Finite Group Databasebeta

← C127×#12681 #44576354

Subgroups — C127×#12681 #44576354

Order 32512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C127C211yesmaximal, characteristic
2C2×#349×C127C214yesmaximal, complemented
3C127×#2112C212yesmaximal, complemented
4C24×C4×C127C411yescharacteristic
5C22×M16×C127—28no—
6C22×M16×C127—24no—
7C22×M16×C127C412yes—
8C22×M16×C127C2212yes—
9C23×C8×C127C411yescharacteristic
10C23×C8×C127C2211yescharacteristic
11C2×M32×C127C2214yescomplemented
12#349×C127—28no—
13C22⋊C16×C127—24no—
14C25×C127C811yescharacteristic
15C23×C4×C127—24no—
16C23×C4×C127—22no—
17C23×C4×C127—22no—
18C23×C4×C127—24no—
19C23×C4×C127C811yescharacteristic
20C23×C4×C127C812yes—
21C23×C4×C127C2×C411yescharacteristic
22C23×C4×C127C2×C412yes—
23C2×M16×C127—232no—
24C2×M16×C127—24no—
25C2×M16×C127—24no—
26C2×M16×C127—24no—
27C2×M16×C127—24no—
28C2×M16×C127—24no—
29C2×M16×C127D418yes—
30C22×C8×C127—22no—
31C22×C8×C127—24no—
32C22×C8×C127—24no—
33C22×C8×C127C2×C412yes—
34C22×C8×C127C2×C411yescharacteristic
35C22×C8×C127D414yes—
36C22×C8×C127C2311yescharacteristic
37M32×C127—28no—
38C2×C16×C127—24no—
39C24×C127—24no—
40C24×C127—22no—
41C24×C127—44no—
42C24×C127C2×C811yescharacteristic
43C24×C127C2×C812yes—
44C22×C4×C127—28no—
45C22×C4×C127—216no—
46C22×C4×C127—28no—
47C22×C4×C127—22no—
48C22×C4×C127—24no—
49C22×C4×C127—24no—
50C22×C4×C127—24no—
51C22×C4×C127—24no—
52C22×C4×C127—22no—
53C22×C4×C127—22no—
54C22×C4×C127—22no—
55C22×C4×C127—22no—
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127C22⋊C414yes—
59C22×C4×C127C22⋊C412yes—
60C22×C4×C127C2×C811yescharacteristic
61C22×C4×C127C2×C812yes—
62C22×C4×C127M1612yes—
63C22×C4×C127M1614yes—
64C22×C4×C127C22×C411yescharacteristic
65M16×C127—216no—
66M16×C127—216no—
67C2×C8×C127—28no—
68C2×C8×C127—22no—
69C2×C8×C127—22no—
70C2×C8×C127—22no—
71C2×C8×C127—28no—
72C2×C8×C127C22⋊C414yes—
73C2×C8×C127C22⋊C412yes—
74C2×C8×C127C2×D414yes—
75C2×C8×C127C2×D412yes—
76C16×C127—44no—
77C23×C127—24no—
78C23×C127—22no—
79C23×C127—28no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—44no—
83C23×C127—416no—
84C23×C127—44no—
85C23×C127—44no—
86C23×C127C22⋊C812yes—
87C23×C127C22⋊C814yes—
88C23×C127C22×C811yescharacteristic
89C2×C4×C127—28no—
90C2×C4×C127—216no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—24no—
96C2×C4×C127—24no—
97C2×C4×C127—22no—
98C2×C4×C127—24no—
99C2×C4×C127—24no—
100C2×C4×C127—24no—
101C2×C4×C127C22⋊C814yes—
102C2×C4×C127C22⋊C812yes—
103C2×C4×C127C2×C22⋊C412yes—
104C2×C4×C127C2×C22⋊C411yescharacteristic
105C2×C4×C127C2×M1611yescharacteristic
106C2×C4×C127C2×M1612yes—
107C8×C127—24no—
108C8×C127—22no—
109C8×C127#12014yes—
110C22×C127—28no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—22no—
114C22×C127—24no—
115C22×C127—44no—
116C22×C127—416no—
117C22×C127—44no—
118C22×C127—44no—
119C22×C127#34914yescomplemented
120C22×C127C2×C22⋊C812yes—
121C22×C127C2×C22⋊C811yescharacteristic
122C4×C127—22no—
123C4×C127—24no—
124C4×C127#37911yescharacteristic
125C4×C127#40712yes—
126C4×C127#40711yescharacteristic
127#12681C12711yesmaximal, characteristic, direct product factor, Sylow
128C2×C127—24no—
129C2×C127—22no—
130C2×C127—44no—
