Finite Group Databasebeta

← C22×#324×C127 #44581017

Subgroups — C22×#324×C127 #44581017

Order 32512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C127C211yesmaximal, characteristic
2C22×C22⋊C8×C127C212yesmaximal
3C2×#324×C127C2112yesmaximal, direct product factor
4C24×C4×C127C411yescharacteristic
5C24×C4×C127C411yescharacteristic
6C24×C4×C127C2211yescharacteristic
7C23×C8×C127—22no—
8C22×C4⋊C4×C127—26no—
9C22×C4⋊C4×C127—22no—
10C22×C4⋊C4×C127C416yes—
11C22×C4⋊C4×C127C2216yes—
12C2×C22⋊C8×C127—26no—
13C2×C22⋊C8×C127C22112yes—
14#324×C127C22116yesdirect product factor
15C25×C127C2×C411yescharacteristic
16C23×C4×C127—23no—
17C23×C4×C127—21no—
18C23×C4×C127—24no—
19C23×C4×C127—23no—
20C23×C4×C127—21no—
21C23×C4×C127—44no—
22C23×C4×C127—44no—
23C23×C4×C127C818yescomplemented
24C23×C4×C127C2×C416yes—
25C23×C4×C127C2×C416yes—
26C23×C4×C127C2×C413yes—
27C23×C4×C127D412yes—
28C23×C4×C127D416yes—
29C23×C4×C127C2313yes—
30C22×C8×C127—212no—
31C22×C8×C127—48no—
32C2×C4⋊C4×C127—212no—
33C2×C4⋊C4×C127—212no—
34C2×C4⋊C4×C127—212no—
35C2×C4⋊C4×C127—212no—
36C2×C4⋊C4×C127C2×C4112yes—
37C2×C4⋊C4×C127C2314yes—
38C22⋊C8×C127—212no—
39C22⋊C8×C127C2318yes—
40C24×C127—24no—
41C24×C127—24no—
42C24×C127C22⋊C413yes—
43C24×C127C22⋊C411yescharacteristic
44C24×C127C2×C814yes—
45C24×C127M1614yes—
46C24×C127C22×C413yes—
47C22×C4×C127—224no—
48C22×C4×C127—28no—
49C22×C4×C127—28no—
50C22×C4×C127—26no—
51C22×C4×C127—26no—
52C22×C4×C127—212no—
53C22×C4×C127—26no—
54C22×C4×C127—26no—
55C22×C4×C127—46no—
56C22×C4×C127—412no—
57C22×C4×C127—46no—
58C22×C4×C127—412no—
59C22×C4×C127—44no—
60C22×C4×C127—412no—
61C22×C4×C127C22⋊C416yes—
62C22×C4×C127C22⋊C416yes—
63C22×C4×C127C2×C8124yescomplemented
64C22×C4×C127C22×C414yes—
65C22×C4×C127C22×C414yes—
66C22×C4×C127C22×C413yes—
67C22×C4×C127C2×D416yes—
68C22×C4×C127C2×D416yes—
69C22×C4×C127C2411yescharacteristic
70C2×C8×C127—28no—
71C2×C8×C127—424no—
72C4⋊C4×C127—224no—
73C4⋊C4×C127—28no—
74C23×C127—216no—
75C23×C127—216no—
76C23×C127—26no—
77C23×C127—26no—
78C23×C127—26no—
79C23×C127—26no—
80C23×C127C22⋊C8112yes—
81C23×C127C22⋊C814yes—
82C23×C127#9914yes—
83C23×C127#10114yes—
84C23×C127C2×C22⋊C413yes—
85C23×C127C2×C22⋊C413yes—
86C23×C127C22×C816yes—
87C23×C127C2×M1616yes—
88C23×C127C23×C411yescharacteristic
89C2×C4×C127—24no—
90C2×C4×C127—24no—
91C2×C4×C127—224no—
92C2×C4×C127—224no—
93C2×C4×C127—212no—
94C2×C4×C127—212no—
95C2×C4×C127—24no—
96C2×C4×C127—46no—
97C2×C4×C127—43no—
98C2×C4×C127—46no—
99C2×C4×C127—41no—
100C2×C4×C127—412no—
101C2×C4×C127—412no—
102C2×C4×C127—44no—
103C2×C4×C127C2×C22⋊C416yes—
104C2×C4×C127C22×C818yescomplemented
105C2×C4×C127C22×D412yes—
106C8×C127—48no—
107C22×C127—23no—
108C22×C127—21no—
109C22×C127—23no—
110C22×C127—21no—
111C22×C127—224no—
112C22×C127—212no—
113C22×C127—212no—
114C22×C127#324116yesdirect product factor
115C22×C127C2×C22⋊C816yes—
116C22×C127C2×C22⋊C816yes—
117C22×C127C2×#9916yes—
118C22×C127C2×#10116yes—
119C22×C127C22×C22⋊C411yescharacteristic
120C22×C127C23×C811yescharacteristic
121C22×C127C22×M1611yescharacteristic
122C4×C127—28no—
123C4×C127—22no—
124C4×C127—26no—
125C4×C127—44no—
126C22×#324C12711yesmaximal, characteristic, direct product factor, Sylow
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—26no—
130C2×C127C2×#324112yesdirect product factor
131C2×C127C22×C22⋊C811yescharacteristic
132C2×C127C22×#9911yescharacteristic
133C2×C127C22×#10111yescharacteristic
134C23×C4⋊C4C2×C12711yescharacteristic
135C22×C22⋊C8C2×C12712yes—
136C2×#324C2×C127112yesdirect product factor
137C127C22×#32411yescharacteristic, direct product factor, Sylow
138C24×C4C4×C12711yescharacteristic
