Finite Group Databasebeta

← C127×#12684 #44576357

Subgroups — C127×#12684 #44576357

Order 32512.

894 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 210 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×#3573C211yesmaximal, characteristic
2C127×#2115C213yesmaximal, complemented
3C127×#2113C213yesmaximal, complemented
4#585×C127C411yescharacteristic
5#568×C127—26no—
6#568×C127C413yes—
7C2×#131×C127—26no—
8C2×#131×C127C2213yes—
9C2×M32×C127C2213yescomplemented
10C22×C16×C127C2211yescharacteristic, complemented
11#349×C127—26no—
12C22⋊C16×C127—26no—
132−1+4×C127—23no—
14H(5,𝔽2)×C127—23no—
15H(5,𝔽2)×C127C814yes—
16C2×#41×C127—26no—
17C2×#41×C127C2×C413yes—
18#131×C127—24no—
19#131×C127—212no—
20#131×C127—212no—
21#131×C127—26no—
22#131×C127—26no—
23C2×M16×C127—212no—
24C2×M16×C127—26no—
25C2×M16×C127C2×C413yes—
26C2×M16×C127D416yes—
27C22×C8×C127—24no—
28C22×C8×C127D416yes—
29C22×C8×C127C2311yescharacteristic
30M32×C127—26no—
31C2×C16×C127—23no—
32C2×C16×C127—23no—
33#41×C127—24no—
34#41×C127—212no—
35#41×C127—212no—
36#41×C127—26no—
37#41×C127—26no—
38C2×Q8×C127—26no—
39C2×Q8×C127M1613yes—
40C2×D4×C127—26no—
41C2×D4×C127—212no—
42C2×D4×C127C2×C816yes—
43C2×D4×C127M1613yes—
44C22×C4×C127—24no—
45C22×C4×C127C22⋊C416yes—
46C22×C4×C127C22×C411yescharacteristic
47M16×C127—212no—
48M16×C127—26no—
49M16×C127—212no—
50C2×C8×C127—212no—
51C2×C8×C127—26no—
52C2×C8×C127—26no—
53C2×C8×C127C22⋊C416yes—
54C2×C8×C127C2×D413yes—
55C2×C8×C127C2×D413yes—
56C16×C127—43no—
57C16×C127—41no—
58C23×C127—24no—
59C23×C127C22⋊C816yes—
60C23×C127C22×C811yescharacteristic
61Q8×C127—24no—
62Q8×C127—26no—
63D4×C127—212no—
64D4×C127—212no—
65D4×C127—26no—
66C2×C4×C127—212no—
67C2×C4×C127—26no—
68C2×C4×C127—26no—
69C2×C4×C127C22⋊C816yes—
70C2×C4×C127C2×C22⋊C413yes—
71C2×C4×C127C2×M1613yes—
72C8×C127—26no—
73C8×C127#12013yes—
74C8×C127#12011yescharacteristic
75C22×C127—26no—
76C22×C127—41no—
77C22×C127—46no—
78C22×C127—84no—
79C22×C127C2×C22⋊C813yes—
80C4×C127—26no—
81C4×C127#37911yescharacteristic
82C4×C127#40713yes—
83#12684C12711yesmaximal, characteristic, direct product factor, Sylow
84C2×C127—23no—
85C2×C127—46no—
86C2×C127#177911yescharacteristic
87#3573C2×C12711yescharacteristic
88#2115C2×C12713yescomplemented
89#2113C2×C12713yescomplemented
90C127#1268411yescharacteristic, direct product factor, Sylow
91#585C4×C12711yescharacteristic
92#568—26no—
93#568C4×C12713yes—
94C2×#131—26no—
95C2×#131C22×C12713yes—
96C2×M32C22×C12713yescomplemented
97C22×C16C22×C12711yescharacteristic, complemented
98#349—26no—
99C22⋊C16—26no—
1002−1+4—23no—
101H(5,𝔽2)—23no—
102H(5,𝔽2)C8×C12714yes—
103C2×#41—26no—
104C2×#41C2×C4×C12713yes—
105#131—24no—
106#131—212no—
107#131—212no—
108#131—26no—
109#131—26no—
110C2×M16—212no—
111C2×M16—26no—
112C2×M16C2×C4×C12713yes—
113C2×M16D4×C12716yes—
114C22×C8—24no—
115C22×C8D4×C12716yes—
116C22×C8C23×C12711yescharacteristic
117M32—26no—
118C2×C16—23no—
119C2×C16—23no—
120#41—24no—
121#41—212no—
122#41—212no—
123#41—26no—
124#41—26no—
125C2×Q8—26no—
126C2×Q8M16×C12713yes—
127C2×D4—26no—
128C2×D4—212no—
129C2×D4C2×C8×C12716yes—
130C2×D4M16×C12713yes—
131C22×C4—24no—
132C22×C4C22⋊C4×C12716yes—
133C22×C4C22×C4×C12711yescharacteristic
134M16—212no—
135M16—26no—
136M16—212no—
137C2×C8—212no—
138C2×C8—26no—
139C2×C8—26no—
140C2×C8C22⋊C4×C12716yes—
141C2×C8C2×D4×C12713yes—
142C2×C8C2×D4×C12713yes—
143C16—41no—
144C16—43no—
145C23—24no—
146C23C22⋊C8×C12716yes—
147C23C22×C8×C12711yescharacteristic
148Q8—24no—
149Q8—26no—
150D4—212no—
151D4—212no—
152D4—26no—
153C2×C4—212no—
154C2×C4—26no—
155C2×C4—26no—
156C2×C4C22⋊C8×C12716yes—
157C2×C4C2×C22⋊C4×C12713yes—
158C2×C4C2×M16×C12713yes—
159C8—26no—
160C8#120×C12711yescharacteristic
161C8#120×C12713yes—
162C22—26no—
163C22—41no—
164C22—46no—
165C22—84no—
166C22C2×C22⋊C8×C12713yes—
167C4—26no—
168C4#379×C12711yescharacteristic
169C4#407×C12713yes—
170C2—23no—
171C2—46no—
172C2C127×#177911yescharacteristic
subgroupG/H
centerC8×C127#120
commutatorC23C22×C8×C127
Frattini, cosocleC22×C8C23×C127
socleC2×C127#1779
prime radical#585×C127C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2115C2semidirect
3. C127×#2113C2semidirect
9. C2×M32×C127C22semidirect
10. C22×C16×C127C22semidirect
83. #12684C127direct (N × H)
88. #2115C2×C127semidirect
89. #2113C2×C127semidirect
90. C127#12684direct (N × H)
96. C2×M32C22×C127semidirect
97. C22×C16C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35732
C127×#21152
C127×#21132
#12684127