Finite Group Databasebeta

← C4×#343×C127 #44574382

Subgroups — C4×#343×C127 #44574382

Order 32512.

994 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×M16×C127C212yesmaximal
2C2×C4×C4⋊C4×C127C211yesmaximal, characteristic
3C2×#343×C127C214yesmaximal
4C22×M16×C127C2214yes—
5C22×C4⋊C4×C127C2212yes—
6C22×C42×C127C412yes—
7C22×C42×C127C2211yescharacteristic
8C4×M16×C127—24no—
9C4×M16×C127C418yescomplemented
10C2×#97×C127—22no—
11C2×C4×C8×C127—22no—
12C4×C4⋊C4×C127—22no—
13C4×C4⋊C4×C127—22no—
14C2×#95×C127C412yes—
15#343×C127C4116yesdirect product factor
16C23×C4×C127C2×C411yescharacteristic
17C23×C4×C127C2×C414yes—
18C23×C4×C127C2×C411yescharacteristic
19C23×C4×C127C2311yescharacteristic
20C2×M16×C127—24no—
21C2×M16×C127—28no—
22C2×M16×C127C2×C4116yes—
23C2×M16×C127C2×C418yes—
24C22×C8×C127—24no—
25C2×C4⋊C4×C127—24no—
26C2×C4⋊C4×C127—24no—
27C2×C4⋊C4×C127C2×C418yes—
28C2×C42×C127—22no—
29C2×C42×C127—22no—
30C2×C42×C127—44no—
31C2×C42×C127C2×C414yes—
32C2×C42×C127D412yes—
33C2×C42×C127D412yes—
34C2×C42×C127D412yes—
35C2×C42×C127Q812yes—
36#97×C127—28no—
37C4×C8×C127—28no—
38#95×C127—24no—
39C24×C127C22×C411yescharacteristic
40C22×C4×C127—22no—
41C22×C4×C127—22no—
42C22×C4×C127—24no—
43C22×C4×C127—24no—
44C22×C4×C127—42no—
45C22×C4×C127—44no—
46C22×C4×C127—44no—
47C22×C4×C127C4212yes—
48C22×C4×C127C4218yes—
49C22×C4×C127C4212yes—
50C22×C4×C127C22⋊C412yes—
51C22×C4×C127C22⋊C411yescharacteristic
52C22×C4×C127C22⋊C411yescharacteristic
53C22×C4×C127C4⋊C412yes—
54C22×C4×C127C4⋊C412yes—
55C22×C4×C127C22×C412yes—
56C22×C4×C127C22×C414yes—
57C22×C4×C127C2×D411yescharacteristic
58C22×C4×C127C2×D412yes—
59C22×C4×C127C2×Q811yescharacteristic
60C22×C4×C127#4114yes—
61M16×C127—28no—
62M16×C127—216no—
63M16×C127C42116yescomplemented
64C2×C8×C127—28no—
65C2×C8×C127—216no—
66C4⋊C4×C127—216no—
67C42×C127—44no—
68C42×C127—44no—
69C42×C127C22⋊C414yes—
70C42×C127C22⋊C414yes—
71C42×C127C4⋊C414yes—
72C42×C127C4⋊C414yes—
73C23×C127—21no—
74C23×C127—21no—
75C23×C127—22no—
76C23×C127C2×C4211yescharacteristic
77C23×C127C2×C4212yes—
78C23×C127C2×C22⋊C411yescharacteristic
79C23×C127C2×C4⋊C411yescharacteristic
80C23×C127#11712yes—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—24no—
84C2×C4×C127—24no—
85C2×C4×C127—24no—
86C2×C4×C127—28no—
87C2×C4×C127—42no—
88C2×C4×C127—42no—
89C2×C4×C127—48no—
90C2×C4×C127—44no—
91C2×C4×C127—44no—
92C2×C4×C127—48no—
93C2×C4×C127#9512yes—
94C2×C4×C127#9514yes—
95C2×C4×C127#9512yes—
96C2×C4×C127#10012yes—
97C2×C4×C127#10112yes—
98C2×C4×C127C2×C4214yes—
99C2×C4×C127C2×C22⋊C412yes—
100C2×C4×C127C2×C4⋊C412yes—
101C2×C4×C127#11714yes—
102C2×C4×C127C4×D414yes—
103C2×C4×C127C4×D418yes—
104C2×C4×C127C4×Q814yes—
105C8×C127—216no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—22no—
113C22×C127—22no—
114C22×C127C4311yescharacteristic
115C22×C127C2×#9511yescharacteristic
116C22×C127#37611yescharacteristic
117C22×C127C4×C22⋊C412yes—
118C22×C127C4×C4⋊C412yes—
119C22×C127C2×#10011yescharacteristic
120C22×C127C2×#10111yescharacteristic
121C22×C127#41312yes—
122C4×C127—22no—
123C4×C127—24no—
124C4×C127—48no—
125C4×C127#34314yesdirect product factor
126C4×C127C4×C22⋊C414yes—
127C4×C127C4×C4⋊C414yes—
128C4×#343C12711yesmaximal, characteristic, direct product factor, Sylow
129C2×C127—21no—
130C2×C127—21no—
131C2×C127—22no—
132C2×C127C4×#9511yescharacteristic
133C2×C127C2×#34311yescharacteristic
134C2×C127#174211yescharacteristic
135C2×C127C4×#10012yes—
136C2×C127C4×#10112yes—
137C2×C4×M16C2×C12712yes—
138C2×C4×C4⋊C4C2×C12711yescharacteristic
