Finite Group Databasebeta

← C127×#16904 #44580577

Subgroups — C127×#16904 #44580577

Order 32512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#1443C214yesmaximal, complemented
2C127×#1441C214yesmaximal, complemented
3C127×#1439C214yesmaximal
4C127×C22⋊#95C212yesmaximal
5C127×C22⋊#95C211yesmaximal, characteristic
6C24×C4×C127C2211yescharacteristic
7C22×C4⋊C4×C127C2214yes—
8C22×C22⋊C4×C127C2214yes—
9C22×C22⋊C4×C127C2212yes—
10C2×#95×C127C2214yes—
11C2×#95×C127C2212yes—
12C2×#95×C127C2214yes—
13C2×#95×C127C2214yes—
14C2×#95×C127C2212yes—
15C2×#95×C127C2214yes—
16C2×#95×C127C2214yes—
17C25×C127C2311yescharacteristic
18C23×C4×C127—42no—
19C23×C4×C127—44no—
20C23×C4×C127Q814yes—
21C23×C4×C127C2314yes—
22C23×C4×C127C2314yes—
23C23×C4×C127C2312yes—
24C23×C4×C127C2314yes—
25C2×C4⋊C4×C127—28no—
26C2×C4⋊C4×C127—44no—
27C2×C4⋊C4×C127—44no—
28C2×C22⋊C4×C127—28no—
29C2×C22⋊C4×C127—24no—
30C2×C22⋊C4×C127—28no—
31C2×C22⋊C4×C127—44no—
32C2×C22⋊C4×C127—42no—
33C2×C22⋊C4×C127—42no—
34#95×C127—28no—
35#95×C127—42no—
36#95×C127—44no—
37#95×C127—44no—
38#95×C127—42no—
39#95×C127—44no—
40#95×C127—44no—
41C24×C127—24no—
42C24×C127—21no—
43C24×C127—22no—
44C24×C127—24no—
45C24×C127—21no—
46C24×C127C2×Q812yes—
47C24×C127#4114yes—
48C24×C127C2411yescharacteristic
49C22×C4×C127—24no—
50C22×C4×C127—24no—
51C22×C4×C127—24no—
52C22×C4×C127—24no—
53C22×C4×C127—22no—
54C22×C4×C127—22no—
55C22×C4×C127—24no—
56C22×C4×C127—24no—
57C22×C4×C127—24no—
58C22×C4×C127—24no—
59C22×C4×C127—24no—
60C22×C4×C127—24no—
61C22×C4×C127—24no—
62C22×C4×C127—24no—
63C22×C4×C127—24no—
64C22×C4×C127—24no—
65C22×C4×C127—24no—
66C22×C4×C127—24no—
67C22×C4×C127—24no—
68C22×C4×C127—24no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—24no—
72C22×C4×C127—24no—
73C22×C4×C127—48no—
74C22×C4×C127—44no—
75C22×C4×C127—416no—
76C22×C4×C127C2×Q814yes—
77C22×C4×C127#4118yes—
78C4⋊C4×C127—416no—
79C22⋊C4×C127—416no—
80C22⋊C4×C127—48no—
81C23×C127—24no—
82C23×C127—24no—
83C23×C127—24no—
84C23×C127—24no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—28no—
88C23×C127—28no—
89C23×C127—44no—
90C23×C127—44no—
91C23×C127—44no—
92C23×C127—44no—
93C23×C127#12514yes—
94C23×C127C22×Q811yescharacteristic
95C23×C127C2×#4112yes—
96C23×C127C2×#4114yes—
97C23×C127H(5,𝔽2)11yescharacteristic
98C23×C127H(5,𝔽2)12yes—
99C23×C127H(5,𝔽2)14yes—
100C23×C1272−1+411yescharacteristic
101C2×C4×C127—28no—
102C2×C4×C127—24no—
103C2×C4×C127—24no—
104C2×C4×C127—24no—
105C2×C4×C127—28no—
106C2×C4×C127—28no—
107C2×C4×C127—416no—
108C2×C4×C127—44no—
109C2×C4×C127—44no—
110C2×C4×C127—44no—
111C2×C4×C127—44no—
112C2×C4×C127—44no—
113C2×C4×C127—44no—
114C2×C4×C127—44no—
115C2×C4×C127—44no—
116C2×C4×C127—44no—
117C2×C4×C127—44no—
118C2×C4×C127—44no—
119C2×C4×C127—44no—
120C2×C4×C127—44no—
121C2×C4×C127—44no—
122C2×C4×C127—44no—
123C2×C4×C127—44no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—44no—
127C2×C4×C127—44no—
128C22×C127—28no—
129C22×C127—24no—
130C22×C127—42no—
131C22×C127—42no—
132C22×C127—44no—
133C22×C127—48no—
134C22×C127—48no—
135C22×C127C2×#12511yescharacteristic
136C22×C127#53312yes—
137C22×C127#53812yes—
138C22×C127#53814yes—
139C22×C127#53912yes—
140C22×C127#54214yes—
