Finite Group Databasebeta

← C127×#20335 #44584008

Subgroups — C127×#20335 #44584008

Order 32512.

2206 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C127C212yesmaximal
2C22×#117×C127C211yesmaximal, characteristic
3C127×#1729C218yesmaximal, complemented
4C127×#1728C214yesmaximal, complemented
5C24×C4×C127C2211yescharacteristic
6C22×M16×C127—28no—
7C22×M16×C127—28no—
8C22×M16×C127C4116yescomplemented
9C22×M16×C127C2214yes—
10C22×M16×C127C2214yes—
11C23×C8×C127C2214yes—
12C2×#117×C127—28no—
13C2×#117×C127C414yes—
14C2×#117×C127C2214yes—
15C22×C4⋊C4×C127C412yes—
16C22×C22⋊C4×C127C412yes—
17C22×C42×C127C2212yes—
18#336×C127C22116yescomplemented
19C25×C127C2×C411yescharacteristic
20C23×C4×C127C2×C414yes—
21C23×C4×C127C2×C411yescharacteristic
22C23×C4×C127C2×C412yes—
23C23×C4×C127C2×C418yes—
24C23×C4×C127D414yes—
25C23×C4×C127D418yes—
26C23×C4×C127Q814yes—
27C23×C4×C127C2311yescharacteristic
28C23×C4×C127C2312yes—
29C2×M16×C127—264no—
30C2×M16×C127—28no—
31C2×M16×C127—28no—
32C2×M16×C127—28no—
33C2×M16×C127—28no—
34C2×M16×C127C2×C4116yes—
35C2×M16×C127C2×C4116yescomplemented
36C22×C8×C127—28no—
37C22×C8×C127—28no—
38C22×C8×C127C2×C4116yescomplemented
39C22×C8×C127C2318yes—
40#117×C127—216no—
41#117×C127—28no—
42#117×C127—28no—
43C2×C4⋊C4×C127—28no—
44C2×C4⋊C4×C127C2×C418yes—
45C2×C22⋊C4×C127—28no—
46C2×C22⋊C4×C127C2×C418yes—
47C2×C42×C127—28no—
48C2×C42×C127C2×C414yes—
49C2×C42×C127C2314yes—
50C24×C127—44no—
51C24×C127C4214yes—
52C24×C127C22⋊C414yes—
53C24×C127C4⋊C414yes—
54C24×C127C22×C412yes—
55C24×C127C22×C411yescharacteristic
56C22×C4×C127—28no—
57C22×C4×C127—48no—
58C22×C4×C127C4214yes—
59C22×C4×C127C4218yes—
60C22×C4×C127C22⋊C414yes—
61C22×C4×C127C22⋊C418yes—
62C22×C4×C127C22⋊C4132yes—
63C22×C4×C127C4⋊C414yes—
64C22×C4×C127C4⋊C418yes—
65C22×C4×C127C4⋊C4132yes—
66C22×C4×C127C22×C418yes—
67C22×C4×C127C22×C411yescharacteristic
68C22×C4×C127C22×C412yes—
69C22×C4×C127C22×C414yes—
70C22×C4×C127C22×C414yes—
71C22×C4×C127C22×C414yes—
72C22×C4×C127C2×D414yes—
73C22×C4×C127C2×D412yes—
74C22×C4×C127C2×D414yes—
75C22×C4×C127C2×D414yes—
76C22×C4×C127C2×D414yes—
77C22×C4×C127C2×Q812yes—
78C22×C4×C127C2×Q814yes—
79C22×C4×C127C2411yescharacteristic
80M16×C127—232no—
81M16×C127—232no—
82C2×C8×C127—232no—
83C2×C8×C127—28no—
84C2×C8×C127—28no—
85C2×C8×C127C22×C4116yes—
86C4⋊C4×C127—216no—
87C22⋊C4×C127—216no—
88C42×C127—28no—
89C42×C127—28no—
90C23×C127—24no—
91C23×C127—28no—
92C23×C127—48no—
93C23×C127—48no—
94C23×C127—48no—
95C23×C127#95116yes—
96C23×C127C2×C4214yes—
97C23×C127C2×C4212yes—
98C23×C127C2×C22⋊C414yes—
99C23×C127C2×C22⋊C412yes—
100C23×C127C2×C4⋊C414yes—
101C23×C127C2×C4⋊C412yes—
102C23×C127C23×C411yescharacteristic
103C2×C4×C127—28no—
104C2×C4×C127—216no—
105C2×C4×C127—416no—