139C2×#343C2×C12714yes—
140C127C4×#34311yescharacteristic, direct product factor, Sylow
141C22×M16C22×C12714yes—
142C22×C4⋊C4C22×C12712yes—
143C22×C42C4×C12712yes—
144C22×C42C22×C12711yescharacteristic
145C4×M16—24no—
146C4×M16C4×C12718yescomplemented
147C2×#97—22no—
148C2×C4×C8—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C2×#95C4×C12712yes—
152#343C4×C127116yesdirect product factor
153C23×C4C2×C4×C12711yescharacteristic
154C23×C4C2×C4×C12714yes—
155C23×C4C2×C4×C12711yescharacteristic
156C23×C4C23×C12711yescharacteristic
157C2×M16—24no—
158C2×M16—28no—
159C2×M16C2×C4×C127116yes—
160C2×M16C2×C4×C12718yes—
161C22×C8—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C2×C4×C12718yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—44no—
168C2×C42C2×C4×C12714yes—
169C2×C42D4×C12712yes—
170C2×C42D4×C12712yes—
171C2×C42D4×C12712yes—
172C2×C42Q8×C12712yes—
173#97—28no—
174C4×C8—28no—
175#95—24no—
176C24C22×C4×C12711yescharacteristic
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4C42×C12712yes—
185C22×C4C42×C12718yes—
186C22×C4C42×C12712yes—
187C22×C4C22⋊C4×C12712yes—
188C22×C4C22⋊C4×C12711yescharacteristic
189C22×C4C22⋊C4×C12711yescharacteristic
190C22×C4C4⋊C4×C12712yes—
191C22×C4C4⋊C4×C12712yes—
192C22×C4C22×C4×C12712yes—
193C22×C4C22×C4×C12714yes—
194C22×C4C2×D4×C12711yescharacteristic
195C22×C4C2×D4×C12712yes—
196C22×C4C2×Q8×C12711yescharacteristic
197C22×C4#41×C12714yes—
198M16—28no—
199M16—216no—
200M16C42×C127116yescomplemented
201C2×C8—28no—
202C2×C8—216no—
203C4⋊C4—216no—
204C42—44no—
205C42—44no—
206C42C22⋊C4×C12714yes—
207C42C22⋊C4×C12714yes—
208C42C4⋊C4×C12714yes—
209C42C4⋊C4×C12714yes—
210C23—21no—
211C23—21no—
212C23—22no—
213C23C2×C42×C12711yescharacteristic
214C23C2×C42×C12712yes—
215C23C2×C22⋊C4×C12711yescharacteristic
216C23C2×C4⋊C4×C12711yescharacteristic
217C23#117×C12712yes—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—28no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—48no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—48no—
230C2×C4#95×C12712yes—
231C2×C4#95×C12714yes—
232C2×C4#95×C12712yes—
233C2×C4#100×C12712yes—
234C2×C4#101×C12712yes—
235C2×C4C2×C42×C12714yes—
236C2×C4C2×C22⋊C4×C12712yes—
237C2×C4C2×C4⋊C4×C12712yes—
238C2×C4#117×C12714yes—
239C2×C4C4×D4×C12714yes—
240C2×C4C4×D4×C12718yes—
241C2×C4C4×Q8×C12714yes—
242C8—216no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22C43×C12711yescharacteristic
252C22C2×#95×C12711yescharacteristic
253C22#376×C12711yescharacteristic
254C22C4×C22⋊C4×C12712yes—
255C22C4×C4⋊C4×C12712yes—
256C22C2×#100×C12711yescharacteristic
257C22C2×#101×C12711yescharacteristic
258C22#413×C12712yes—
259C4—22no—
260C4—24no—
261C4—48no—
262C4#343×C12714yesdirect product factor
263C4C4×C22⋊C4×C12714yes—
264C4C4×C4⋊C4×C12714yes—
265C2—21no—
266C2—21no—
267C2—22no—
268C2C4×#95×C12711yescharacteristic
269C2C2×#343×C12711yescharacteristic
270C2C127×#174211yescharacteristic
271C2C4×#100×C12712yes—
272C2C4×#101×C12712yes—
subgroupG/H
centerC22×C4×C127C22⋊C4
commutatorC22C43×C127
Frattini, cosocleC23×C4C23×C127
socleC23×C127C2×C22⋊C4
prime radicalC24×C127C22×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
9. C4×M16×C127C4semidirect
15. #343×C127C4direct (N × H)
63. M16×C127C42semidirect
125. C4×C127#343direct (N × H)
128. C4×#343C127direct (N × H)
140. C127C4×#343direct (N × H)
146. C4×M16C4×C127semidirect
152. #343C4×C127direct (N × H)
200. M16C42×C127semidirect
262. C4#343×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×M16×C1272
C2×C4×C4⋊C4×C1272
C2×#343×C1272
C4×#343127