Finite Group Databasebeta

← C127×#7234 #44570907

Subgroups — C127×#7234 #44570907

Order 32512.

264 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C127×#2112C211yesmaximal, characteristic
2C127×#1810C212yesmaximal
3C23×C8×C127C2211yescharacteristic
4C2×M32×C127C412yes—
5C2×#108×C127—22no—
6#407×C127—22no—
7C22⋊C16×C127—22no—
8#341×C127C414yes—
9C23×C4×C127D411yescharacteristic
10C2×M16×C127—24no—
11C22×C8×C127—42no—
12C22×C8×C127C2×C411yescharacteristic
13C22×C8×C127C2×C412yes—
14C22×C8×C127D412yes—
15C22×C8×C127Q811yescharacteristic
16M32×C127—42no—
17C2×C16×C127—22no—
18#108×C127—44no—
19C22⋊C8×C127—42no—
20C22⋊C8×C127—42no—
21C24×C127Q1611yescharacteristic
22C22×C4×C127—41no—
23C22×C4×C127—41no—
24C22×C4×C127C22⋊C412yes—
25C22×C4×C127C4⋊C411yescharacteristic
26C22×C4×C127D811yescharacteristic
27C22×C4×C127SD1612yes—
28M16×C127—44no—
29M16×C127—44no—
30C2×C8×C127—24no—
31C2×C8×C127—22no—
32C2×C8×C127—42no—
33C2×C8×C127—41no—
34C2×C8×C127—44no—
35C2×C8×C127C4211yescharacteristic
36C2×C8×C127C22⋊C411yescharacteristic
37C2×C8×C127C4⋊C412yes—
38C16×C127—42no—
39C23×C127—41no—
40C23×C127—42no—
41C23×C127#10312yes—
42C23×C127#10711yescharacteristic
43C2×C4×C127—22no—
44C2×C4×C127—21no—
45C2×C4×C127—22no—
46C2×C4×C127—21no—
47C2×C4×C127—41no—
48C2×C4×C127—42no—
49C2×C4×C127#9511yescharacteristic
50C2×C4×C127#10212yes—
51C2×C4×C127#10611yescharacteristic
52C8×C127—21no—
53C8×C127—22no—
54C8×C127—84no—
55C8×C127#9912yes—
56C22×C127—22no—
57C22×C127—21no—
58C22×C127—24no—
59C22×C127—22no—
60C22×C127—44no—
61C22×C127#34011yescharacteristic
62C4×C127—22no—
63C4×C127—21no—
64C4×C127H(3,ℤ/4)11yescharacteristic
65C4×C127#34211yescharacteristic
66#7234C12711yesmaximal, characteristic, direct product factor, Sylow
67C2×C127—42no—
68C2×C127—41no—
69C2×C127#128911yescharacteristic
70C2×C127#138812yes—
71#2112C2×C12711yescharacteristic
72#1810C2×C12712yes—
73C127#723411yescharacteristic, direct product factor, Sylow
74C23×C8C22×C12711yescharacteristic
75C2×M32C4×C12712yes—
76C2×#108—22no—
77#407—22no—
78C22⋊C16—22no—
79#341C4×C12714yes—
80C23×C4D4×C12711yescharacteristic
81C2×M16—24no—
82C22×C8—42no—
83C22×C8C2×C4×C12711yescharacteristic
84C22×C8C2×C4×C12712yes—
85C22×C8D4×C12712yes—
86C22×C8Q8×C12711yescharacteristic
87M32—42no—
88C2×C16—22no—
89#108—44no—
90C22⋊C8—42no—
91C22⋊C8—42no—
92C24Q16×C12711yescharacteristic
93C22×C4—41no—
94C22×C4—41no—
95C22×C4C22⋊C4×C12712yes—
96C22×C4C4⋊C4×C12711yescharacteristic
97C22×C4D8×C12711yescharacteristic
98C22×C4SD16×C12712yes—
99M16—44no—
100M16—44no—
101C2×C8—24no—
102C2×C8—22no—
103C2×C8—42no—
104C2×C8—41no—
105C2×C8—44no—
106C2×C8C42×C12711yescharacteristic
107C2×C8C22⋊C4×C12711yescharacteristic
108C2×C8C4⋊C4×C12712yes—
109C16—42no—
110C23—41no—
111C23—42no—
112C23#103×C12712yes—
113C23#107×C12711yescharacteristic
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4#95×C12711yescharacteristic
121C2×C4#102×C12712yes—
122C2×C4#106×C12711yescharacteristic
123C8—21no—
124C8—22no—
125C8—84no—
126C8#99×C12712yes—
127C22—22no—
128C22—21no—
129C22—24no—
130C22—22no—
131C22—44no—
132C22#340×C12711yescharacteristic
133C4—22no—
134C4—21no—
135C4H(3,ℤ/4)×C12711yescharacteristic
136C4#342×C12711yescharacteristic
137C2—42no—
138C2—41no—
139C2C127×#128911yescharacteristic
140C2C127×#138812yes—
subgroupG/H
center, socleC22×C127#340
commutatorC2×C8C42×C127
Frattini, cosocleC23×C8C22×C127
prime radicalC24×C127Q16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
66. #7234C127direct (N × H)
73. C127#7234direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#21122
C127×#18102
#7234127