Finite Group Databasebeta

← C127×#20701 #44584374

Subgroups — C127×#20701 #44584374

Order 32512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C127C211yesmaximal, characteristic
2C2×#407×C127C211yesmaximal, characteristic
3C127×#2280C211yesmaximal, characteristic
4C127×#1825C218yesmaximal, complemented
5C127×#1736C214yesmaximal, complemented
6C24×C4×C127C2211yescharacteristic
7C22×M16×C127C2212yes—
8C2×#117×C127—24no—
9C2×#117×C127—22no—
10C2×#117×C127C418yescomplemented
11C2×#117×C127C2212yes—
12C2×#117×C127C2214yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C22⋊C4×C127—21no—
15C22×C42×C127—21no—
16#407×C127—24no—
17C2×C22⋊C8×C127C2214yes—
18#381×C127—21no—
19#380×C127C2214yes—
20#379×C127—21no—
21C4×C22⋊C4×C127—22no—
22#340×C127C22116yescomplemented
23C25×C127D411yescharacteristic
24C23×C4×C127—28no—
25C23×C4×C127—22no—
26C23×C4×C127C2×C414yes—
27C23×C4×C127D411yescharacteristic
28C23×C4×C127D412yes—
29C23×C4×C127D414yes—
30C23×C4×C127C2312yes—
31C23×C4×C127C2311yescharacteristic
32C2×M16×C127—24no—
33C2×M16×C127—28no—
34C22×C8×C127C2314yes—
35#117×C127—216no—
36#117×C127—28no—
37#117×C127—28no—
38#117×C127C2×C4116yes—
39C2×C4⋊C4×C127—22no—
40C2×C4⋊C4×C127—24no—
41C2×C4⋊C4×C127C2×C418yes—
42C2×C4⋊C4×C127C2314yes—
43C2×C4⋊C4×C127C2314yes—
44C2×C22⋊C4×C127—28no—
45C2×C22⋊C4×C127—24no—
46C2×C22⋊C4×C127—22no—
47C2×C22⋊C4×C127—24no—
48C2×C42×C127—28no—
49C2×C42×C127—24no—
50C2×C42×C127—22no—
51C22⋊C8×C127—28no—
52#95×C127—24no—
53C24×C127—24no—
54C24×C127—44no—
55C24×C127C22⋊C414yes—
56C24×C127C2×D412yes—
57C24×C127C2×D411yescharacteristic
58C22×C4×C127—24no—
59C22×C4×C127—216no—
60C22×C4×C127—28no—
61C22×C4×C127—216no—
62C22×C4×C127—28no—
63C22×C4×C127—28no—
64C22×C4×C127—24no—
65C22×C4×C127—24no—
66C22×C4×C127—48no—
67C22×C4×C127C22⋊C414yes—
68C22×C4×C127C22⋊C418yes—
69C22×C4×C127C22×C414yes—
70C22×C4×C127C22×C412yes—
71C22×C4×C127C2×D412yes—
72C22×C4×C127C2×D414yes—
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D412yes—
75C22×C4×C127#4118yes—
76C22×C4×C127C2411yescharacteristic
77M16×C127—48no—
78C2×C8×C127—24no—
79C2×C8×C127—28no—
80C4⋊C4×C127—28no—
81C4⋊C4×C127—24no—
82C4⋊C4×C127—28no—
83C4⋊C4×C127C22×C418yes—
84C22⋊C4×C127—216no—
85C22⋊C4×C127—48no—
86C22⋊C4×C127—44no—
87C42×C127—216no—
88C42×C127—44no—
89C23×C127—24no—
90C23×C127—24no—
91C23×C127—24no—
92C23×C127—24no—
93C23×C127—28no—
94C23×C127—48no—
95C23×C127—416no—
96C23×C127C2×C22⋊C414yes—
97C23×C127C2×C22⋊C412yes—
98C23×C127#12314yes—
99C23×C127C22×D411yescharacteristic
100C2×C4×C127—24no—
101C2×C4×C127—28no—
102C2×C4×C127—28no—
103C2×C4×C127—28no—
104C2×C4×C127—216no—
105C2×C4×C127—28no—
106C2×C4×C127—416no—
107C2×C4×C127—48no—
108C2×C4×C127—44no—
109C2×C4×C127—48no—
110C2×C4×C127C2×C22⋊C414yes—
111C2×C4×C127C2×C22⋊C412yes—
112C2×C4×C127#11718yes—
113C2×C4×C127#12318yes—
114C2×C4×C127#12314yes—
115C2×C4×C127C23×C411yescharacteristic
116C2×C4×C127C22×D411yescharacteristic
117C2×C4×C127C2×#4114yes—
118C8×C127—44no—
119C22×C127—28no—
120C22×C127—24no—
121C22×C127—24no—
122C22×C127—24no—
123C22×C127—24no—
124C22×C127—416no—
125C22×C127—48no—
126C22×C127#38114yes—
127C22×C127C22×C22⋊C411yescharacteristic
128C22×C127C2×#12312yes—
129C22×C127#57514yes—
130C4×C127—24no—
131C4×C127—48no—
132C4×C127—84no—
133C4×C127#38114yes—
134C4×C127C2×#11712yes—
135C4×C127C2×#12312yes—
136#20701C12711yesmaximal, characteristic, direct product factor, Sylow
137C2×C127—24no—
138C2×C127—44no—
139C2×C127C2×#38111yescharacteristic
140C2×C127#289612yes—
