Finite Group Databasebeta

← C127×#3202 #30636154

Subgroups — C127×#3202 #30636154

Order 16256.

378 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 168 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
1#536×C127C212yesmaximal
2#485×C127C214yesmaximal, complemented
3#470×C127C212yesmaximal, complemented
4#467×C127C212yesmaximal, complemented
5#451×C127C214yesmaximal, complemented
6#408×C127C211yesmaximal, characteristic
7C2×#41×C127C2211yescharacteristic
8C22×Q8×C127C2212yes—
9C2×Q16×C127C2212yes—
10C2×Q16×C127C2212yes—
11C2×M16×C127C2211yescharacteristic
12C22×C8×C127C2211yescharacteristic
13C4⋊Q8×C127—22no—
14#124×C127—22no—
15#123×C127—22no—
16#123×C127—22no—
17C22⋊Q8×C127C2214yes—
18C22⋊Q8×C127C2214yes—
19#117×C127C2212yes—
20#107×C127C2212yes—
21#106×C127C2212yes—
22#103×C127C2214yes—
23#103×C127C2214yes—
24C22⋊C8×C127C2214yes—
25#41×C127—41no—
26#41×C127—41no—
27C2×Q8×C127—22no—
28C2×Q8×C127—42no—
29C2×Q8×C127—42no—
30C2×Q8×C127D411yescharacteristic
31C2×Q8×C127C2314yes—
32C2×D4×C127D411yescharacteristic
33C2×D4×C127D412yes—
34C22×C4×C127—22no—
35C22×C4×C127C2312yes—
36C22×C4×C127C2311yescharacteristic
37Q16×C127—42no—
38Q16×C127—42no—
39M16×C127—41no—
40C2×C8×C127—41no—
41C2×C8×C127C2312yes—
42C2×C8×C127C2312yes—
43C4⋊C4×C127—24no—
44C4⋊C4×C127—24no—
45C4⋊C4×C127C2314yes—
46C22⋊C4×C127—22no—
47C22⋊C4×C127—24no—
48C22⋊C4×C127—24no—
49C42×C127—22no—
50C23×C127C2×D411yescharacteristic
51C23×C127C2×D412yes—
52Q8×C127—24no—
53Q8×C127—24no—
54Q8×C127—41no—
55Q8×C127—44no—
56D4×C127—41no—
57D4×C127—42no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—24no—
63C2×C4×C127—24no—
64C2×C4×C127—44no—
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2411yescharacteristic
68C8×C127—22no—
69C8×C127—22no—
70C22×C127—22no—
71C22×C127—21no—
72C22×C127—22no—
73C22×C127—24no—
74C22×C127C22×D411yescharacteristic
75C4×C127—21no—
76C4×C127—22no—
77C4×C127—44no—
78C4×C127—44no—
79C4×C127H(5,𝔽2)11yescharacteristic
80C4×C127H(5,𝔽2)11yescharacteristic
81C2×C127—42no—
82C2×C127—41no—
83C2×C127#53411yescharacteristic
84C2×C127#57812yes—
85#3202C12711yesmaximal, characteristic, direct product factor, Sylow
86C127#320211yescharacteristic, direct product factor, Sylow
87#536C2×C12712yes—
88#485C2×C12714yescomplemented
89#470C2×C12712yescomplemented
90#467C2×C12712yescomplemented
91#451C2×C12714yescomplemented
92#408C2×C12711yescharacteristic
93C2×#41C22×C12711yescharacteristic
94C22×Q8C22×C12712yes—
95C2×Q16C22×C12712yes—
96C2×Q16C22×C12712yes—
97C2×M16C22×C12711yescharacteristic
98C22×C8C22×C12711yescharacteristic
99C4⋊Q8—22no—
100#124—22no—
101#123—22no—
102#123—22no—
103C22⋊Q8C22×C12714yes—
104C22⋊Q8C22×C12714yes—
105#117C22×C12712yes—
106#107C22×C12712yes—
107#106C22×C12712yes—
108#103C22×C12714yes—
109#103C22×C12714yes—
110C22⋊C8C22×C12714yes—
111#41—41no—
112#41—41no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8D4×C12711yescharacteristic
117C2×Q8C23×C12714yes—
118C2×D4D4×C12711yescharacteristic
119C2×D4D4×C12712yes—
120C22×C4—22no—
121C22×C4C23×C12712yes—
122C22×C4C23×C12711yescharacteristic
123Q16—42no—
124Q16—42no—
125M16—41no—
126C2×C8—41no—
127C2×C8C23×C12712yes—
128C2×C8C23×C12712yes—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4C23×C12714yes—
132C22⋊C4—22no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C42—22no—
136C23C2×D4×C12711yescharacteristic
137C23C2×D4×C12712yes—
138Q8—24no—
139Q8—24no—
140Q8—41no—
141Q8—44no—
142D4—42no—
143D4—41no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—44no—
151C2×C4C2×D4×C12712yes—
152C2×C4C2×D4×C12711yescharacteristic
153C2×C4C24×C12711yescharacteristic
154C8—22no—
155C8—22no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—24no—
160C22C22×D4×C12711yescharacteristic
161C4—22no—
162C4—21no—
163C4—44no—
164C4—44no—
165C4H(5,𝔽2)×C12711yescharacteristic
166C4H(5,𝔽2)×C12711yescharacteristic
167C2—42no—
168C2—41no—
169C2#534×C12711yescharacteristic
170C2#578×C12712yes—
subgroupG/H
center, socleC22×C127C22×D4
commutator, Frattini, cosocleC2×C4C24×C127
prime radicalC2×#41×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. #485×C127C2semidirect
3. #470×C127C2semidirect
4. #467×C127C2semidirect
5. #451×C127C2semidirect
85. #3202C127direct (N × H)
86. C127#3202direct (N × H)
88. #485C2×C127semidirect
89. #470C2×C127semidirect
90. #467C2×C127semidirect
91. #451C2×C127semidirect

Maximal subgroups

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

subgroupindex
#536×C1272
#485×C1272
#470×C1272
#467×C1272
#451×C1272
#408×C1272
#3202127