Finite Group Databasebeta

← C127×#7232 #44570905

Subgroups — C127×#7232 #44570905

Order 32512.

292 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×C22⋊C16×C127C211yesmaximal, characteristic
2C127×#1810C211yesmaximal, characteristic
3C127×#1807C211yesmaximal, characteristic
4C23×C8×C127C2211yescharacteristic
5C22×C16×C127C412yescomplemented
6C2×#108×C127—21no—
7C2×#106×C127—21no—
8#407×C127—21no—
9#380×C127—21no—
10C22⋊C16×C127—21no—
11C22⋊C16×C127—21no—
12#341×C127C412yescomplemented
13#340×C127C412yes—
14C23×C4×C127D411yescharacteristic
15C2×M16×C127—22no—
16C22×C8×C127—42no—
17C22×C8×C127C2×C411yescharacteristic
18C22×C8×C127C2×C411yescharacteristic
19C22×C8×C127C2×C411yescharacteristic
20C22×C8×C127D411yescharacteristic
21C22×C8×C127D411yescharacteristic
22C22×C8×C127Q811yescharacteristic
23C2×C4⋊C4×C127—22no—
24C2×C22⋊C4×C127—41no—
25C2×C16×C127—22no—
26C2×C16×C127—42no—
27#108×C127—42no—
28#106×C127—42no—
29C22⋊C8×C127—41no—
30C22⋊C8×C127—41no—
31#95×C127—41no—
32C24×C127SD1611yescharacteristic
33C22×C4×C127—41no—
34C22×C4×C127—41no—
35C22×C4×C127—42no—
36C22×C4×C127C22⋊C411yescharacteristic
37C22×C4×C127C22⋊C411yescharacteristic
38C22×C4×C127C4⋊C411yescharacteristic
39C22×C4×C127D811yescharacteristic
40C22×C4×C127SD1611yescharacteristic
41C22×C4×C127Q1611yescharacteristic
42M16×C127—42no—
43M16×C127—42no—
44C2×C8×C127—22no—
45C2×C8×C127—22no—
46C2×C8×C127—22no—
47C2×C8×C127—41no—
48C2×C8×C127—41no—
49C2×C8×C127—41no—
50C2×C8×C127—42no—
51C2×C8×C127C4211yescharacteristic
52C2×C8×C127C22⋊C411yescharacteristic
53C2×C8×C127C4⋊C411yescharacteristic
54C2×C8×C127C4⋊C411yescharacteristic
55C4⋊C4×C127—42no—
56C4⋊C4×C127—42no—
57C22⋊C4×C127—82no—
58C16×C127—42no—
59C23×C127—22no—
60C23×C127—22no—
61C23×C127—41no—
62C23×C127#10211yescharacteristic
63C23×C127#10311yescharacteristic
64C23×C127#10611yescharacteristic
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—82no—
75C2×C4×C127—82no—
76C2×C4×C127—82no—
77C2×C4×C127#9511yescharacteristic
78C2×C4×C127#10211yescharacteristic
79C2×C4×C127#10311yescharacteristic
80C2×C4×C127#10711yescharacteristic
81C8×C127—21no—
82C8×C127—21no—
83C8×C127—21no—
84C8×C127—82no—
85C8×C127#9912yes—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—44no—
94C22×C127#34011yescharacteristic
95C22×C127#35912yes—
96C22×C127#36512yes—
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—162no—
101C4×C127H(3,ℤ/4)11yescharacteristic
102C4×C127#34211yescharacteristic
103#7232C12711yesmaximal, characteristic, direct product factor, Sylow
104C2×C127—22no—
105C2×C127—22no—
106C2×C127—41no—
107C2×C127#128911yescharacteristic
108C2×C127#138211yescharacteristic
109C2×C127#138311yescharacteristic
110C2×C22⋊C16C2×C12711yescharacteristic
111#1810C2×C12711yescharacteristic
112#1807C2×C12711yescharacteristic
113C127#723211yescharacteristic, direct product factor, Sylow
114C23×C8C22×C12711yescharacteristic
115C22×C16C4×C12712yescomplemented
116C2×#108—21no—
117C2×#106—21no—
118#407—21no—
119#380—21no—
120C22⋊C16—21no—
121C22⋊C16—21no—
122#341C4×C12712yescomplemented
123#340C4×C12712yes—
124C23×C4D4×C12711yescharacteristic
125C2×M16—22no—
126C22×C8—42no—
127C22×C8C2×C4×C12711yescharacteristic
128C22×C8C2×C4×C12711yescharacteristic
129C22×C8C2×C4×C12711yescharacteristic
130C22×C8D4×C12711yescharacteristic
131C22×C8D4×C12711yescharacteristic
132C22×C8Q8×C12711yescharacteristic
133C2×C4⋊C4—22no—
134C2×C22⋊C4—41no—
135C2×C16—22no—
136C2×C16—42no—
137#108—42no—
138#106—42no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#95—41no—
142C24SD16×C12711yescharacteristic
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4C22⋊C4×C12711yescharacteristic
147C22×C4C22⋊C4×C12711yescharacteristic
148C22×C4C4⋊C4×C12711yescharacteristic
149C22×C4D8×C12711yescharacteristic
150C22×C4SD16×C12711yescharacteristic
151C22×C4Q16×C12711yescharacteristic
152M16—42no—
153M16—42no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—42no—
161C2×C8C42×C12711yescharacteristic
162C2×C8C22⋊C4×C12711yescharacteristic
163C2×C8C4⋊C4×C12711yescharacteristic
164C2×C8C4⋊C4×C12711yescharacteristic
165C4⋊C4—42no—
166C4⋊C4—42no—
167C22⋊C4—82no—
168C16—42no—
169C23—22no—
170C23—22no—
171C23—41no—
172C23#102×C12711yescharacteristic
173C23#103×C12711yescharacteristic
174C23#106×C12711yescharacteristic
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4#95×C12711yescharacteristic
188C2×C4#102×C12711yescharacteristic
189C2×C4#103×C12711yescharacteristic
190C2×C4#107×C12711yescharacteristic
191C8—21no—
192C8—21no—
193C8—21no—
194C8—82no—
195C8#99×C12712yes—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22#340×C12711yescharacteristic
205C22#359×C12712yes—
206C22#365×C12712yes—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—162no—
211C4H(3,ℤ/4)×C12711yescharacteristic
212C4#342×C12711yescharacteristic
213C2—22no—
214C2—22no—
215C2—41no—
216C2C127×#128911yescharacteristic
217C2C127×#138211yescharacteristic
218C2C127×#138311yescharacteristic
subgroupG/H
center, socleC22×C127#340
commutatorC2×C8C42×C127
Frattini, cosocleC23×C8C22×C127
prime radicalC24×C127SD16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
5. C22×C16×C127C4semidirect
12. #341×C127C4semidirect
103. #7232C127direct (N × H)
113. C127#7232direct (N × H)
115. C22×C16C4×C127semidirect
122. #341C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C16×C1272
C127×#18102
C127×#18072
#7232127