Finite Group Databasebeta

← C127×#2640 #30635592

Subgroups — C127×#2640 #30635592

Order 16256.

500 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 200 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×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
2C2×#124×C127C211yesmaximal, characteristic
3#397×C127C214yesmaximal, complemented
4#393×C127C214yesmaximal, complemented
5#391×C127C212yesmaximal, complemented
6#390×C127C212yesmaximal
7#376×C127C211yesmaximal, characteristic, complemented
8C22×Q8×C127C2211yescharacteristic
9C22×Q8×C127C2212yes—
10C22×D4×C127C2211yescharacteristic
11C4⋊Q8×C127—24no—
12#124×C127—24no—
13C2×C4⋊C4×C127C2212yes—
14C2×C4⋊C4×C127C2212yes—
15C2×C22⋊C4×C127C2212yes—
16C2×C22⋊C4×C127C2212yes—
17C2×C22⋊C4×C127C2218yes—
18C2×C42×C127C2212yes—
19C2×C42×C127C2211yescharacteristic
20#95×C127C2212yes—
21#95×C127C2212yes—
22#95×C127C2218yes—
23C24×C127C2312yes—
24C2×Q8×C127—22no—
25C2×Q8×C127—24no—
26C2×Q8×C127—24no—
27C2×Q8×C127—44no—
28C2×D4×C127—22no—
29C2×D4×C127—24no—
30C22×C4×C127C2314yes—
31C22×C4×C127C2312yes—
32C22×C4×C127C2314yes—
33C22×C4×C127C2311yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—24no—
37C4⋊C4×C127—24no—
38C22⋊C4×C127—24no—
39C22⋊C4×C127—24no—
40C22⋊C4×C127—48no—
41C42×C127—24no—
42C42×C127D414yes—
43C23×C127—22no—
44C23×C127—24no—
45C23×C127—24no—
46C23×C127—24no—
47C23×C127C2411yescharacteristic
48Q8×C127—48no—
49Q8×C127—44no—
50D4×C127—44no—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—24no—
55C2×C4×C127—24no—
56C2×C4×C127—24no—
57C2×C4×C127—24no—
58C2×C4×C127—28no—
59C2×C4×C127—24no—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D412yes—
62C2×C4×C127C2×D412yes—
63C2×C4×C127#4114yes—
64C22×C127—42no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—48no—
68C22×C127C22×D411yescharacteristic
69C22×C127C2×#4112yes—
70C22×C127H(5,𝔽2)12yes—
71C22×C1272−1+412yes—
72C4×C127—22no—
73C4×C127—22no—
74C4×C127—42no—
75C4×C127—44no—
76C4×C127—44no—
77C4×C127#12414yes—
78C2×C127—82no—
79C2×C127C2×#12411yescharacteristic
80C2×C127#53511yescharacteristic
81C2×C127#53611yescharacteristic
82C2×C127#54214yes—
83#2640C12711yesmaximal, characteristic, direct product factor, Sylow
84C127#264011yescharacteristic, direct product factor, Sylow
85C2×C4⋊Q8C2×C12711yescharacteristic, complemented
86C2×#124C2×C12711yescharacteristic
87#397C2×C12714yescomplemented
88#393C2×C12714yescomplemented
89#391C2×C12712yescomplemented
90#390C2×C12712yes—
91#376C2×C12711yescharacteristic, complemented
92C22×Q8C22×C12711yescharacteristic
93C22×Q8C22×C12712yes—
94C22×D4C22×C12711yescharacteristic
95C4⋊Q8—24no—
96#124—24no—
97C2×C4⋊C4C22×C12712yes—
98C2×C4⋊C4C22×C12712yes—
99C2×C22⋊C4C22×C12712yes—
100C2×C22⋊C4C22×C12712yes—
101C2×C22⋊C4C22×C12718yes—
102C2×C42C22×C12712yes—
103C2×C42C22×C12711yescharacteristic
104#95C22×C12712yes—
105#95C22×C12712yes—
106#95C22×C12718yes—
107C24C23×C12712yes—
108C2×Q8—22no—
109C2×Q8—24no—
110C2×Q8—24no—
111C2×Q8—44no—
112C2×D4—22no—
113C2×D4—24no—
114C22×C4C23×C12714yes—
115C22×C4C23×C12712yes—
116C22×C4C23×C12714yes—
117C22×C4C23×C12711yescharacteristic
118C22×C4C23×C12711yescharacteristic
119C22×C4C23×C12711yescharacteristic
120C4⋊C4—24no—
121C4⋊C4—24no—
122C22⋊C4—24no—
123C22⋊C4—24no—
124C22⋊C4—48no—
125C42—24no—
126C42D4×C12714yes—
127C23—22no—
128C23—24no—
129C23—24no—
130C23—24no—
131C23C24×C12711yescharacteristic
132Q8—48no—
133Q8—44no—
134D4—44no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—28no—
143C2×C4—24no—
144C2×C4C2×D4×C12712yes—
145C2×C4C2×D4×C12712yes—
146C2×C4C2×D4×C12712yes—
147C2×C4#41×C12714yes—
148C22—42no—
149C22—42no—
150C22—42no—
151C22—48no—
152C22C22×D4×C12711yescharacteristic
153C22C2×#41×C12712yes—
154C22H(5,𝔽2)×C12712yes—
155C222−1+4×C12712yes—
156C4—22no—
157C4—22no—
158C4—42no—
159C4—44no—
160C4—44no—
161C4#124×C12714yes—
162C2—82no—
163C2C2×#124×C12711yescharacteristic
164C2#535×C12711yescharacteristic
165C2#536×C12711yescharacteristic
166C2#542×C12714yes—
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC22×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×C127C2semidirect
3. #397×C127C2semidirect
4. #393×C127C2semidirect
5. #391×C127C2semidirect
7. #376×C127C2semidirect
83. #2640C127direct (N × H)
84. C127#2640direct (N × H)
85. C2×C4⋊Q8C2×C127semidirect
87. #397C2×C127semidirect
88. #393C2×C127semidirect
89. #391C2×C127semidirect
91. #376C2×C127semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×C1272
C2×#124×C1272
#397×C1272
#393×C1272
#391×C1272
#390×C1272
#376×C1272
#2640127