Finite Group Databasebeta

← C2×#458×C127 #30635976

Subgroups — C2×#458×C127 #30635976

Order 16256.

794 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 216 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×H(5,𝔽2)×C127C211yesmaximal, characteristic
2C2×#123×C127C213yesmaximal, complemented
3#458×C127C218yesmaximal, direct product factor
4C2×#99×C127C213yesmaximal, complemented
5H(5,𝔽2)×C127—26no—
6H(5,𝔽2)×C127C2214yes—
7C2×#41×C127—23no—
8C22×D4×C127—23no—
9C22×D4×C127C2213yes—
10C23×C4×C127C2211yescharacteristic, complemented
11#123×C127—26no—
12#123×C127C22112yescomplemented
13C2×C4⋊C4×C127—23no—
14C2×C22⋊C4×C127—23no—
15C2×C22⋊C4×C127C2213yescomplemented
16#99×C127C22112yescomplemented
17C24×C127—21no—
18C24×C127D413yes—
19C24×C127C2311yescharacteristic
20#41×C127—212no—
21#41×C127—212no—
22C2×Q8×C127—21no—
23C2×D4×C127—23no—
24C2×D4×C127—212no—
25C2×D4×C127—26no—
26C2×D4×C127—212no—
27C2×D4×C127—26no—
28C2×D4×C127D416yes—
29C2×D4×C127C2316yes—
30C22×C4×C127—23no—
31C22×C4×C127—23no—
32C22×C4×C127—23no—
33C22×C4×C127—23no—
34C22×C4×C127D413yes—
35C22×C4×C127C2312yescomplemented
36C4⋊C4×C127—212no—
37C22⋊C4×C127—212no—
38C22⋊C4×C127—23no—
39C22⋊C4×C127C2316yescomplemented
40C23×C127—24no—
41C23×C127—23no—
42C23×C127—23no—
43C23×C127—42no—
44C23×C127—46no—
45C23×C127—82no—
46C23×C127C2×D413yes—
47C23×C127C2×D416yes—
48C23×C127C2×D413yes—
49C23×C127C2411yescharacteristic
50Q8×C127—24no—
51D4×C127—212no—
52D4×C127—212no—
53D4×C127—212no—
54C2×C4×C127—212no—
55C2×C4×C127—23no—
56C2×C4×C127—23no—
57C2×C4×C127—26no—
58C2×C4×C127—26no—
59C2×C4×C127—46no—
60C2×C4×C127—43no—
61C2×C4×C127—41no—
62C2×C4×C127C2×D416yes—
63C22×C127—26no—
64C22×C127—26no—
65C22×C127—43no—
66C22×C127—41no—
67C22×C127—46no—
68C22×C127—46no—
69C22×C127—83no—
70C22×C127—84no—
71C22×C127#12013yes—
72C22×C127#12011yescharacteristic
73C22×C127C22×D413yes—
74C4×C127—26no—
75C4×C127—46no—
76C4×C127—42no—
77C2×C127—23no—
78C2×C127—23no—
79C2×C127—46no—
80C2×C127#45812yesdirect product factor
81C2×C127C2×#12011yescharacteristic
82C2×#458C12711yesmaximal, characteristic, direct product factor, Sylow
83C127C2×#45811yescharacteristic, direct product factor, Sylow
84C2×H(5,𝔽2)C2×C12711yescharacteristic
85C2×#123C2×C12713yescomplemented
86#458C2×C12718yesdirect product factor
87C2×#99C2×C12713yescomplemented
88H(5,𝔽2)—26no—
89H(5,𝔽2)C22×C12714yes—
90C2×#41—23no—
91C22×D4—23no—
92C22×D4C22×C12713yes—
93C23×C4C22×C12711yescharacteristic, complemented
94#123—26no—
95#123C22×C127112yescomplemented
96C2×C4⋊C4—23no—
97C2×C22⋊C4—23no—
98C2×C22⋊C4C22×C12713yescomplemented
99#99C22×C127112yescomplemented
100C24—21no—
101C24D4×C12713yes—
102C24C23×C12711yescharacteristic
103#41—212no—
104#41—212no—
105C2×Q8—21no—
106C2×D4—23no—
107C2×D4—212no—
108C2×D4—26no—
109C2×D4—212no—
110C2×D4—26no—
111C2×D4D4×C12716yes—
112C2×D4C23×C12716yes—
113C22×C4—23no—
114C22×C4—23no—
115C22×C4—23no—
116C22×C4—23no—
117C22×C4D4×C12713yes—
118C22×C4C23×C12712yescomplemented
119C4⋊C4—212no—
120C22⋊C4—212no—
121C22⋊C4—23no—
122C22⋊C4C23×C12716yescomplemented
123C23—24no—
124C23—23no—
125C23—23no—
126C23—42no—
127C23—46no—
128C23—82no—
129C23C2×D4×C12713yes—
130C23C2×D4×C12716yes—
131C23C2×D4×C12713yes—
132C23C24×C12711yescharacteristic
133Q8—24no—
134D4—212no—
135D4—212no—
136D4—212no—
137C2×C4—212no—
138C2×C4—23no—
139C2×C4—23no—
140C2×C4—26no—
141C2×C4—26no—
142C2×C4—46no—
143C2×C4—43no—
144C2×C4—41no—
145C2×C4C2×D4×C12716yes—
146C22—26no—
147C22—26no—
148C22—43no—
149C22—41no—
150C22—46no—
151C22—46no—
152C22—83no—
153C22—84no—
154C22#120×C12713yes—
155C22#120×C12711yescharacteristic
156C22C22×D4×C12713yes—
157C4—26no—
158C4—46no—
159C4—42no—
160C2—23no—
161C2—23no—
162C2—46no—
163C2#458×C12712yesdirect product factor
164C2C2×#120×C12711yescharacteristic
subgroupG/H
center, socleC22×C127#120
commutator, Frattini, cosocleC23C24×C127
prime radicalC2×H(5,𝔽2)×C127C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#123×C127C2semidirect
3. #458×C127C2direct (N × H)
4. C2×#99×C127C2semidirect
10. C23×C4×C127C22semidirect
12. #123×C127C22semidirect
15. C2×C22⋊C4×C127C22semidirect
16. #99×C127C22semidirect
35. C22×C4×C127C23semidirect
39. C22⋊C4×C127C23semidirect
80. C2×C127#458direct (N × H)
82. C2×#458C127direct (N × H)
83. C127C2×#458direct (N × H)
85. C2×#123C2×C127semidirect
86. #458C2×C127direct (N × H)
87. C2×#99C2×C127semidirect
93. C23×C4C22×C127semidirect
95. #123C22×C127semidirect
98. C2×C22⋊C4C22×C127semidirect
99. #99C22×C127semidirect
118. C22×C4C23×C127semidirect
122. C22⋊C4C23×C127semidirect
163. C2#458×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C1272
C2×#123×C1272
#458×C1272
C2×#99×C1272
C2×#458127