Finite Group Databasebeta

← C127×#12312 #44575985

Subgroups — C127×#12312 #44575985

Order 32512.

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

normal subgroups only

This group has 372 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
2C127×#2214C212yesmaximal, complemented
3C127×#1709C211yesmaximal, characteristic, complemented
4C127×#1357C211yesmaximal, characteristic, complemented
5C127×#1335C212yesmaximal, complemented
6#576×C127—22no—
7#510×C127—22no—
8#494×C127C2211yescharacteristic, complemented
9#493×C127C2211yescharacteristic
10#486×C127—22no—
11#472×C127—22no—
12#453×C127—22no—
13#443×C127C2212yes—
14#364×C127C2212yescomplemented
15#362×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×Q16×C127—22no—
23C2×D8×C127—22no—
24C2×D8×C127—42no—
25#131×C127—22no—
26C4⋊Q8×C127—21no—
27#127×C127D412yes—
28Q32×C127—42no—
29SD32×C127—42no—
30M32×C127—22no—
31#108×C127D412yes—
32#104×C127—22no—
33#102×C127—44no—
34#100×C127—22no—
35C4×C8×C127—21no—
36C4×C8×C127C2311yescharacteristic
37#41×C127—22no—
38#41×C127—44no—
39C2×Q8×C127—41no—
40C2×D4×C127—22no—
41C2×D4×C127—42no—
42C2×D4×C127—44no—
43C2×D4×C127—82no—
44Q16×C127—42no—
45Q16×C127—42no—
46Q16×C127D812yes—
47SD16×C127—22no—
48SD16×C127—44no—
49D8×C127—44no—
50D8×C127—42no—
51D8×C127—42no—
52D8×C127D812yes—
53M16×C127—22no—
54M16×C127—42no—
55C2×C8×C127—22no—
56C2×C8×C127—42no—
57C2×C8×C127C2×D412yes—
58C4⋊C4×C127—42no—
59C42×C127C2×D411yescharacteristic
60C16×C127—42no—
61C23×C127—42no—
62C23×C127—82no—
63Q8×C127—22no—
64Q8×C127—82no—
65D4×C127—22no—
66D4×C127—42no—
67D4×C127—44no—
68D4×C127—84no—
69D4×C127—84no—
70C2×C4×C127—21no—
71C2×C4×C127—42no—
72C2×C4×C127—82no—
73C2×C4×C127—81no—
74C2×C4×C127#12011yescharacteristic
75C8×C127—42no—
76C8×C127—42no—
77C8×C127C2×D812yes—
78C8×C127AGL(1,ℤ/8)12yes—
79C22×C127—42no—
80C22×C127—82no—
81C22×C127—162no—
82C22×C127—322no—
83C22×C127#45311yescharacteristic
84C4×C127—22no—
85C4×C127—42no—
86C4×C127—161no—
87C4×C127#44712yes—
88#12312C12711yesmaximal, characteristic, direct product factor, Sylow
89C2×C127—21no—
90C2×C127—82no—
91C2×C127—162no—
92C2×C127#161911yescharacteristic
93#3292C2×C12711yescharacteristic
94#2214C2×C12712yescomplemented
95#1709C2×C12711yescharacteristic, complemented
96#1357C2×C12711yescharacteristic, complemented
97#1335C2×C12712yescomplemented
98C127#1231211yescharacteristic, direct product factor, Sylow
99#576—22no—
100#510—22no—
101#494C22×C12711yescharacteristic, complemented
102#493C22×C12711yescharacteristic
103#486—22no—
104#472—22no—
105#453—22no—
106#443C22×C12712yes—
107#364C22×C12712yescomplemented
108#362—22no—
109C82C22×C12711yescharacteristic, complemented
110H(5,𝔽2)—22no—
111AGL(1,ℤ/8)—24no—
112AGL(1,ℤ/8)—44no—
113#135—42no—
114#135D4×C12712yes—
115C2×Q16—22no—
116C2×D8—22no—
117C2×D8—42no—
118#131—22no—
119C4⋊Q8—21no—
120#127D4×C12712yes—
121Q32—42no—
122SD32—42no—
123M32—22no—
124#108D4×C12712yes—
125#104—22no—
126#102—44no—
127#100—22no—
128C4×C8—21no—
129C4×C8C23×C12711yescharacteristic
130#41—22no—
131#41—44no—
132C2×Q8—41no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—82no—
137Q16—42no—
138Q16—42no—
139Q16D8×C12712yes—
140SD16—22no—
141SD16—44no—
142D8—44no—
143D8—42no—
144D8—42no—
145D8D8×C12712yes—
146M16—22no—
147M16—42no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8C2×D4×C12712yes—
151C4⋊C4—42no—
152C42C2×D4×C12711yescharacteristic
153C16—42no—
154C23—42no—
155C23—82no—
156Q8—22no—
157Q8—82no—
158D4—22no—
159D4—42no—
160D4—44no—
161D4—84no—
162D4—84no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4#120×C12711yescharacteristic
168C8—42no—
169C8—42no—
170C8C2×D8×C12712yes—
171C8AGL(1,ℤ/8)×C12712yes—
172C22—42no—
173C22—82no—
174C22—162no—
175C22—322no—
176C22#453×C12711yescharacteristic
177C4—22no—
178C4—42no—
179C4—161no—
180C4#447×C12712yes—
181C2—21no—
182C2—82no—
183C2—162no—
184C2C127×#161911yescharacteristic
subgroupG/H
center, socleC2×C127#1619
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3292C2
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
2. C127×#2214C2semidirect
3. C127×#1709C2semidirect
4. C127×#1357C2semidirect
5. C127×#1335C2semidirect
8. #494×C127C22semidirect
14. #364×C127C22semidirect
16. C82×C127C22semidirect
88. #12312C127direct (N × H)
94. #2214C2×C127semidirect
95. #1709C2×C127semidirect
96. #1357C2×C127semidirect
97. #1335C2×C127semidirect
98. C127#12312direct (N × H)
101. #494C22×C127semidirect
107. #364C22×C127semidirect
109. C82C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32922
C127×#22142
C127×#17092
C127×#13572
C127×#13352
#12312127