Finite Group Databasebeta

← C127×#11465 #44575138

Subgroups — C127×#11465 #44575138

Order 32512.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×D4×C127C211yesmaximal, characteristic
2C2×C22⋊C16×C127C212yesmaximal, complemented
3C127×#1322C212yesmaximal, complemented
4C127×#1314C212yesmaximal, complemented
5C23×C8×C127C411yescharacteristic
6C23×C8×C127C2211yescharacteristic
7C2×C4×D4×C127C411yescharacteristic
8C22×C16×C127—22no—
9C8×D4×C127—22no—
10C8×D4×C127—22no—
11C2×C4⋊C8×C127C2211yescharacteristic
12C2×C22⋊C8×C127C411yescharacteristic
13C2×C22⋊C8×C127C2211yescharacteristic
14C2×C4×C8×C127C411yescharacteristic
15C22⋊C16×C127—22no—
16C22⋊C16×C127C2214yes—
17C22×D4×C127C811yescharacteristic
18C23×C4×C127C2×C411yescharacteristic
19C23×C4×C127C2×C411yescharacteristic
20C22×C8×C127—21no—
21C22×C8×C127—21no—
22C22×C8×C127—42no—
23C22×C8×C127—42no—
24C22×C8×C127C2×C411yescharacteristic
25C22×C8×C127C2×C411yescharacteristic
26C22×C8×C127C2×C411yescharacteristic
27C22×C8×C127D412yes—
28C22×C8×C127D412yes—
29C22×C8×C127C2311yescharacteristic
30C4×D4×C127—22no—
31C4×D4×C127—22no—
32C2×C4⋊C4×C127C811yescharacteristic
33C2×C22⋊C4×C127C811yescharacteristic
34C2×C22⋊C4×C127C811yescharacteristic
35C2×C42×C127C2×C411yescharacteristic
36C2×C16×C127—24no—
37C2×C16×C127—44no—
38C4⋊C8×C127—22no—
39C22⋊C8×C127—22no—
40C22⋊C8×C127—22no—
41C4×C8×C127—21no—
42C4×C8×C127—21no—
43C24×C127C2×C811yescharacteristic
44C24×C127C2×C811yescharacteristic
45C2×D4×C127—22no—
46C2×D4×C127—22no—
47C2×D4×C127C1614yescomplemented
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—41no—
51C22×C4×C127—41no—
52C22×C4×C127—41no—
53C22×C4×C127—41no—
54C22×C4×C127C22⋊C412yes—
55C22×C4×C127C2×C811yescharacteristic
56C22×C4×C127C2×C811yescharacteristic
57C22×C4×C127C2×C811yescharacteristic
58C22×C4×C127C2×C811yescharacteristic
59C22×C4×C127M1612yes—
60C22×C4×C127C22×C411yescharacteristic
61C2×C8×C127—24no—
62C2×C8×C127—22no—
63C2×C8×C127—21no—
64C2×C8×C127—21no—
65C2×C8×C127—42no—
66C2×C8×C127—41no—
67C2×C8×C127—42no—
68C2×C8×C127—42no—
69C2×C8×C127—41no—
70C2×C8×C127—41no—
71C2×C8×C127C22⋊C411yescharacteristic
72C2×C8×C127C22⋊C411yescharacteristic
73C2×C8×C127C2×D412yes—
74C4⋊C4×C127—22no—
75C22⋊C4×C127—22no—
76C22⋊C4×C127—22no—
77C42×C127—21no—
78C42×C127—21no—
79C16×C127—44no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—41no—
86C23×C127C22⋊C812yes—
87C23×C127C2×C1612yes—
88C23×C127C2×C1612yescomplemented
89C23×C127M3212yes—
90C23×C127C22×C811yescharacteristic
91D4×C127—28no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—41no—
101C2×C4×C127—42no—
102C2×C4×C127—42no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127C22⋊C812yes—
108C2×C4×C127C2×C1612yescomplemented
109C2×C4×C127C2×C22⋊C411yescharacteristic
110C2×C4×C127C2×M1611yescharacteristic
111C8×C127—22no—
112C8×C127—21no—
113C8×C127—41no—
114C8×C127—41no—
115C8×C127#9912yes—
116C22×C127—21no—
117C22×C127—22no—
118C22×C127—21no—
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—42no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127C22⋊C1614yes—
128C22×C127C2×C22⋊C811yescharacteristic
129C22×C127C22×C1611yescharacteristic
130C22×C127C2×M3211yescharacteristic
131C4×C127—22no—
132C4×C127—21no—
133C4×C127—22no—
134C4×C127—41no—
135C4×C127—41no—
136C4×C127C2×#9911yescharacteristic
137C4×C127#41311yescharacteristic
138#11465C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—21no—
140C2×C127—21no—
141C2×C127—24no—
142C2×C127—42no—
143C2×C127#145911yescharacteristic
144C2×C127C2×C22⋊C1611yescharacteristic
145C2×C127#211511yescharacteristic
146C2×C8×D4C2×C12711yescharacteristic
147C2×C22⋊C16C2×C12712yescomplemented
