Finite Group Databasebeta

← C127×#12310 #44575983

Subgroups — C127×#12310 #44575983

Order 32512.

406 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3292C211yesmaximal, characteristic, complemented
2C127×#2213C212yesmaximal, complemented
3C127×#1712C211yesmaximal, characteristic, complemented
4C127×#1357C211yesmaximal, characteristic, complemented
5C127×#1335C212yesmaximal, complemented
6#576×C127—22no—
7#509×C127—22no—
8#493×C127—22no—
9#493×C127C2211yescharacteristic
10#493×C127C2211yescharacteristic, complemented
11#472×C127—22no—
12#453×C127—22no—
13#443×C127C2212yes—
14#364×C127C2212yescomplemented
15#361×C127—22no—
16C82×C127C2211yescharacteristic, complemented
17H(5,𝔽2)×C127—22no—
18AGL(1,ℤ/8)×C127—24no—
19AGL(1,ℤ/8)×C127—44no—
20#135×C127—42no—
21#135×C127D412yes—
22C2×D8×C127—22no—
23C2×D8×C127—22no—
24C2×D8×C127—44no—
25C2×D8×C127—42no—
26#131×C127—22no—
27#127×C127—21no—
28#127×C127D412yes—
29SD32×C127—42no—
30D16×C127—42no—
31M32×C127—22no—
32#108×C127D412yes—
33#104×C127—22no—
34#100×C127—22no—
35C4×C8×C127—21no—
36C4×C8×C127C2311yescharacteristic
37#41×C127—22no—
38#41×C127—44no—
39C2×D4×C127—22no—
40C2×D4×C127—42no—
41C2×D4×C127—42no—
42C2×D4×C127—44no—
43C2×D4×C127—41no—
44C2×D4×C127—82no—
45Q16×C127D812yescomplemented
46SD16×C127—22no—
47SD16×C127—44no—
48D8×C127—44no—
49D8×C127—42no—
50D8×C127—42no—
51D8×C127—42no—
52D8×C127—42no—
53D8×C127—88no—
54D8×C127D812yescomplemented
55M16×C127—22no—
56M16×C127—42no—
57C2×C8×C127—22no—
58C2×C8×C127—42no—
59C2×C8×C127C2×D412yes—
60C42×C127C2×D411yescharacteristic
61C16×C127—42no—
62C23×C127—42no—
63C23×C127—82no—
64C23×C127—81no—
65Q8×C127—22no—
66D4×C127—22no—
67D4×C127—42no—
68D4×C127—44no—
69D4×C127—84no—
70D4×C127—84no—
71D4×C127—84no—
72D4×C127—82no—
73C2×C4×C127—21no—
74C2×C4×C127—42no—
75C2×C4×C127—82no—
76C2×C4×C127#12011yescharacteristic
77C8×C127—24no—
78C8×C127—42no—
79C8×C127C2×D812yes—
80C8×C127AGL(1,ℤ/8)12yes—
81C22×C127—42no—
82C22×C127—82no—
83C22×C127—162no—
84C22×C127—162no—
85C22×C127—161no—
86C22×C127—322no—
87C22×C127#45311yescharacteristic
88C4×C127—22no—
89C4×C127—42no—
90C4×C127#44712yes—
91#12310C12711yesmaximal, characteristic, direct product factor, Sylow
92C2×C127—21no—
93C2×C127—82no—
94C2×C127—162no—
95C2×C127—321no—
96C2×C127#161911yescharacteristic
97#3292C2×C12711yescharacteristic, complemented
98#2213C2×C12712yescomplemented
99#1712C2×C12711yescharacteristic, complemented
100#1357C2×C12711yescharacteristic, complemented
101#1335C2×C12712yescomplemented
102C127#1231011yescharacteristic, direct product factor, Sylow
103#576—22no—
104#509—22no—
105#493—22no—
106#493C22×C12711yescharacteristic
107#493C22×C12711yescharacteristic, complemented
108#472—22no—
109#453—22no—
110#443C22×C12712yes—
111#364C22×C12712yescomplemented
112#361—22no—
113C82C22×C12711yescharacteristic, complemented
114H(5,𝔽2)—22no—
115AGL(1,ℤ/8)—24no—
116AGL(1,ℤ/8)—44no—
117#135—42no—
118#135D4×C12712yes—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—44no—
122C2×D8—42no—
123#131—22no—
124#127—21no—
125#127D4×C12712yes—
126SD32—42no—
127D16—42no—
128M32—22no—
129#108D4×C12712yes—
130#104—22no—
131#100—22no—
132C4×C8—21no—
133C4×C8C23×C12711yescharacteristic
134#41—22no—
135#41—44no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—44no—
140C2×D4—41no—
141C2×D4—82no—
142Q16D8×C12712yescomplemented
143SD16—22no—
144SD16—44no—
145D8—44no—
146D8—42no—
147D8—42no—
148D8—42no—
149D8—42no—
150D8—88no—
151D8D8×C12712yescomplemented
152M16—22no—
153M16—42no—
154C2×C8—22no—
155C2×C8—42no—
156C2×C8C2×D4×C12712yes—
157C42C2×D4×C12711yescharacteristic
158C16—42no—
159C23—42no—
160C23—82no—
161C23—81no—
162Q8—22no—
163D4—22no—
164D4—42no—
165D4—44no—
166D4—84no—
167D4—84no—
168D4—84no—
169D4—82no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—82no—
173C2×C4#120×C12711yescharacteristic
174C8—24no—
175C8—42no—
176C8C2×D8×C12712yes—
177C8AGL(1,ℤ/8)×C12712yes—
178C22—42no—
179C22—82no—
180C22—162no—
181C22—162no—
182C22—161no—
183C22—322no—
184C22#453×C12711yescharacteristic
185C4—22no—
186C4—42no—
187C4#447×C12712yes—
188C2—21no—
189C2—82no—
190C2—162no—
191C2—321no—
192C2C127×#161911yescharacteristic
subgroupG/H
center, socleC2×C127#1619
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3292C2semidirect
2. C127×#2213C2semidirect
3. C127×#1712C2semidirect
4. C127×#1357C2semidirect
5. C127×#1335C2semidirect
10. #493×C127C22semidirect
14. #364×C127C22semidirect
16. C82×C127C22semidirect
45. Q16×C127D8semidirect
54. D8×C127D8semidirect
91. #12310C127direct (N × H)
97. #3292C2×C127semidirect
98. #2213C2×C127semidirect
99. #1712C2×C127semidirect
100. #1357C2×C127semidirect
101. #1335C2×C127semidirect
102. C127#12310direct (N × H)
107. #493C22×C127semidirect
111. #364C22×C127semidirect
113. C82C22×C127semidirect
142. Q16D8×C127semidirect
151. D8D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32922
C127×#22132
C127×#17122
C127×#13572
C127×#13352
#12310127