Finite Group Databasebeta

← C127×#2537 #30635489

Subgroups — C127×#2537 #30635489

Order 16256.

788 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 248 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×#121×C127C214yesmaximal, complemented
2C2×C4×D4×C127C212yesmaximal, complemented
3#394×C127C214yesmaximal, complemented
4#384×C127C211yesmaximal, characteristic, complemented
5#380×C127C214yesmaximal, complemented
6C22×D4×C127C2214yes—
7C22×D4×C127C2212yes—
8C23×C4×C127C2214yes—
9#121×C127—216no—
10C4×D4×C127—28no—
11C2×C4⋊C4×C127C2212yes—
12C2×C4⋊C4×C127C2214yes—
13C2×C4⋊C4×C127C2212yes—
14C2×C22⋊C4×C127C2214yes—
15C2×C22⋊C4×C127C2218yes—
16C2×C42×C127C2211yescharacteristic
17#95×C127C2214yes—
18C24×C127C2314yes—
19C2×D4×C127—216no—
20C2×D4×C127—24no—
21C2×D4×C127—24no—
22C2×D4×C127—44no—
23C2×D4×C127D418yescomplemented
24C22×C4×C127—28no—
25C22×C4×C127—28no—
26C22×C4×C127—28no—
27C22×C4×C127C2314yes—
28C22×C4×C127C2314yes—
29C22×C4×C127C2311yescharacteristic
30C22×C4×C127C2312yes—
31C4⋊C4×C127—24no—
32C4⋊C4×C127—28no—
33C4⋊C4×C127—42no—
34C22⋊C4×C127—216no—
35C22⋊C4×C127—28no—
36C42×C127D414yescomplemented
37C23×C127—24no—
38C23×C127—24no—
39C23×C127—24no—
40C23×C127—24no—
41C23×C127—24no—
42C23×C127—24no—
43C23×C127C2×D418yes—
44C23×C127C2411yescharacteristic
45D4×C127—216no—
46D4×C127—416no—
47C2×C4×C127—21no—
48C2×C4×C127—21no—
49C2×C4×C127—24no—
50C2×C4×C127—24no—
51C2×C4×C127—28no—
52C2×C4×C127—24no—
53C2×C4×C127—24no—
54C2×C4×C127—416no—
55C2×C4×C127C2×D412yes—
56C2×C4×C127C2×D414yes—
57C2×C4×C127C2×D414yes—
58C2×C4×C127#4114yes—
59C22×C127—28no—
60C22×C127—28no—
61C22×C127—28no—
62C22×C127—44no—
63C22×C127—44no—
64C22×C127—44no—
65C22×C127—44no—
66C22×C127C22×D411yescharacteristic
67C22×C127C22×D412yes—
68C22×C127C2×#4112yes—
69C22×C127H(5,𝔽2)11yescharacteristic
70C22×C1272−1+411yescharacteristic
71C4×C127—22no—
72C4×C127—44no—
73C4×C127—44no—
74C4×C127#12118yescomplemented
75C2×C127—48no—
76C2×C127C2×#12112yes—
77C2×C127#53711yescharacteristic
78C2×C127D4211yescharacteristic
79C2×C127#54712yes—
80C2×C127#55511yescharacteristic
81#2537C12711yesmaximal, characteristic, direct product factor, Sylow
82C127#253711yescharacteristic, direct product factor, Sylow
83C2×#121C2×C12714yescomplemented
84C2×C4×D4C2×C12712yescomplemented
85#394C2×C12714yescomplemented
86#384C2×C12711yescharacteristic, complemented
87#380C2×C12714yescomplemented
88C22×D4C22×C12714yes—
89C22×D4C22×C12712yes—
90C23×C4C22×C12714yes—
91#121—216no—
92C4×D4—28no—
93C2×C4⋊C4C22×C12712yes—
94C2×C4⋊C4C22×C12714yes—
95C2×C4⋊C4C22×C12712yes—
96C2×C22⋊C4C22×C12714yes—
97C2×C22⋊C4C22×C12718yes—
98C2×C42C22×C12711yescharacteristic
99#95C22×C12714yes—
100C24C23×C12714yes—
101C2×D4—216no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—44no—
105C2×D4D4×C12718yescomplemented
106C22×C4—28no—
107C22×C4—28no—
108C22×C4—28no—
109C22×C4C23×C12714yes—
110C22×C4C23×C12714yes—
111C22×C4C23×C12711yescharacteristic
112C22×C4C23×C12712yes—
113C4⋊C4—24no—
114C4⋊C4—28no—
115C4⋊C4—42no—
116C22⋊C4—216no—
117C22⋊C4—28no—
118C42D4×C12714yescomplemented
119C23—24no—
120C23—24no—
121C23—24no—
122C23—24no—
123C23—24no—
124C23—24no—
125C23C2×D4×C12718yes—
126C23C24×C12711yescharacteristic
127D4—216no—
128D4—416no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—28no—
134C2×C4—24no—
135C2×C4—24no—
136C2×C4—416no—
137C2×C4C2×D4×C12712yes—
138C2×C4C2×D4×C12714yes—
139C2×C4C2×D4×C12714yes—
140C2×C4#41×C12714yes—
141C22—28no—
142C22—28no—
143C22—28no—
144C22—44no—
145C22—44no—
146C22—44no—
147C22—44no—
148C22C22×D4×C12711yescharacteristic
149C22C22×D4×C12712yes—
150C22C2×#41×C12712yes—
151C22H(5,𝔽2)×C12711yescharacteristic
152C222−1+4×C12711yescharacteristic
153C4—22no—
154C4—44no—
155C4—44no—
156C4#121×C12718yescomplemented
157C2—48no—
158C2C2×#121×C12712yes—
159C2#537×C12711yescharacteristic
160C2D42×C12711yescharacteristic
161C2#547×C12712yes—
162C2#555×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#121×C127C2semidirect
2. C2×C4×D4×C127C2semidirect
3. #394×C127C2semidirect
4. #384×C127C2semidirect
5. #380×C127C2semidirect
23. C2×D4×C127D4semidirect
36. C42×C127D4semidirect
74. C4×C127#121semidirect
81. #2537C127direct (N × H)
82. C127#2537direct (N × H)
83. C2×#121C2×C127semidirect
84. C2×C4×D4C2×C127semidirect
85. #394C2×C127semidirect
86. #384C2×C127semidirect
87. #380C2×C127semidirect
105. C2×D4D4×C127semidirect
118. C42D4×C127semidirect
156. C4#121×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C1272
C2×C4×D4×C1272
#394×C1272
#384×C1272
#380×C1272
#2537127