Finite Group Databasebeta

← C127×#7260 #44570933

Subgroups — C127×#7260 #44570933

Order 32512.

336 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 106 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×#1735C211yesmaximal, characteristic
4C23×C8×C127C2211yescharacteristic
5C2×#117×C127—21no—
6C22×C16×C127—21no—
7C2×#108×C127C412yescomplemented
8#407×C127—21no—
9C22⋊C16×C127—21no—
10C22⋊C16×C127C412yescomplemented
11#341×C127—21no—
12#340×C127—21no—
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
23#117×C127—24no—
24#117×C127—42no—
25C2×C4⋊C4×C127—22no—
26C2×C22⋊C4×C127—41no—
27C2×C42×C127—41no—
28C2×C16×C127—22no—
29C2×C16×C127—42no—
30#108×C127—42no—
31C22⋊C8×C127—41no—
32C22⋊C8×C127—41no—
33C24×C127SD1611yescharacteristic
34C22×C4×C127—22no—
35C22×C4×C127—22no—
36C22×C4×C127—42no—
37C22×C4×C127C22⋊C411yescharacteristic
38C22×C4×C127C22⋊C411yescharacteristic
39C22×C4×C127C4⋊C411yescharacteristic
40C22×C4×C127D811yescharacteristic
41C22×C4×C127SD1611yescharacteristic
42C22×C4×C127Q1611yescharacteristic
43M16×C127—42no—
44M16×C127—42no—
45C2×C8×C127—22no—
46C2×C8×C127—22no—
47C2×C8×C127—41no—
48C2×C8×C127—41no—
49C2×C8×C127—41no—
50C2×C8×C127—41no—
51C2×C8×C127—42no—
52C2×C8×C127C4211yescharacteristic
53C2×C8×C127C22⋊C411yescharacteristic
54C2×C8×C127C4⋊C411yescharacteristic
55C2×C8×C127C4⋊C411yescharacteristic
56C4⋊C4×C127—24no—
57C4⋊C4×C127—42no—
58C22⋊C4×C127—44no—
59C42×C127—44no—
60C16×C127—42no—
61C23×C127—22no—
62C23×C127—22no—
63C23×C127—41no—
64C23×C127#10211yescharacteristic
65C23×C127#10311yescharacteristic
66C23×C127#10611yescharacteristic
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—44no—
75C2×C4×C127—82no—
76C2×C4×C127—82no—
77C2×C4×C127#9511yescharacteristic
78C2×C4×C127#9912yes—
79C2×C4×C127#10211yescharacteristic
80C2×C4×C127#10311yescharacteristic
81C2×C4×C127#10412yes—
82C2×C4×C127#10711yescharacteristic
83C8×C127—21no—
84C8×C127—21no—
85C8×C127—21no—
86C8×C127—21no—
87C8×C127—82no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—44no—
96C22×C127#32812yes—
97C22×C127#34011yescharacteristic
98C22×C127#35912yes—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—84no—
102C4×C127#32712yes—
103C4×C127#33911yescharacteristic
104C4×C127#34211yescharacteristic
105#7260C12711yesmaximal, characteristic, direct product factor, Sylow
106C2×C127—22no—
107C2×C127—22no—
108C2×C127—41no—
109C2×C127#129411yescharacteristic
110C2×C127#138211yescharacteristic
111C2×C127#138911yescharacteristic
112C2×C22⋊C16C2×C12711yescharacteristic
113#1810C2×C12711yescharacteristic
114#1735C2×C12711yescharacteristic
115C127#726011yescharacteristic, direct product factor, Sylow
116C23×C8C22×C12711yescharacteristic
117C2×#117—21no—
118C22×C16—21no—
119C2×#108C4×C12712yescomplemented
120#407—21no—
121C22⋊C16—21no—
122C22⋊C16C4×C12712yescomplemented
123#341—21no—
124#340—21no—
125#340C4×C12712yes—
126C23×C4D4×C12711yescharacteristic
127C2×M16—22no—
128C22×C8—42no—
129C22×C8C2×C4×C12711yescharacteristic
130C22×C8C2×C4×C12711yescharacteristic
131C22×C8C2×C4×C12711yescharacteristic
132C22×C8D4×C12711yescharacteristic
133C22×C8D4×C12711yescharacteristic
134C22×C8Q8×C12711yescharacteristic
135#117—24no—
136#117—42no—
137C2×C4⋊C4—22no—
138C2×C22⋊C4—41no—
139C2×C42—41no—
140C2×C16—22no—
141C2×C16—42no—
142#108—42no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C24SD16×C12711yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4C22⋊C4×C12711yescharacteristic
150C22×C4C22⋊C4×C12711yescharacteristic
151C22×C4C4⋊C4×C12711yescharacteristic
152C22×C4D8×C12711yescharacteristic
153C22×C4SD16×C12711yescharacteristic
154C22×C4Q16×C12711yescharacteristic
155M16—42no—
156M16—42no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8C42×C12711yescharacteristic
165C2×C8C22⋊C4×C12711yescharacteristic
166C2×C8C4⋊C4×C12711yescharacteristic
167C2×C8C4⋊C4×C12711yescharacteristic
168C4⋊C4—24no—
169C4⋊C4—42no—
170C22⋊C4—44no—
171C42—44no—
172C16—42no—
173C23—22no—
174C23—22no—
175C23—41no—
176C23#102×C12711yescharacteristic
177C23#103×C12711yescharacteristic
178C23#106×C12711yescharacteristic
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—44no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4#95×C12711yescharacteristic
190C2×C4#99×C12712yes—
191C2×C4#102×C12711yescharacteristic
192C2×C4#103×C12711yescharacteristic
193C2×C4#104×C12712yes—
194C2×C4#107×C12711yescharacteristic
195C8—21no—
196C8—21no—
197C8—21no—
198C8—21no—
199C8—82no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—44no—
208C22#328×C12712yes—
209C22#340×C12711yescharacteristic
210C22#359×C12712yes—
211C4—21no—
212C4—21no—
213C4—84no—
214C4#327×C12712yes—
215C4#339×C12711yescharacteristic
216C4#342×C12711yescharacteristic
217C2—22no—
218C2—22no—
219C2—41no—
220C2C127×#129411yescharacteristic
221C2C127×#138211yescharacteristic
222C2C127×#138911yescharacteristic
subgroupG/H
centerC2×C4×C127#102
commutatorC2×C8C42×C127
Frattini, cosocleC23×C8C22×C127
socleC22×C127#340
prime radicalC24×C127SD16
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×#108×C127C4semidirect
10. C22⋊C16×C127C4semidirect
105. #7260C127direct (N × H)
115. C127#7260direct (N × H)
119. C2×#108C4×C127semidirect
122. C22⋊C16C4×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×#17352
#7260127