Finite Group Databasebeta

← C127×#14658 #44578331

Subgroups — C127×#14658 #44578331

Order 32512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C127C211yesmaximal, characteristic
2C2×#380×C127C212yesmaximal, complemented
3C127×#1441C214yesmaximal, complemented
4C127×#1440C212yesmaximal, complemented
5C127×#1439C214yesmaximal, complemented
6C127×#1438C212yesmaximal
7C23×D4×C127C2211yescharacteristic
8C24×C4×C127C2211yescharacteristic
9C2×#123×C127—28no—
10C2×#123×C127C418yescomplemented
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2214yes—
13C22×C22⋊C4×C127C2212yes—
14C22×C22⋊C4×C127C2211yescharacteristic
15C22×C22⋊C4×C127C2214yes—
16C22×C22⋊C4×C127C2214yes—
17#380×C127—28no—
18C2×#95×C127C2214yes—
19C2×#95×C127C2214yes—
20C2×#95×C127C2218yes—
21C25×C127C2311yescharacteristic
22C25×C127C2311yescharacteristic
23C22×D4×C127—24no—
24C22×D4×C127—44no—
25C22×D4×C127C2×C414yes—
26C23×C4×C127—44no—
27C23×C4×C127C2×C414yes—
28C23×C4×C127D418yes—
29C23×C4×C127C2318yes—
30C23×C4×C127C2312yes—
31C23×C4×C127C2311yescharacteristic
32C23×C4×C127C2312yes—
33#123×C127—232no—
34C2×C4⋊C4×C127—24no—
35C2×C4⋊C4×C127—24no—
36C2×C4⋊C4×C127—28no—
37C2×C4⋊C4×C127—48no—
38C2×C4⋊C4×C127C2×C418yes—
39C2×C22⋊C4×C127—24no—
40C2×C22⋊C4×C127—24no—
41C2×C22⋊C4×C127—24no—
42C2×C22⋊C4×C127—28no—
43C2×C22⋊C4×C127—48no—
44C2×C22⋊C4×C127—48no—
45C2×C22⋊C4×C127—44no—
46C2×C22⋊C4×C127C2×C414yes—
47C2×C22⋊C4×C127C2×C418yes—
48#95×C127—28no—
49#95×C127—48no—
50#95×C127—44no—
51C24×C127—24no—
52C24×C127—24no—
53C24×C127—24no—
54C24×C127—22no—
55C24×C127—24no—
56C24×C127—24no—
57C24×C127—24no—
58C24×C127C22×C412yes—
59C24×C127C22×C412yes—
60C24×C127C2×D414yes—
61C24×C127C2411yescharacteristic
62C2×D4×C127—28no—
63C2×D4×C127—48no—
64C2×D4×C127—416no—
65C22×C4×C127—28no—
66C22×C4×C127—22no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—24no—
72C22×C4×C127—28no—
73C22×C4×C127—28no—
74C22×C4×C127—28no—
75C22×C4×C127—28no—
76C22×C4×C127—28no—
77C22×C4×C127—28no—
78C22×C4×C127—28no—
79C22×C4×C127—48no—
80C22×C4×C127—416no—
81C22×C4×C127C22⋊C4116yescomplemented
82C22×C4×C127C22×C414yes—
83C22×C4×C127C22×C412yes—
84C22×C4×C127C22×C414yes—
85C22×C4×C127C2×D414yes—
86C22×C4×C127C2×D414yes—
87C4⋊C4×C127—216no—
88C4⋊C4×C127—416no—
89C22⋊C4×C127—28no—
90C22⋊C4×C127—216no—
91C22⋊C4×C127—416no—
92C22⋊C4×C127—416no—
93C23×C127—24no—
94C23×C127—28no—
95C23×C127—28no—
96C23×C127—24no—
97C23×C127—24no—
98C23×C127—28no—
99C23×C127—22no—
100C23×C127—24no—
101C23×C127—28no—
102C23×C127—44no—
103C23×C127—48no—
104C23×C127—44no—
105C23×C127—44no—
106C23×C127—42no—
107C23×C127—42no—
108C23×C127—44no—
109C23×C127—44no—
110C23×C127—48no—
111C23×C127—44no—
112C23×C127C2×C22⋊C414yes—
113C23×C127C23×C411yescharacteristic
114C23×C127C22×D412yes—
115C23×C127H(5,𝔽2)14yes—
116C23×C127H(5,𝔽2)14yes—
117C23×C1272−1+414yes—
118D4×C127—416no—
119C2×C4×C127—28no—
120C2×C4×C127—28no—
121C2×C4×C127—28no—
122C2×C4×C127—24no—
123C2×C4×C127—22no—
124C2×C4×C127—24no—
125C2×C4×C127—28no—
126C2×C4×C127—416no—
127C2×C4×C127—44no—
128C2×C4×C127—42no—
129C2×C4×C127—42no—
130C2×C4×C127—44no—
131C2×C4×C127—48no—
132C2×C4×C127—48no—
133C2×C4×C127—48no—
134C2×C4×C127—48no—
135C2×C4×C127—48no—
136C2×C4×C127—48no—
137C2×C4×C127—48no—
138C2×C4×C127C2×C22⋊C418yes—
139C22×C127—24no—
140C22×C127—28no—
141C22×C127—48no—
142C22×C127—48no—
143C22×C127—48no—
144C22×C127—42no—
145C22×C127—44no—
146C22×C127—48no—
147C22×C127—84no—
148C22×C127—84no—
149C22×C127C22×C22⋊C411yescharacteristic
150C22×C127#51812yes—
151C22×C127#52014yes—
152C22×C127#53414yes—
153C22×C127#53512yes—
154C22×C127#53612yes—
155C22×C127#53714yes—
156C22×C127#56014yes—
157C22×C127#56218yes—
158C22×C127#56314yes—
159C4×C127—28no—
160C4×C127—42no—
161C4×C127—44no—
