121
commutator, Frattini, cosocleC42C23×C127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #493×C127C2semidirect
2. #459×C127C2semidirect
3. #436×C127C2semidirect
4. #334×C127C2semidirect
5. #331×C127C2semidirect
50. C8×C127D8semidirect
68. #1687C127direct (N × H)
69. C127#1687direct (N × H)
70. #493C2×C127semidirect
71. #459C2×C127semidirect
72. #436C2×C127semidirect
73. #334C2×C127semidirect
74. #331C2×C127semidirect
119. C8D8×C127semidirect

Maximal subgroups

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

subgroupindex
#493×C1272
#459×C1272
#436×C1272
#334×C1272
#331×C1272
#1687127