148#1322C2×C12712yescomplemented
149#1314C2×C12712yescomplemented
150C127#1146511yescharacteristic, direct product factor, Sylow
151C23×C8C4×C12711yescharacteristic
152C23×C8C22×C12711yescharacteristic
153C2×C4×D4C4×C12711yescharacteristic
154C22×C16—22no—
155C8×D4—22no—
156C8×D4—22no—
157C2×C4⋊C8C22×C12711yescharacteristic
158C2×C22⋊C8C4×C12711yescharacteristic
159C2×C22⋊C8C22×C12711yescharacteristic
160C2×C4×C8C4×C12711yescharacteristic
161C22⋊C16—22no—
162C22⋊C16C22×C12714yes—
163C22×D4C8×C12711yescharacteristic
164C23×C4C2×C4×C12711yescharacteristic
165C23×C4C2×C4×C12711yescharacteristic
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8C2×C4×C12711yescharacteristic
171C22×C8C2×C4×C12711yescharacteristic
172C22×C8C2×C4×C12711yescharacteristic
173C22×C8D4×C12712yes—
174C22×C8D4×C12712yes—
175C22×C8C23×C12711yescharacteristic
176C4×D4—22no—
177C4×D4—22no—
178C2×C4⋊C4C8×C12711yescharacteristic
179C2×C22⋊C4C8×C12711yescharacteristic
180C2×C22⋊C4C8×C12711yescharacteristic
181C2×C42C2×C4×C12711yescharacteristic
182C2×C16—24no—
183C2×C16—44no—
184C4⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C4×C8—21no—
188C4×C8—21no—
189C24C2×C8×C12711yescharacteristic
190C24C2×C8×C12711yescharacteristic
191C2×D4—22no—
192C2×D4—22no—
193C2×D4C16×C12714yescomplemented
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4C22⋊C4×C12712yes—
201C22×C4C2×C8×C12711yescharacteristic
202C22×C4C2×C8×C12711yescharacteristic
203C22×C4C2×C8×C12711yescharacteristic
204C22×C4C2×C8×C12711yescharacteristic
205C22×C4M16×C12712yes—
206C22×C4C22×C4×C12711yescharacteristic
207C2×C8—24no—
208C2×C8—22no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8C22⋊C4×C12711yescharacteristic
218C2×C8C22⋊C4×C12711yescharacteristic
219C2×C8C2×D4×C12712yes—
220C4⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C42—21no—
224C42—21no—
225C16—44no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C22⋊C8×C12712yes—
233C23C2×C16×C12712yes—
234C23C2×C16×C12712yescomplemented
235C23M32×C12712yes—
236C23C22×C8×C12711yescharacteristic
237D4—28no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4C22⋊C8×C12712yes—
254C2×C4C2×C16×C12712yescomplemented
255C2×C4C2×C22⋊C4×C12711yescharacteristic
256C2×C4C2×M16×C12711yescharacteristic
257C8—22no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8#99×C12712yes—
262C22—21no—
263C22—22no—
264C22—21no—
265C22—22no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22C22⋊C16×C12714yes—
274C22C2×C22⋊C8×C12711yescharacteristic
275C22C22×C16×C12711yescharacteristic
276C22C2×M32×C12711yescharacteristic
277C4—22no—
278C4—21no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4C2×#99×C12711yescharacteristic
283C4#413×C12711yescharacteristic
284C2—21no—
285C2—21no—
286C2—24no—
287C2—42no—
288C2C127×#145911yescharacteristic
289C2C2×C22⋊C16×C12711yescharacteristic
290C2C127×#211511yescharacteristic
subgroupG/H
centerC2×C8×C127C22⋊C4
commutatorC22C22×C16×C127
Frattini, cosocleC22×C8C23×C127
socleC22×C127C2×C22⋊C8
prime radicalC22×D4×C127C8
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C16×C127C2semidirect
3. C127×#1322C2semidirect
4. C127×#1314C2semidirect
47. C2×D4×C127C16semidirect
88. C23×C127C2×C16semidirect
108. C2×C4×C127C2×C16semidirect
138. #11465C127direct (N × H)
147. C2×C22⋊C16C2×C127semidirect
148. #1322C2×C127semidirect
149. #1314C2×C127semidirect
150. C127#11465direct (N × H)
193. C2×D4C16×C127semidirect
234. C23C2×C16×C127semidirect
254. C2×C4C2×C16×C127semidirect

Maximal subgroups

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

subgroupindex
C2×C8×D4×C1272
C2×C22⋊C16×C1272
C127×#13222
C127×#13142
#11465127