106C2×C4×C127#95116yes—
107C2×C4×C127#95132yes—
108C2×C4×C127C2×C4212yes—
109C2×C4×C127C2×C4214yes—
110C2×C4×C127C2×C22⋊C418yes—
111C2×C4×C127C2×C22⋊C418yes—
112C2×C4×C127C2×C22⋊C418yes—
113C2×C4×C127C2×C22⋊C412yes—
114C2×C4×C127C2×C22⋊C414yes—
115C2×C4×C127C2×C4⋊C418yes—
116C2×C4×C127C2×C4⋊C418yes—
117C2×C4×C127C2×C4⋊C418yes—
118C2×C4×C127C2×C4⋊C412yes—
119C2×C4×C127C2×C4⋊C414yes—
120C2×C4×C127C23×C412yes—
121C2×C4×C127C22×D411yescharacteristic
122C2×C4×C127C22×D412yes—
123C2×C4×C127C22×Q811yescharacteristic
124C8×C127—216no—
125C22×C127—24no—
126C22×C127—28no—
127C22×C127—24no—
128C22×C127—28no—
129C22×C127—24no—
130C22×C127—28no—
131C22×C127—416no—
132C22×C127C2×#9518yes—
133C22×C127C2×#9514yes—
134C22×C127C22×C4211yescharacteristic
135C22×C127C22×C22⋊C411yescharacteristic
136C22×C127C22×C4⋊C411yescharacteristic
137C22×C127#56814yes—
138C4×C127—48no—
139C4×C127C2×#9518yes—
140C4×C127C2×#9514yes—
141C4×C127C22×C22⋊C412yes—
142C4×C127C22×C4⋊C412yes—
143#20335C12711yesmaximal, characteristic, direct product factor, Sylow
144C2×C127—24no—
145C2×C127—28no—
146C2×C127C22×#9511yescharacteristic
147C2×C127#287312yes—
148C2×C127#287912yes—
149C2×C127#290512yes—
150C23×M16C2×C12712yes—
151C22×#117C2×C12711yescharacteristic
152#1729C2×C12718yescomplemented
153#1728C2×C12714yescomplemented
154C127#2033511yescharacteristic, direct product factor, Sylow
155C24×C4C22×C12711yescharacteristic
156C22×M16—28no—
157C22×M16—28no—
158C22×M16C4×C127116yescomplemented
159C22×M16C22×C12714yes—
160C22×M16C22×C12714yes—
161C23×C8C22×C12714yes—
162C2×#117—28no—
163C2×#117C4×C12714yes—
164C2×#117C22×C12714yes—
165C22×C4⋊C4C4×C12712yes—
166C22×C22⋊C4C4×C12712yes—
167C22×C42C22×C12712yes—
168#336C22×C127116yescomplemented
169C25C2×C4×C12711yescharacteristic
170C23×C4C2×C4×C12714yes—
171C23×C4C2×C4×C12711yescharacteristic
172C23×C4C2×C4×C12712yes—
173C23×C4C2×C4×C12718yes—
174C23×C4D4×C12714yes—
175C23×C4D4×C12718yes—
176C23×C4Q8×C12714yes—
177C23×C4C23×C12712yes—
178C23×C4C23×C12711yescharacteristic
179C2×M16—264no—
180C2×M16—28no—
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C2×M16C2×C4×C127116yes—
185C2×M16C2×C4×C127116yescomplemented
186C22×C8—28no—
187C22×C8—28no—
188C22×C8C2×C4×C127116yescomplemented
189C22×C8C23×C12718yes—
190#117—216no—
191#117—28no—
192#117—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4C2×C4×C12718yes—
195C2×C22⋊C4—28no—
196C2×C22⋊C4C2×C4×C12718yes—
197C2×C42—28no—
198C2×C42C2×C4×C12714yes—
199C2×C42C23×C12714yes—
200C24—44no—
201C24C42×C12714yes—
202C24C22⋊C4×C12714yes—
203C24C4⋊C4×C12714yes—
204C24C22×C4×C12712yes—
205C24C22×C4×C12711yescharacteristic
206C22×C4—28no—
207C22×C4—48no—
208C22×C4C42×C12714yes—
209C22×C4C42×C12718yes—
210C22×C4C22⋊C4×C12714yes—
