Finite Group Databasebeta

← C127×#2013 #30634965

Subgroups — C127×#2013 #30634965

Order 16256.

406 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C22×SD16×C127C212yesmaximal
2C2×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
3C2×#103×C127C212yesmaximal, complemented
4#390×C127C211yesmaximal, characteristic
5#336×C127C211yesmaximal, characteristic, complemented
6C22×Q8×C127C2212yes—
7C22×D4×C127C2211yescharacteristic
8C2×SD16×C127—24no—
9C2×SD16×C127—24no—
10C2×SD16×C127—24no—
11C22×C8×C127C2212yes—
12C4⋊Q8×C127—24no—
13C2×C4⋊C4×C127—22no—
14C2×C4⋊C4×C127C2211yescharacteristic
15C2×C22⋊C4×C127—22no—
16C2×C42×C127C2211yescharacteristic
17#103×C127—24no—
18C24×C127—21no—
19C2×Q8×C127—22no—
20C2×Q8×C127—22no—
21C2×Q8×C127—22no—
22C2×Q8×C127—22no—
23C2×Q8×C127—22no—
24C2×Q8×C127D414yes—
25C2×D4×C127—22no—
26C2×D4×C127—22no—
27C2×D4×C127—21no—
28C2×D4×C127D412yes—
29C22×C4×C127—22no—
30C22×C4×C127—21no—
31C22×C4×C127D412yes—
32C22×C4×C127C2311yescharacteristic
33SD16×C127—216no—
34C2×C8×C127—22no—
35C2×C8×C127—22no—
36C2×C8×C127D414yes—
37C4⋊C4×C127—28no—
38C4⋊C4×C127—22no—
39C22⋊C4×C127—44no—
40C42×C127—22no—
41C23×C127—22no—
42C23×C127—41no—
43C23×C127—41no—
44C23×C127—42no—
45C23×C127—42no—
46C23×C127C2×D411yescharacteristic
47Q8×C127—24no—
48Q8×C127—24no—
49Q8×C127—42no—
50Q8×C127—42no—
51D4×C127—24no—
52D4×C127—42no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—24no—
56C2×C4×C127—42no—
57C2×C4×C127—42no—
58C2×C4×C127—41no—
59C2×C4×C127—42no—
60C2×C4×C127SD1614yescomplemented
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D412yes—
64C2×C4×C127#4111yescharacteristic
65C8×C127—24no—
66C22×C127—42no—
67C22×C127—44no—
68C22×C127—82no—
69C22×C127—81no—
70C22×C127—81no—
71C22×C127#12011yescharacteristic
72C22×C127#12111yescharacteristic
73C22×C127#12711yescharacteristic
74C22×C127C2×SD1612yes—
75C22×C127#13712yes—
76C4×C127—24no—
77C4×C127—44no—
78C4×C127—81no—
79C4×C127#12012yes—
80C4×C127#12112yes—
81C2×C127—82no—
82C2×C127#39211yescharacteristic
83C2×C127#44812yes—
84C2×C127#46112yes—
85C2×C127#49211yescharacteristic
86C2×C127#49711yescharacteristic
87#2013C12711yesmaximal, characteristic, direct product factor, Sylow
88C127#201311yescharacteristic, direct product factor, Sylow
89C22×SD16C2×C12712yes—
90C2×C4⋊Q8C2×C12711yescharacteristic, complemented
91C2×#103C2×C12712yescomplemented
92#390C2×C12711yescharacteristic
93#336C2×C12711yescharacteristic, complemented
94C22×Q8C22×C12712yes—
95C22×D4C22×C12711yescharacteristic
96C2×SD16—24no—
97C2×SD16—24no—
98C2×SD16—24no—
99C22×C8C22×C12712yes—
100C4⋊Q8—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4C22×C12711yescharacteristic
103C2×C22⋊C4—22no—
104C2×C42C22×C12711yescharacteristic
105#103—24no—
106C24—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8D4×C12714yes—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4D4×C12712yes—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4D4×C12712yes—
120C22×C4C23×C12711yescharacteristic
121SD16—216no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8D4×C12714yes—
125C4⋊C4—28no—
126C4⋊C4—22no—
127C22⋊C4—44no—
128C42—22no—
129C23—22no—
130C23—41no—
131C23—41no—
132C23—42no—
133C23—42no—
134C23C2×D4×C12711yescharacteristic
135Q8—24no—
136Q8—24no—
137Q8—42no—
138Q8—42no—
139D4—24no—
140D4—42no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4SD16×C12714yescomplemented
149C2×C4C2×D4×C12711yescharacteristic
150C2×C4C2×D4×C12712yes—
151C2×C4C2×D4×C12712yes—
152C2×C4#41×C12711yescharacteristic
153C8—24no—
154C22—42no—
155C22—44no—
156C22—81no—
157C22—82no—
158C22—81no—
159C22#120×C12711yescharacteristic
160C22#121×C12711yescharacteristic
161C22#127×C12711yescharacteristic
162C22C2×SD16×C12712yes—
163C22#137×C12712yes—
164C4—24no—
165C4—44no—
166C4—81no—
167C4#120×C12712yes—
168C4#121×C12712yes—
169C2—82no—
170C2#392×C12711yescharacteristic
171C2#448×C12712yes—
172C2#461×C12712yes—
173C2#492×C12711yescharacteristic
174C2#497×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C2×D4
commutator, Frattini, cosocleC22×C4C23×C127
prime radicalC22×D4×C127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊Q8×C127C2semidirect
3. C2×#103×C127C2semidirect
5. #336×C127C2semidirect
60. C2×C4×C127SD16semidirect
87. #2013C127direct (N × H)
88. C127#2013direct (N × H)
90. C2×C4⋊Q8C2×C127semidirect
91. C2×#103C2×C127semidirect
93. #336C2×C127semidirect
148. C2×C4SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C22×SD16×C1272
C2×C4⋊Q8×C1272
C2×#103×C1272
#390×C1272
#336×C1272
#2013127