141C2×C127#309314yes—
142C22×#117C2×C12711yescharacteristic
143C2×#407C2×C12711yescharacteristic
144#2280C2×C12711yescharacteristic
145#1825C2×C12718yescomplemented
146#1736C2×C12714yescomplemented
147C127#2070111yescharacteristic, direct product factor, Sylow
148C24×C4C22×C12711yescharacteristic
149C22×M16C22×C12712yes—
150C2×#117—24no—
151C2×#117—22no—
152C2×#117C4×C12718yescomplemented
153C2×#117C22×C12712yes—
154C2×#117C22×C12714yes—
155C22×C4⋊C4C22×C12712yes—
156C22×C22⋊C4—21no—
157C22×C42—21no—
158#407—24no—
159C2×C22⋊C8C22×C12714yes—
160#381—21no—
161#380C22×C12714yes—
162#379—21no—
163C4×C22⋊C4—22no—
164#340C22×C127116yescomplemented
165C25D4×C12711yescharacteristic
166C23×C4—28no—
167C23×C4—22no—
168C23×C4C2×C4×C12714yes—
169C23×C4D4×C12712yes—
170C23×C4D4×C12711yescharacteristic
171C23×C4D4×C12714yes—
172C23×C4C23×C12712yes—
173C23×C4C23×C12711yescharacteristic
174C2×M16—24no—
175C2×M16—28no—
176C22×C8C23×C12714yes—
177#117—216no—
178#117—28no—
179#117—28no—
180#117C2×C4×C127116yes—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4C2×C4×C12718yes—
184C2×C4⋊C4C23×C12714yes—
185C2×C4⋊C4C23×C12714yes—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C42—28no—
191C2×C42—24no—
192C2×C42—22no—
193C22⋊C8—28no—
194#95—24no—
195C24—24no—
196C24—44no—
197C24C22⋊C4×C12714yes—
198C24C2×D4×C12712yes—
199C24C2×D4×C12711yescharacteristic
200C22×C4—24no—
201C22×C4—216no—
202C22×C4—28no—
203C22×C4—216no—
204C22×C4—28no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—48no—
209C22×C4C22⋊C4×C12714yes—
210C22×C4C22⋊C4×C12718yes—
211C22×C4C22×C4×C12714yes—
212C22×C4C22×C4×C12712yes—
213C22×C4C2×D4×C12712yes—
214C22×C4C2×D4×C12714yes—
215C22×C4C2×D4×C12711yescharacteristic
216C22×C4C2×D4×C12712yes—
217C22×C4#41×C12718yes—
218C22×C4C24×C12711yescharacteristic
219M16—48no—
220C2×C8—24no—
221C2×C8—28no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—28no—
225C4⋊C4C22×C4×C12718yes—
226C22⋊C4—216no—
227C22⋊C4—48no—
228C22⋊C4—44no—
229C42—216no—
230C42—44no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—28no—
236C23—48no—
237C23—416no—
238C23C2×C22⋊C4×C12714yes—
239C23C2×C22⋊C4×C12712yes—
240C23#123×C12714yes—
241C23C22×D4×C12711yescharacteristic
242C2×C4—24no—
243C2×C4—28no—
244C2×C4—28no—
245C2×C4—28no—
246C2×C4—216no—
247C2×C4—28no—
248C2×C4—416no—
249C2×C4—48no—
250C2×C4—44no—
251C2×C4—48no—
252C2×C4C2×C22⋊C4×C12714yes—
253C2×C4C2×C22⋊C4×C12712yes—
254C2×C4#117×C12718yes—
255C2×C4#123×C12718yes—
256C2×C4#123×C12714yes—
257C2×C4C23×C4×C12711yescharacteristic
258C2×C4C22×D4×C12711yescharacteristic
259C2×C4C2×#41×C12714yes—
260C8—44no—
261C22—28no—
262C22—24no—
263C22—24no—
264C22—24no—
265C22—24no—
266C22—416no—
267C22—48no—
268C22#381×C12714yes—
269C22C22×C22⋊C4×C12711yescharacteristic
270C22C2×#123×C12712yes—
271C22#575×C12714yes—
272C4—24no—
273C4—48no—
274C4—84no—
275C4#381×C12714yes—
276C4C2×#117×C12712yes—
277C4C2×#123×C12712yes—
278C2—24no—
279C2—44no—
280C2C2×#381×C12711yescharacteristic
281C2C127×#289612yes—
282C2C127×#309314yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC25×C127D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1825C2semidirect
5. C127×#1736C2semidirect
10. C2×#117×C127C4semidirect
22. #340×C127C22semidirect
136. #20701C127direct (N × H)
145. #1825C2×C127semidirect
146. #1736C2×C127semidirect
147. C127#20701direct (N × H)
152. C2×#117C4×C127semidirect
164. #340C22×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C1272
C2×#407×C1272
C127×#22802
C127×#18252
C127×#17362
#20701127