211C22×C4C22⋊C4×C12718yes—
212C22×C4C22⋊C4×C127132yes—
213C22×C4C4⋊C4×C12714yes—
214C22×C4C4⋊C4×C12718yes—
215C22×C4C4⋊C4×C127132yes—
216C22×C4C22×C4×C12718yes—
217C22×C4C22×C4×C12711yescharacteristic
218C22×C4C22×C4×C12712yes—
219C22×C4C22×C4×C12714yes—
220C22×C4C22×C4×C12714yes—
221C22×C4C22×C4×C12714yes—
222C22×C4C2×D4×C12714yes—
223C22×C4C2×D4×C12712yes—
224C22×C4C2×D4×C12714yes—
225C22×C4C2×D4×C12714yes—
226C22×C4C2×D4×C12714yes—
227C22×C4C2×Q8×C12714yes—
228C22×C4C2×Q8×C12712yes—
229C22×C4C24×C12711yescharacteristic
230M16—232no—
231M16—232no—
232C2×C8—232no—
233C2×C8—28no—
234C2×C8—28no—
235C2×C8C22×C4×C127116yes—
236C4⋊C4—216no—
237C22⋊C4—216no—
238C42—28no—
239C42—28no—
240C23—24no—
241C23—28no—
242C23—48no—
243C23—48no—
244C23—48no—
245C23#95×C127116yes—
246C23C2×C42×C12714yes—
247C23C2×C42×C12712yes—
248C23C2×C22⋊C4×C12714yes—
249C23C2×C22⋊C4×C12712yes—
250C23C2×C4⋊C4×C12714yes—
251C23C2×C4⋊C4×C12712yes—
252C23C23×C4×C12711yescharacteristic
253C2×C4—28no—
254C2×C4—216no—
255C2×C4—416no—
256C2×C4#95×C127116yes—
257C2×C4#95×C127132yes—
258C2×C4C2×C42×C12712yes—
259C2×C4C2×C42×C12714yes—
260C2×C4C2×C22⋊C4×C12718yes—
261C2×C4C2×C22⋊C4×C12718yes—
262C2×C4C2×C22⋊C4×C12718yes—
263C2×C4C2×C22⋊C4×C12712yes—
264C2×C4C2×C22⋊C4×C12714yes—
265C2×C4C2×C4⋊C4×C12718yes—
266C2×C4C2×C4⋊C4×C12718yes—
267C2×C4C2×C4⋊C4×C12718yes—
268C2×C4C2×C4⋊C4×C12712yes—
269C2×C4C2×C4⋊C4×C12714yes—
270C2×C4C23×C4×C12712yes—
271C2×C4C22×D4×C12711yescharacteristic
272C2×C4C22×D4×C12712yes—
273C2×C4C22×Q8×C12711yescharacteristic
274C8—216no—
275C22—24no—
276C22—28no—
277C22—24no—
278C22—28no—
279C22—24no—
280C22—28no—
281C22—416no—
282C22C2×#95×C12718yes—
283C22C2×#95×C12714yes—
284C22C22×C42×C12711yescharacteristic
285C22C22×C22⋊C4×C12711yescharacteristic
286C22C22×C4⋊C4×C12711yescharacteristic
287C22#568×C12714yes—
288C4—48no—
289C4C2×#95×C12718yes—
290C4C2×#95×C12714yes—
291C4C22×C22⋊C4×C12712yes—
292C4C22×C4⋊C4×C12712yes—
293C2—24no—
294C2—28no—
295C2C22×#95×C12711yescharacteristic
296C2C127×#287312yes—
297C2C127×#287912yes—
298C2C127×#290512yes—
subgroupG/H
centerC22×C4×C127C24
commutatorC22C22×C42×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C23×C4
prime radicalC25×C127C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1729C2semidirect
4. C127×#1728C2semidirect
8. C22×M16×C127C4semidirect
18. #336×C127C22semidirect
35. C2×M16×C127C2×C4semidirect
38. C22×C8×C127C2×C4semidirect
143. #20335C127direct (N × H)
152. #1729C2×C127semidirect
153. #1728C2×C127semidirect
154. C127#20335direct (N × H)
158. C22×M16C4×C127semidirect
168. #336C22×C127semidirect
185. C2×M16C2×C4×C127semidirect
188. C22×C8C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C23×M16×C1272
C22×#117×C1272
C127×#17292
C127×#17282
#20335127