141C22×C127#54312yes—
142C22×C127#54311yescharacteristic
143C22×C127#54411yescharacteristic
144C22×C127#55114yes—
145C22×C127#55314yes—
146C22×C127#55314yes—
147C22×C127#56014yes—
148C4×C127—44no—
149C4×C127—48no—
150C4×C127—84no—
151C4×C127—84no—
152#16904C12711yesmaximal, characteristic, direct product factor, Sylow
153C2×C127—44no—
154C2×C127#256912yes—
155C2×C127#260811yescharacteristic
156C2×C127#264611yescharacteristic
157C2×C127#266512yes—
158C2×C127#275714yes—
159C2×C127#277814yes—
160C2×C127#284211yescharacteristic
161#1443C2×C12714yescomplemented
162#1441C2×C12714yescomplemented
163#1439C2×C12714yes—
164C22⋊#95C2×C12712yes—
165C22⋊#95C2×C12711yescharacteristic
166C127#1690411yescharacteristic, direct product factor, Sylow
167C24×C4C22×C12711yescharacteristic
168C22×C4⋊C4C22×C12714yes—
169C22×C22⋊C4C22×C12714yes—
170C22×C22⋊C4C22×C12712yes—
171C2×#95C22×C12714yes—
172C2×#95C22×C12712yes—
173C2×#95C22×C12714yes—
174C2×#95C22×C12714yes—
175C2×#95C22×C12712yes—
176C2×#95C22×C12714yes—
177C2×#95C22×C12714yes—
178C25C23×C12711yescharacteristic
179C23×C4—42no—
180C23×C4—44no—
181C23×C4Q8×C12714yes—
182C23×C4C23×C12714yes—
183C23×C4C23×C12714yes—
184C23×C4C23×C12712yes—
185C23×C4C23×C12714yes—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195#95—28no—
196#95—42no—
197#95—44no—
198#95—44no—
199#95—42no—
200#95—44no—
201#95—44no—
202C24—24no—
203C24—21no—
204C24—22no—
205C24—24no—
206C24—21no—
207C24C2×Q8×C12712yes—
208C24#41×C12714yes—
209C24C24×C12711yescharacteristic
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—416no—
237C22×C4C2×Q8×C12714yes—
238C22×C4#41×C12718yes—
239C4⋊C4—416no—
240C22⋊C4—416no—
241C22⋊C4—48no—
242C23—24no—
243C23—24no—
244C23—24no—
245C23—24no—
246C23—22no—
247C23—22no—
248C23—28no—
249C23—28no—
250C23—44no—
251C23—44no—
252C23—44no—
253C23—44no—
254C23#125×C12714yes—
255C23C22×Q8×C12711yescharacteristic
256C23C2×#41×C12712yes—
257C23C2×#41×C12714yes—
258C23H(5,𝔽2)×C12711yescharacteristic
259C23H(5,𝔽2)×C12712yes—
260C23H(5,𝔽2)×C12714yes—
261C232−1+4×C12711yescharacteristic
262C2×C4—28no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—416no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C22—28no—
290C22—24no—
291C22—42no—
292C22—42no—
293C22—44no—
294C22—48no—
295C22—48no—
296C22C2×#125×C12711yescharacteristic
297C22#533×C12712yes—
298C22#538×C12712yes—
299C22#538×C12714yes—
300C22#539×C12712yes—
301C22#542×C12714yes—
302C22#543×C12712yes—
303C22#543×C12711yescharacteristic
304C22#544×C12711yescharacteristic
305C22#551×C12714yes—
306C22#553×C12714yes—
307C22#553×C12714yes—
308C22#560×C12714yes—
309C4—44no—
310C4—48no—
311C4—84no—
312C4—84no—
313C2—44no—
314C2C127×#256912yes—
315C2C127×#260811yescharacteristic
316C2C127×#264611yescharacteristic
317C2C127×#266512yes—
318C2C127×#275714yes—
319C2C127×#277814yes—
320C2C127×#284211yescharacteristic
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
1. C127×#1443C2semidirect
2. C127×#1441C2semidirect
152. #16904C127direct (N × H)
161. #1443C2×C127semidirect
162. #1441C2×C127semidirect
166. C127#16904direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#14432
C127×#14412
C127×#14392
C127×C22⋊#952
#16904127