Finite Group Databasebeta

← C127×#7233 #44570906

Subgroups — C127×#7233 #44570906

Order 32512.

308 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1807C212yesmaximal
3C23×C8×C127C2211yescharacteristic
4C2×M32×C127C412yescomplemented
5C2×#106×C127—22no—
6#380×C127—22no—
7C22⋊C16×C127—22no—
8#340×C127C414yescomplemented
9C23×C4×C127D411yescharacteristic
10C22×C8×C127—42no—
11C22×C8×C127C2×C411yescharacteristic
12C22×C8×C127C2×C412yes—
13C22×C8×C127D412yes—
14C22×C8×C127Q811yescharacteristic
15C2×C4⋊C4×C127—24no—
16C2×C22⋊C4×C127—42no—
17M32×C127—42no—
18C2×C16×C127—22no—
19#106×C127—44no—
20#95×C127—42no—
21C24×C127D811yescharacteristic
22C22×C4×C127—41no—
23C22×C4×C127—41no—
24C22×C4×C127—44no—
25C22×C4×C127C22⋊C412yes—
26C22×C4×C127C4⋊C411yescharacteristic
27C22×C4×C127SD1612yes—
28C22×C4×C127Q1611yescharacteristic
29C2×C8×C127—24no—
30C2×C8×C127—22no—
31C2×C8×C127—42no—
32C2×C8×C127—41no—
33C2×C8×C127C4211yescharacteristic
34C2×C8×C127C22⋊C411yescharacteristic
35C2×C8×C127C4⋊C412yes—
36C4⋊C4×C127—44no—
37C4⋊C4×C127—44no—
38C22⋊C4×C127—84no—
39C16×C127—42no—
40C23×C127—24no—
41C23×C127—41no—
42C23×C127#10212yes—
43C23×C127#10711yescharacteristic
44C2×C4×C127—22no—
45C2×C4×C127—21no—
46C2×C4×C127—22no—
47C2×C4×C127—21no—
48C2×C4×C127—42no—
49C2×C4×C127—41no—
50C2×C4×C127—84no—
51C2×C4×C127—84no—
52C2×C4×C127—84no—
53C2×C4×C127#9511yescharacteristic
54C2×C4×C127#10312yes—
55C2×C4×C127#10611yescharacteristic
56C8×C127—21no—
57C8×C127—22no—
58C8×C127#9912yes—
59C22×C127—22no—
60C22×C127—22no—
61C22×C127—21no—
62C22×C127—22no—
63C22×C127—44no—
64C22×C127#34011yescharacteristic
65C22×C127#36014yes—
66C4×C127—22no—
67C4×C127—21no—
68C4×C127—164no—
69C4×C127H(3,ℤ/4)11yescharacteristic
70C4×C127#34211yescharacteristic
71#7233C12711yesmaximal, characteristic, direct product factor, Sylow
72C2×C127—24no—
73C2×C127—41no—
74C2×C127#128911yescharacteristic
75C2×C127#138612yes—
76#2112C2×C12711yescharacteristic
77#1807C2×C12712yes—
78C127#723311yescharacteristic, direct product factor, Sylow
79C23×C8C22×C12711yescharacteristic
80C2×M32C4×C12712yescomplemented
81C2×#106—22no—
82#380—22no—
83C22⋊C16—22no—
84#340C4×C12714yescomplemented
85C23×C4D4×C12711yescharacteristic
86C22×C8—42no—
87C22×C8C2×C4×C12711yescharacteristic
88C22×C8C2×C4×C12712yes—
89C22×C8D4×C12712yes—
90C22×C8Q8×C12711yescharacteristic
91C2×C4⋊C4—24no—
92C2×C22⋊C4—42no—
93M32—42no—
94C2×C16—22no—
95#106—44no—
96#95—42no—
97C24D8×C12711yescharacteristic
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—44no—
101C22×C4C22⋊C4×C12712yes—
102C22×C4C4⋊C4×C12711yescharacteristic
103C22×C4SD16×C12712yes—
104C22×C4Q16×C12711yescharacteristic
105C2×C8—24no—
106C2×C8—22no—
107C2×C8—42no—
108C2×C8—41no—
109C2×C8C42×C12711yescharacteristic
110C2×C8C22⋊C4×C12711yescharacteristic
111C2×C8C4⋊C4×C12712yes—
112C4⋊C4—44no—
113C4⋊C4—44no—
114C22⋊C4—84no—
115C16—42no—
116C23—24no—
117C23—41no—
118C23#102×C12712yes—
119C23#107×C12711yescharacteristic
120C2×C4—22no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—21no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—84no—
127C2×C4—84no—
128C2×C4—84no—
129C2×C4#95×C12711yescharacteristic
130C2×C4#103×C12712yes—
131C2×C4#106×C12711yescharacteristic
132C8—21no—
133C8—22no—
134C8#99×C12712yes—
135C22—22no—
136C22—22no—
137C22—21no—
138C22—22no—
139C22—44no—
140C22#340×C12711yescharacteristic
141C22#360×C12714yes—
142C4—22no—
143C4—21no—
144C4—164no—
145C4H(3,ℤ/4)×C12711yescharacteristic
146C4#342×C12711yescharacteristic
147C2—24no—
148C2—41no—
149C2C127×#128911yescharacteristic
150C2C127×#138612yes—
subgroupG/H
center, socleC22×C127#340
commutatorC2×C8C42×C127
Frattini, cosocleC23×C8C22×C127
prime radicalC24×C127D8
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
4. C2×M32×C127C4semidirect
8. #340×C127C4semidirect
71. #7233C127direct (N × H)
78. C127#7233direct (N × H)
80. C2×M32C4×C127semidirect
84. #340C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#21122
C127×#18072
#7233127