Finite Group Databasebeta

← C127×#17059 #44580732

Subgroups — C127×#17059 #44580732

Order 32512.

1530 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×#397×C127C214yesmaximal
2C2×#380×C127C212yesmaximal
3C127×#1445C214yesmaximal, complemented
4C127×#1444C214yesmaximal, complemented
5C127×C22⋊#95C211yesmaximal, characteristic
6C24×C4×C127C2211yescharacteristic
7C22×C4⋊C4×C127C2212yes—
8C22×C4⋊C4×C127C2214yes—
9C22×C4⋊C4×C127C2212yes—
10C22×C22⋊C4×C127C2214yes—
11C22×C22⋊C4×C127C2212yes—
12#397×C127—216no—
13#380×C127—28no—
14C2×#95×C127C2218yes—
15C2×#95×C127C2214yes—
16C2×#95×C127C2214yes—
17C2×#95×C127C2214yes—
18C25×C127C2311yescharacteristic
19C23×C4×C127—28no—
20C23×C4×C127—42no—
21C23×C4×C127D414yes—
22C23×C4×C127C2314yes—
23C23×C4×C127C2314yes—
24C23×C4×C127C2314yes—
25C23×C4×C127C2312yes—
26C2×C4⋊C4×C127—24no—
27C2×C4⋊C4×C127—24no—
28C2×C4⋊C4×C127—28no—
29C2×C4⋊C4×C127—28no—
30C2×C4⋊C4×C127—44no—
31C2×C4⋊C4×C127—44no—
32C2×C4⋊C4×C127—44no—
33C2×C22⋊C4×C127—28no—
34C2×C22⋊C4×C127—216no—
35C2×C22⋊C4×C127—42no—
36C2×C22⋊C4×C127—42no—
37C2×C22⋊C4×C127D418yes—
38#95×C127—28no—
39#95×C127—28no—
40#95×C127—216no—
41#95×C127—28no—
42C24×C127—24no—
43C24×C127—21no—
44C24×C127—22no—
45C24×C127—24no—
46C24×C127—21no—
47C24×C127C2×D414yes—
48C24×C127C2×D412yes—
49C24×C127C2411yescharacteristic
50C22×C4×C127—24no—
51C22×C4×C127—24no—
52C22×C4×C127—28no—
53C22×C4×C127—24no—
54C22×C4×C127—24no—
55C22×C4×C127—28no—
56C22×C4×C127—216no—
57C22×C4×C127—28no—
58C22×C4×C127—24no—
59C22×C4×C127—24no—
60C22×C4×C127—28no—
61C22×C4×C127—28no—
62C22×C4×C127—24no—
63C22×C4×C127—28no—
64C22×C4×C127—28no—
65C22×C4×C127—24no—
66C22×C4×C127—44no—
67C22×C4×C127—416no—
68C22×C4×C127C2×D418yes—
69C22×C4×C127C2×D414yes—
70C22×C4×C127#4118yes—
71C4⋊C4×C127—48no—
72C4⋊C4×C127—416no—
73C4⋊C4×C127—48no—
74C22⋊C4×C127—48no—
75C22⋊C4×C127—416no—
76C23×C127—24no—
77C23×C127—24no—
78C23×C127—22no—
79C23×C127—24no—
80C23×C127—22no—
81C23×C127—24no—
82C23×C127—216no—
83C23×C127—44no—
84C23×C127—44no—
85C23×C127—44no—
86C23×C127—44no—
87C23×C127#12714yes—
88C23×C127C22×D411yescharacteristic
89C23×C127C22×D412yes—
90C23×C127C2×#4114yes—
91C23×C127H(5,𝔽2)11yescharacteristic
92C23×C127H(5,𝔽2)12yes—
93C23×C1272−1+414yes—
94C23×C1272−1+411yescharacteristic
95C2×C4×C127—24no—
96C2×C4×C127—24no—
97C2×C4×C127—216no—
98C2×C4×C127—28no—
99C2×C4×C127—216no—
100C2×C4×C127—416no—
101C2×C4×C127—48no—
102C2×C4×C127—44no—
103C2×C4×C127—44no—
104C2×C4×C127—44no—
105C2×C4×C127—44no—
106C2×C4×C127—44no—
107C2×C4×C127—44no—
108C2×C4×C127—48no—
109C2×C4×C127—44no—
110C2×C4×C127—44no—
111C2×C4×C127—48no—
112C2×C4×C127—416no—
113C2×C4×C127#12318yes—
114C22×C127—24no—
115C22×C127—28no—
116C22×C127—42no—
117C22×C127—44no—
118C22×C127—48no—
119C22×C127—48no—
120C22×C127—42no—
121C22×C127C2×#12312yes—
122C22×C127C2×#12711yescharacteristic
123C22×C127#53411yescharacteristic
124C22×C127#53412yes—
125C22×C127#53612yes—
126C22×C127#53711yescharacteristic
127C22×C127#53712yes—
128C22×C127#54114yes—
129C22×C127#54214yes—
130C22×C127#54718yes—
131C22×C127#55114yes—
132C22×C127#56314yes—
133C4×C127—28no—
134C4×C127—48no—
135C4×C127—84no—
136C4×C127—84no—