162C4×C127—88no—
163#14658C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—44no—
165C2×C127—82no—
166C2×C127#230811yescharacteristic
167C2×C127#231112yes—
168C2×C127#237614yes—
169C2×C127#281314yes—
170C2×C127#281414yes—
171C22×#123C2×C12711yescharacteristic
172C2×#380C2×C12712yescomplemented
173#1441C2×C12714yescomplemented
174#1440C2×C12712yescomplemented
175#1439C2×C12714yescomplemented
176#1438C2×C12712yes—
177C127#1465811yescharacteristic, direct product factor, Sylow
178C23×D4C22×C12711yescharacteristic
179C24×C4C22×C12711yescharacteristic
180C2×#123—28no—
181C2×#123C4×C12718yescomplemented
182C22×C4⋊C4C22×C12712yes—
183C22×C4⋊C4C22×C12714yes—
184C22×C22⋊C4C22×C12712yes—
185C22×C22⋊C4C22×C12711yescharacteristic
186C22×C22⋊C4C22×C12714yes—
187C22×C22⋊C4C22×C12714yes—
188#380—28no—
189C2×#95C22×C12714yes—
190C2×#95C22×C12714yes—
191C2×#95C22×C12718yes—
192C25C23×C12711yescharacteristic
193C25C23×C12711yescharacteristic
194C22×D4—24no—
195C22×D4—44no—
196C22×D4C2×C4×C12714yes—
197C23×C4—44no—
198C23×C4C2×C4×C12714yes—
199C23×C4D4×C12718yes—
200C23×C4C23×C12718yes—
201C23×C4C23×C12712yes—
202C23×C4C23×C12711yescharacteristic
203C23×C4C23×C12712yes—
204#123—232no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4C2×C4×C12718yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—48no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4C2×C4×C12714yes—
218C2×C22⋊C4C2×C4×C12718yes—
219#95—28no—
220#95—48no—
221#95—44no—
222C24—24no—
223C24—24no—
224C24—24no—
225C24—22no—
226C24—24no—
227C24—24no—
228C24—24no—
229C24C22×C4×C12712yes—
230C24C22×C4×C12712yes—
231C24C2×D4×C12714yes—
232C24C24×C12711yescharacteristic
233C2×D4—28no—
234C2×D4—48no—
235C2×D4—416no—
236C22×C4—28no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—48no—
251C22×C4—416no—
252C22×C4C22⋊C4×C127116yescomplemented
253C22×C4C22×C4×C12714yes—
254C22×C4C22×C4×C12712yes—
255C22×C4C22×C4×C12714yes—
256C22×C4C2×D4×C12714yes—
257C22×C4C2×D4×C12714yes—
258C4⋊C4—216no—
259C4⋊C4—416no—
260C22⋊C4—28no—
261C22⋊C4—216no—
262C22⋊C4—416no—
263C22⋊C4—416no—
264C23—24no—
265C23—28no—
266C23—28no—
267C23—24no—
268C23—24no—
269C23—28no—
270C23—22no—
271C23—24no—
272C23—28no—
273C23—44no—
274C23—48no—
275C23—44no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—44no—
281C23—48no—
282C23—44no—
283C23C2×C22⋊C4×C12714yes—
284C23C23×C4×C12711yescharacteristic
285C23C22×D4×C12712yes—
286C23H(5,𝔽2)×C12714yes—
287C23H(5,𝔽2)×C12714yes—
288C232−1+4×C12714yes—
289D4—416no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—416no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4C2×C22⋊C4×C12718yes—
310C22—24no—
311C22—28no—
312C22—48no—
313C22—48no—
314C22—48no—
315C22—48no—
316C22—42no—
317C22—44no—
318C22—84no—
319C22—84no—
320C22C22×C22⋊C4×C12711yescharacteristic
321C22#518×C12712yes—
322C22#520×C12714yes—
323C22#534×C12714yes—
324C22#535×C12712yes—
325C22#536×C12712yes—
326C22#537×C12714yes—
327C22#560×C12714yes—
328C22#562×C12718yes—
329C22#563×C12714yes—
330C4—28no—
331C4—42no—
332C4—44no—
333C4—88no—
334C2—44no—
335C2—82no—
336C2C127×#230811yescharacteristic
337C2C127×#231112yes—
338C2C127×#237614yes—
339C2C127×#281314yes—
340C2C127×#281414yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#380×C127C2semidirect
3. C127×#1441C2semidirect
4. C127×#1440C2semidirect
5. C127×#1439C2semidirect
10. C2×#123×C127C4semidirect
81. C22×C4×C127C22⋊C4semidirect
163. #14658C127direct (N × H)
172. C2×#380C2×C127semidirect
173. #1441C2×C127semidirect
174. #1440C2×C127semidirect
175. #1439C2×C127semidirect
177. C127#14658direct (N × H)
181. C2×#123C4×C127semidirect
252. C22×C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C1272
C2×#380×C1272
C127×#14412
C127×#14402
C127×#14392
C127×#14382
#14658127