131C2×C127#177911yescharacteristic
132C2×C127C2×#34912yes—
133C23×M16C2×C12711yescharacteristic
134C2×#349C2×C12714yescomplemented
135#2112C2×C12712yescomplemented
136C127#1268111yescharacteristic, direct product factor, Sylow
137C24×C4C4×C12711yescharacteristic
138C22×M16—28no—
139C22×M16—24no—
140C22×M16C4×C12712yes—
141C22×M16C22×C12712yes—
142C23×C8C4×C12711yescharacteristic
143C23×C8C22×C12711yescharacteristic
144C2×M32C22×C12714yescomplemented
145#349—28no—
146C22⋊C16—24no—
147C25C8×C12711yescharacteristic
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4C8×C12711yescharacteristic
153C23×C4C8×C12712yes—
154C23×C4C2×C4×C12711yescharacteristic
155C23×C4C2×C4×C12712yes—
156C2×M16—232no—
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16D4×C12718yes—
163C22×C8—22no—
164C22×C8—24no—
165C22×C8—24no—
166C22×C8C2×C4×C12712yes—
167C22×C8C2×C4×C12711yescharacteristic
168C22×C8D4×C12714yes—
169C22×C8C23×C12711yescharacteristic
170M32—28no—
171C2×C16—24no—
172C24—24no—
173C24—22no—
174C24—44no—
175C24C2×C8×C12711yescharacteristic
176C24C2×C8×C12712yes—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—216no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4C22⋊C4×C12712yes—
192C22×C4C22⋊C4×C12714yes—
193C22×C4C2×C8×C12711yescharacteristic
194C22×C4C2×C8×C12712yes—
195C22×C4M16×C12712yes—
196C22×C4M16×C12714yes—
197C22×C4C22×C4×C12711yescharacteristic
198M16—216no—
199M16—216no—
200C2×C8—28no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—28no—
205C2×C8C22⋊C4×C12714yes—
206C2×C8C22⋊C4×C12712yes—
207C2×C8C2×D4×C12712yes—
208C2×C8C2×D4×C12714yes—
209C16—44no—
210C23—24no—
211C23—22no—
212C23—28no—
213C23—22no—
214C23—22no—
215C23—44no—
216C23—416no—
217C23—44no—
218C23—44no—
219C23C22⋊C8×C12712yes—
220C23C22⋊C8×C12714yes—
221C23C22×C8×C12711yescharacteristic
222C2×C4—28no—
223C2×C4—216no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4C22⋊C8×C12714yes—
235C2×C4C22⋊C8×C12712yes—
236C2×C4C2×C22⋊C4×C12712yes—
237C2×C4C2×C22⋊C4×C12711yescharacteristic
238C2×C4C2×M16×C12712yes—
239C2×C4C2×M16×C12711yescharacteristic
240C8—24no—
241C8—22no—
242C8#120×C12714yes—
243C22—28no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—24no—
248C22—44no—
249C22—416no—
250C22—44no—
251C22—44no—
252C22#349×C12714yescomplemented
253C22C2×C22⋊C8×C12712yes—
254C22C2×C22⋊C8×C12711yescharacteristic
255C4—22no—
256C4—24no—
257C4#379×C12711yescharacteristic
258C4#407×C12712yes—
259C4#407×C12711yescharacteristic
260C2—24no—
261C2—22no—
262C2—44no—
263C2C127×#177911yescharacteristic
264C2C2×#349×C12712yes—
subgroupG/H
centerC2×C4×C127C2×C22⋊C4
commutatorC23C22×C8×C127
Frattini, cosocleC22×C8C23×C127
socleC22×C127C2×C22⋊C8
prime radicalC25×C127C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#349×C127C2semidirect
3. C127×#2112C2semidirect
11. C2×M32×C127C22semidirect
119. C22×C127#349semidirect
127. #12681C127direct (N × H)
134. C2×#349C2×C127semidirect
135. #2112C2×C127semidirect
136. C127#12681direct (N × H)
144. C2×M32C22×C127semidirect
252. C22#349×C127semidirect

Maximal subgroups

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

subgroupindex
C23×M16×C1272
C2×#349×C1272
C127×#21122
#12681127