137#17059C12711yesmaximal, characteristic, direct product factor, Sylow
138C2×C127—44no—
139C2×C127#248311yescharacteristic
140C2×C127#250112yes—
141C2×C127#262512yes—
142C2×C127#265611yescharacteristic
143C2×C127#271714yes—
144C2×C127#272214yes—
145C2×C127#281711yescharacteristic
146C2×#397C2×C12714yes—
147C2×#380C2×C12712yes—
148#1445C2×C12714yescomplemented
149#1444C2×C12714yescomplemented
150C22⋊#95C2×C12711yescharacteristic
151C127#1705911yescharacteristic, direct product factor, Sylow
152C24×C4C22×C12711yescharacteristic
153C22×C4⋊C4C22×C12712yes—
154C22×C4⋊C4C22×C12714yes—
155C22×C4⋊C4C22×C12712yes—
156C22×C22⋊C4C22×C12714yes—
157C22×C22⋊C4C22×C12712yes—
158#397—216no—
159#380—28no—
160C2×#95C22×C12718yes—
161C2×#95C22×C12714yes—
162C2×#95C22×C12714yes—
163C2×#95C22×C12714yes—
164C25C23×C12711yescharacteristic
165C23×C4—28no—
166C23×C4—42no—
167C23×C4D4×C12714yes—
168C23×C4C23×C12714yes—
169C23×C4C23×C12714yes—
170C23×C4C23×C12714yes—
171C23×C4C23×C12712yes—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—216no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4D4×C12718yes—
184#95—28no—
185#95—28no—
186#95—216no—
187#95—28no—
188C24—24no—
189C24—21no—
190C24—22no—
191C24—24no—
192C24—21no—
193C24C2×D4×C12714yes—
194C24C2×D4×C12712yes—
195C24C24×C12711yescharacteristic
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—216no—
203C22×C4—28no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—44no—
213C22×C4—416no—
214C22×C4C2×D4×C12718yes—
215C22×C4C2×D4×C12714yes—
216C22×C4#41×C12718yes—
217C4⋊C4—48no—
218C4⋊C4—416no—
219C4⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—416no—
222C23—24no—
223C23—24no—
224C23—22no—
225C23—24no—
226C23—22no—
227C23—24no—
228C23—216no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23#127×C12714yes—
234C23C22×D4×C12711yescharacteristic
235C23C22×D4×C12712yes—
236C23C2×#41×C12714yes—
237C23H(5,𝔽2)×C12711yescharacteristic
238C23H(5,𝔽2)×C12712yes—
239C232−1+4×C12714yes—
240C232−1+4×C12711yescharacteristic
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—216no—
244C2×C4—28no—
245C2×C4—216no—
246C2×C4—416no—
247C2×C4—48no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—416no—
259C2×C4#123×C12718yes—
260C22—24no—
261C22—28no—
262C22—44no—
263C22—48no—
264C22—48no—
265C22—42no—
266C22—42no—
267C22C2×#123×C12712yes—
268C22C2×#127×C12711yescharacteristic
269C22#534×C12712yes—
270C22#534×C12711yescharacteristic
271C22#536×C12712yes—
272C22#537×C12711yescharacteristic
273C22#537×C12712yes—
274C22#541×C12714yes—
275C22#542×C12714yes—
276C22#547×C12718yes—
277C22#551×C12714yes—
278C22#563×C12714yes—
279C4—28no—
280C4—48no—
281C4—84no—
282C4—84no—
283C2—44no—
284C2C127×#248311yescharacteristic
285C2C127×#250112yes—
286C2C127×#262512yes—
287C2C127×#265611yescharacteristic
288C2C127×#271714yes—
289C2C127×#272214yes—
290C2C127×#281711yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1445C2semidirect
4. C127×#1444C2semidirect
137. #17059C127direct (N × H)
148. #1445C2×C127semidirect
149. #1444C2×C127semidirect
151. C127#17059direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#397×C1272
C2×#380×C1272
C127×#14452
C127×#14442
C127×C22⋊#952
#17059127