Finite Group Databasebeta

← C127×#12451 #44576124

Subgroups — C127×#12451 #44576124

Order 32512.

224 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C127×#2252C211yesmaximal, characteristic, complemented
2D4×C16×C127C211yesmaximal, characteristic
3C127×#1673C212yesmaximal
4C127×#1371C211yesmaximal, characteristic, complemented
5C127×#1364C212yesmaximal, complemented
6C22×C16×C127—21no—
7#500×C127C2212yes—
8#461×C127—22no—
9C8×D4×C127C2211yescharacteristic
10#366×C127—21no—
11#366×C127—21no—
12C4⋊C16×C127C2211yescharacteristic
13C22⋊C16×C127—21no—
14C4×C16×C127C2211yescharacteristic
15#335×C127C2212yes—
16#332×C127—22no—
17C2×SD16×C127—42no—
18C22×C8×C127—21no—
19C4⋊Q8×C127—22no—
20C4×D4×C127D411yescharacteristic
21C2×C16×C127—21no—
22C2×C16×C127—42no—
23C2×C16×C127D411yescharacteristic
24C2×C16×C127D411yescharacteristic
25#107×C127—22no—
26#107×C127—22no—
27C4⋊C8×C127—22no—
28C4⋊C8×C127D411yescharacteristic
29#103×C127—42no—
30C22⋊C8×C127—21no—
31C4×C8×C127C2311yescharacteristic
32C2×Q8×C127—42no—
33C2×D4×C127D811yescharacteristic
34C22×C4×C127—21no—
35SD16×C127—84no—
36C2×C8×C127—21no—
37C2×C8×C127—41no—
38C2×C8×C127—41no—
39C2×C8×C127—42no—
40C2×C8×C127C2×D411yescharacteristic
41C2×C8×C127#4111yescharacteristic
42C4⋊C4×C127—42no—
43C4⋊C4×C127—42no—
44C4⋊C4×C127D811yescharacteristic
45C22⋊C4×C127—21no—
46C42×C127C2×D411yescharacteristic
47C16×C127—21no—
48C16×C127—41no—
49C16×C127SD1612yescomplemented
50C23×C127—21no—
51Q8×C127—82no—
52Q8×C127—82no—
53D4×C127—21no—
54D4×C127D1612yes—
55C2×C4×C127—21no—
56C2×C4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127—82no—
59C2×C4×C127#12111yescharacteristic
60C2×C4×C127C2×D811yescharacteristic
61C2×C4×C127#13511yescharacteristic
62C8×C127—21no—
63C8×C127—41no—
64C8×C127—82no—
65C8×C127C2×SD1611yescharacteristic
66C8×C127#13711yescharacteristic
67C22×C127—22no—
68C22×C127—41no—
69C22×C127—41no—
70C22×C127#46611yescharacteristic
71C4×C127—21no—
72C4×C127—41no—
73C4×C127—162no—
74C4×C127#46111yescharacteristic
75C4×C127#47311yescharacteristic
76C4×C127C2×D1611yescharacteristic
77C4×C127#50811yescharacteristic
78#12451C12711yesmaximal, characteristic, direct product factor, Sylow
79C2×C127—42no—
80C2×C127#167311yescharacteristic
81C2×C127#221511yescharacteristic
82C2×C127#222211yescharacteristic
83#2252C2×C12711yescharacteristic, complemented
84D4×C16C2×C12711yescharacteristic
85#1673C2×C12712yes—
86#1371C2×C12711yescharacteristic, complemented
87#1364C2×C12712yescomplemented
88C127#1245111yescharacteristic, direct product factor, Sylow
89C22×C16—21no—
90#500C22×C12712yes—
91#461—22no—
92C8×D4C22×C12711yescharacteristic
93#366—21no—
94#366—21no—
95C4⋊C16C22×C12711yescharacteristic
96C22⋊C16—21no—
97C4×C16C22×C12711yescharacteristic
98#335C22×C12712yes—
99#332—22no—
100C2×SD16—42no—
101C22×C8—21no—
102C4⋊Q8—22no—
103C4×D4D4×C12711yescharacteristic
104C2×C16—21no—
105C2×C16—42no—
106C2×C16D4×C12711yescharacteristic
107C2×C16D4×C12711yescharacteristic
108#107—22no—
109#107—22no—
110C4⋊C8—22no—
111C4⋊C8D4×C12711yescharacteristic
112#103—42no—
113C22⋊C8—21no—
114C4×C8C23×C12711yescharacteristic
115C2×Q8—42no—
116C2×D4D8×C12711yescharacteristic
117C22×C4—21no—
118SD16—84no—
119C2×C8—21no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—42no—
123C2×C8C2×D4×C12711yescharacteristic
124C2×C8#41×C12711yescharacteristic
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4D8×C12711yescharacteristic
128C22⋊C4—21no—
129C42C2×D4×C12711yescharacteristic
130C16—21no—
131C16—41no—
132C16SD16×C12712yescomplemented
133C23—21no—
134Q8—82no—
135Q8—82no—
136D4—21no—
137D4D16×C12712yes—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—82no—
142C2×C4#121×C12711yescharacteristic
143C2×C4C2×D8×C12711yescharacteristic
144C2×C4#135×C12711yescharacteristic
145C8—21no—
146C8—41no—
147C8—82no—
148C8C2×SD16×C12711yescharacteristic
149C8#137×C12711yescharacteristic
150C22—22no—
151C22—41no—
152C22—41no—
153C22#466×C12711yescharacteristic
154C4—21no—
155C4—41no—
156C4—162no—
157C4#461×C12711yescharacteristic
158C4#473×C12711yescharacteristic
159C4C2×D16×C12711yescharacteristic
160C4#508×C12711yescharacteristic
161C2—42no—
162C2C127×#167311yescharacteristic
163C2C127×#221511yescharacteristic
164C2C127×#222211yescharacteristic
subgroupG/H
center, socleC22×C127#466
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC2×D4×C127D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2252C2semidirect
4. C127×#1371C2semidirect
5. C127×#1364C2semidirect
49. C16×C127SD16semidirect
78. #12451C127direct (N × H)
83. #2252C2×C127semidirect
86. #1371C2×C127semidirect
87. #1364C2×C127semidirect
88. C127#12451direct (N × H)
132. C16SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#22522
D4×C16×C1272
C127×#16732
C127×#13712
C127×#13642
#12451127