139C24×C4C4×C12711yescharacteristic
140C24×C4C22×C12711yescharacteristic
141C23×C8—22no—
142C22×C4⋊C4—22no—
143C22×C4⋊C4—26no—
144C22×C4⋊C4C4×C12716yes—
145C22×C4⋊C4C22×C12716yes—
146C2×C22⋊C8—26no—
147C2×C22⋊C8C22×C127112yes—
148#324C22×C127116yesdirect product factor
149C25C2×C4×C12711yescharacteristic
150C23×C4—23no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—23no—
154C23×C4—21no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4C8×C12718yescomplemented
158C23×C4C2×C4×C12716yes—
159C23×C4C2×C4×C12716yes—
160C23×C4C2×C4×C12713yes—
161C23×C4D4×C12712yes—
162C23×C4D4×C12716yes—
163C23×C4C23×C12713yes—
164C22×C8—212no—
165C22×C8—48no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—212no—
168C2×C4⋊C4—212no—
169C2×C4⋊C4—212no—
170C2×C4⋊C4C2×C4×C127112yes—
171C2×C4⋊C4C23×C12714yes—
172C22⋊C8—212no—
173C22⋊C8C23×C12718yes—
174C24—24no—
175C24—24no—
176C24C22⋊C4×C12713yes—
177C24C22⋊C4×C12711yescharacteristic
178C24C2×C8×C12714yes—
179C24M16×C12714yes—
180C24C22×C4×C12713yes—
181C22×C4—224no—
182C22×C4—28no—
183C22×C4—28no—
184C22×C4—26no—
185C22×C4—26no—
186C22×C4—212no—
187C22×C4—26no—
188C22×C4—26no—
189C22×C4—46no—
190C22×C4—412no—
191C22×C4—46no—
192C22×C4—412no—
193C22×C4—44no—
194C22×C4—412no—
195C22×C4C22⋊C4×C12716yes—
196C22×C4C22⋊C4×C12716yes—
197C22×C4C2×C8×C127124yescomplemented
198C22×C4C22×C4×C12714yes—
199C22×C4C22×C4×C12714yes—
200C22×C4C22×C4×C12713yes—
201C22×C4C2×D4×C12716yes—
202C22×C4C2×D4×C12716yes—
203C22×C4C24×C12711yescharacteristic
204C2×C8—28no—
205C2×C8—424no—
206C4⋊C4—224no—
207C4⋊C4—28no—
208C23—216no—
209C23—216no—
210C23—26no—
211C23—26no—
212C23—26no—
213C23—26no—
214C23C22⋊C8×C127112yes—
215C23C22⋊C8×C12714yes—
216C23#99×C12714yes—
217C23#101×C12714yes—
218C23C2×C22⋊C4×C12713yes—
219C23C2×C22⋊C4×C12713yes—
220C23C22×C8×C12716yes—
221C23C2×M16×C12716yes—
222C23C23×C4×C12711yescharacteristic
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—224no—
226C2×C4—224no—
227C2×C4—212no—
228C2×C4—212no—
229C2×C4—24no—
230C2×C4—46no—
231C2×C4—43no—
232C2×C4—46no—
233C2×C4—41no—
234C2×C4—412no—
235C2×C4—412no—
236C2×C4—44no—
237C2×C4C2×C22⋊C4×C12716yes—
238C2×C4C22×C8×C12718yescomplemented
239C2×C4C22×D4×C12712yes—
240C8—48no—
241C22—23no—
242C22—21no—
243C22—23no—
244C22—21no—
245C22—224no—
246C22—212no—
247C22—212no—
248C22#324×C127116yesdirect product factor
249C22C2×C22⋊C8×C12716yes—
250C22C2×C22⋊C8×C12716yes—
251C22C2×#99×C12716yes—
252C22C2×#101×C12716yes—
253C22C22×C22⋊C4×C12711yescharacteristic
254C22C23×C8×C12711yescharacteristic
255C22C22×M16×C12711yescharacteristic
256C4—28no—
257C4—22no—
258C4—26no—
259C4—44no—
260C2—21no—
261C2—21no—
262C2—26no—
263C2C2×#324×C127112yesdirect product factor
264C2C22×C22⋊C8×C12711yescharacteristic
265C2C22×#99×C12711yescharacteristic
266C2C22×#101×C12711yescharacteristic
subgroupG/H
center, socleC24×C127C22⋊C4
commutatorC22C23×C8×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC25×C127C2×C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C24×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#324×C127C2direct (N × H)
14. #324×C127C22direct (N × H)
23. C23×C4×C127C8semidirect
63. C22×C4×C127C2×C8semidirect
104. C2×C4×C127C22×C8semidirect
114. C22×C127#324direct (N × H)
126. C22×#324C127direct (N × H)
130. C2×C127C2×#324direct (N × H)
136. C2×#324C2×C127direct (N × H)
137. C127C22×#324direct (N × H)
148. #324C22×C127direct (N × H)
157. C23×C4C8×C127semidirect
197. C22×C4C2×C8×C127semidirect
238. C2×C4C22×C8×C127semidirect
248. C22#324×C127direct (N × H)
263. C2C2×#324×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C1272
C22×C22⋊C8×C1272
C2×#324×C1272
C22×#324127