Finite Group Databasebeta

← C127×#7266 #44570939

Subgroups — C127×#7266 #44570939

Order 32512.

218 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#347×C127C212yesmaximal
2C127×#1836C211yesmaximal, characteristic
3C2×M32×C127—22no—
4C2×#97×C127C2211yescharacteristic
5#384×C127C411yescharacteristic
6#382×C127C411yescharacteristic
7#347×C127—22no—
8#347×C127C414yescomplemented
9#340×C127—22no—
10C22×C8×C127D412yes—
11C2×C4⋊C4×C127—21no—
12C2×C4⋊C4×C127—21no—
13C2×C4⋊C4×C127—41no—
14C2×C42×C127C2×C411yescharacteristic
15M32×C127—24no—
16M32×C127—24no—
17C2×C16×C127—24no—
18#97×C127C2×C412yes—
19#97×C127D411yescharacteristic
20#97×C127Q811yescharacteristic
21#95×C127—41no—
22C22×C4×C127—21no—
23C22×C4×C127—41no—
24C22×C4×C127—41no—
25C22×C4×C127C22⋊C411yescharacteristic
26C2×C8×C127—22no—
27C2×C8×C127—22no—
28C2×C8×C127C22⋊C412yes—
29C2×C8×C127C4⋊C412yes—
30C4⋊C4×C127—42no—
31C4⋊C4×C127—42no—
32C4⋊C4×C127—82no—
33C42×C127—21no—
34C42×C127C4211yescharacteristic
35C42×C127C4⋊C411yescharacteristic
36C16×C127—28no—
37C23×C127#10011yescharacteristic
38C2×C4×C127—21no—
39C2×C4×C127—21no—
40C2×C4×C127—41no—
41C2×C4×C127—41no—
42C2×C4×C127—82no—
43C2×C4×C127—81no—
44C2×C4×C127—81no—
45C2×C4×C127—81no—
46C2×C4×C127—81no—
47C2×C4×C127#9511yescharacteristic
48C2×C4×C127#10111yescharacteristic
49C2×C4×C127#10412yes—
50C2×C4×C127#10412yes—
51C8×C127—22no—
52C8×C127—22no—
53C22×C127—21no—
54C22×C127—21no—
55C22×C127#32911yescharacteristic
56C22×C127#32911yescharacteristic
57C22×C127#34311yescharacteristic
58C4×C127—41no—
59C4×C127—41no—
60C4×C127—161no—
61C4×C127—161no—
62C4×C127#33012yes—
63C4×C127#33912yes—
64#7266C12711yesmaximal, characteristic, direct product factor, Sylow
65C2×C127—21no—
66C2×C127—21no—
67C2×C127#129511yescharacteristic
68C2×C127#135511yescharacteristic
69C2×C127#135611yescharacteristic
70C2×#347C2×C12712yes—
71#1836C2×C12711yescharacteristic
72C127#726611yescharacteristic, direct product factor, Sylow
73C2×M32—22no—
74C2×#97C22×C12711yescharacteristic
75#384C4×C12711yescharacteristic
76#382C4×C12711yescharacteristic
77#347—22no—
78#347C4×C12714yescomplemented
79#340—22no—
80C22×C8D4×C12712yes—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C42C2×C4×C12711yescharacteristic
85M32—24no—
86M32—24no—
87C2×C16—24no—
88#97C2×C4×C12712yes—
89#97D4×C12711yescharacteristic
90#97Q8×C12711yescharacteristic
91#95—41no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4C22⋊C4×C12711yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8C22⋊C4×C12712yes—
99C2×C8C4⋊C4×C12712yes—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—82no—
103C42—21no—
104C42C42×C12711yescharacteristic
105C42C4⋊C4×C12711yescharacteristic
106C16—28no—
107C23#100×C12711yescharacteristic
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—82no—
113C2×C4—81no—
114C2×C4—81no—
115C2×C4—81no—
116C2×C4—81no—
117C2×C4#95×C12711yescharacteristic
118C2×C4#101×C12711yescharacteristic
119C2×C4#104×C12712yes—
120C2×C4#104×C12712yes—
121C8—22no—
122C8—22no—
123C22—21no—
124C22—21no—
125C22#329×C12711yescharacteristic
126C22#329×C12711yescharacteristic
127C22#343×C12711yescharacteristic
128C4—41no—
129C4—41no—
130C4—161no—
131C4—161no—
132C4#330×C12712yes—
133C4#339×C12712yes—
134C2—21no—
135C2—21no—
136C2C127×#129511yescharacteristic
137C2C127×#135511yescharacteristic
138C2C127×#135611yescharacteristic
subgroupG/H
center, socleC22×C127#329
commutatorC42C42×C127
Frattini, cosocleC2×#97C22×C127
prime radicalC23×C127#100
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#97×C127 ◁ G

Complements

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

NH ≅ G/Ntype
8. #347×C127C4semidirect
64. #7266C127direct (N × H)
72. C127#7266direct (N × H)
78. #347C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#347×C1272
C127×#18362
#7266127