Finite Group Databasebeta

← C127×#12582 #44576255

Subgroups — C127×#12582 #44576255

Order 32512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#99×C127C212yesmaximal
2C127×#2309C211yesmaximal, characteristic
3C127×#1393C214yesmaximal, complemented
4C23×D4×C127C2211yescharacteristic
5C2×#117×C127C411yescharacteristic, complemented
6C2×#117×C127C411yescharacteristic, complemented
7C22×C22⋊C4×C127—22no—
8C2×#99×C127—22no—
9C2×#99×C127—24no—
10C2×#99×C127C418yescomplemented
11C2×#99×C127C2214yes—
12#390×C127—22no—
13#390×C127C412yescomplemented
14#390×C127C2212yes—
15#379×C127—22no—
16C25×C127D412yes—
17C22×D4×C127—42no—
18C22×D4×C127—42no—
19C22×D4×C127C2×C414yes—
20C22×D4×C127C2×C411yescharacteristic
21C22×D4×C127D412yes—
22C22×D4×C127D412yes—
23C22×D4×C127Q812yes—
24C22×D4×C127C2311yescharacteristic
25C23×C4×C127—42no—
26C23×C4×C127C2×C411yescharacteristic
27#117×C127—42no—
28#117×C127—42no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127C2×C412yescomplemented
31C2×C22⋊C4×C127—28no—
32C2×C22⋊C4×C127—24no—
33C2×C22⋊C4×C127—21no—
34C2×C22⋊C4×C127—21no—
35C2×C22⋊C4×C127—24no—
36C2×C22⋊C4×C127—24no—
37C2×C22⋊C4×C127—42no—
38C2×C22⋊C4×C127—44no—
39C2×C42×C127—21no—
40C2×C42×C127C2×C412yescomplemented
41#99×C127—28no—
42#99×C127—24no—
43#99×C127C2×C418yescomplemented
44C24×C127—22no—
45C24×C127—24no—
46C24×C127—22no—
47C24×C127—84no—
48C24×C127C22⋊C412yes—
49C24×C127C22⋊C414yes—
50C24×C127C22⋊C411yescharacteristic
51C24×C127C4⋊C414yes—
52C24×C127C2×D412yes—
53C2×D4×C127—44no—
54C2×D4×C127—44no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—44no—
58C2×D4×C127—42no—
59C2×D4×C127—42no—
60C2×D4×C127—44no—
61C2×D4×C127—41no—
62C2×D4×C127—41no—
63C2×D4×C127C4212yes—
64C2×D4×C127C22⋊C414yes—
65C2×D4×C127C4⋊C414yes—
66C2×D4×C127C4⋊C412yes—
67C2×D4×C127C22×C412yes—
68C2×D4×C127C2×D411yescharacteristic
69C2×D4×C127C2×Q811yescharacteristic
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—48no—
75C22×C4×C127—44no—
76C22×C4×C127—88no—
77C22×C4×C127C4212yes—
78C22×C4×C127C22⋊C411yescharacteristic
79C22×C4×C127C4⋊C412yes—
80C22×C4×C127C22×C411yescharacteristic
81C4⋊C4×C127—42no—
82C4⋊C4×C127—42no—
83C22⋊C4×C127—28no—
84C22⋊C4×C127—48no—
85C22⋊C4×C127—44no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C22⋊C4×C127—84no—
89C22⋊C4×C127—84no—
90C42×C127—42no—
91C42×C127—42no—
92C23×C127—24no—
93C23×C127—24no—
94C23×C127—22no—
95C23×C127—22no—
96C23×C127—24no—
97C23×C127—24no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—42no—
101C23×C127—48no—
102C23×C127—44no—
103C23×C127—44no—
104C23×C127—84no—
105C23×C127—84no—
106C23×C127—84no—
107C23×C127—88no—
108C23×C127#9514yes—
109C23×C127#9512yes—
110C23×C127#9912yes—
111C23×C127#9912yes—
112C23×C127C2×C22⋊C411yescharacteristic
113C23×C127C2×C22⋊C412yes—
114C23×C127C2×C4⋊C412yes—
115D4×C127—42no—
116D4×C127—44no—
117D4×C127—44no—
118D4×C127—42no—
119D4×C127—42no—
120D4×C127—41no—
121D4×C127—41no—
122C2×C4×C127—24no—
123C2×C4×C127—24no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—48no—
129C2×C4×C127—42no—
130C2×C4×C127—42no—
131C2×C4×C127—44no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—816no—
135C2×C4×C127—88no—
136C2×C4×C127#9512yes—
137C2×C4×C127C2×C4211yescharacteristic
138C2×C4×C127C2×C4⋊C411yescharacteristic
139C22×C127—22no—
140C22×C127—24no—
141C22×C127—22no—
142C22×C127—24no—
143C22×C127—22no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—44no—
147C22×C127—42no—
148C22×C127—42no—
149C22×C127—48no—
150C22×C127—48no—
151C22×C127—44no—
152C22×C127—44no—
153C22×C127—82no—
154C22×C127—82no—
155C22×C127—84no—
156C22×C127—84no—
157C22×C127#34214yes—
158C22×C127C2×#9511yescharacteristic
159C22×C127C2×#9911yescharacteristic
160C22×C127C2×#9911yescharacteristic
161C4×C127—22no—
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—82no—
165C4×C127—88no—
166C4×C127—82no—
167#12582C12711yesmaximal, characteristic, direct product factor, Sylow
168C2×C127—21no—
169C2×C127—21no—
170C2×C127—24no—
171C2×C127—42no—
172C2×C127—44no—
173C2×C127—42no—
174C2×C127C2×#34211yescharacteristic
175C2×C127#212712yes—
176C22×#99C2×C12712yes—
177#2309C2×C12711yescharacteristic
178#1393C2×C12714yescomplemented
179C127#1258211yescharacteristic, direct product factor, Sylow
180C23×D4C22×C12711yescharacteristic
181C2×#117C4×C12711yescharacteristic, complemented
182C2×#117C4×C12711yescharacteristic, complemented
183C22×C22⋊C4—22no—
184C2×#99—22no—
185C2×#99—24no—
186C2×#99C4×C12718yescomplemented
187C2×#99C22×C12714yes—
188#390—22no—
189#390C4×C12712yescomplemented
190#390C22×C12712yes—
191#379—22no—
192C25D4×C12712yes—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4C2×C4×C12714yes—
196C22×D4C2×C4×C12711yescharacteristic
197C22×D4D4×C12712yes—
198C22×D4D4×C12712yes—
199C22×D4Q8×C12712yes—
200C22×D4C23×C12711yescharacteristic
201C23×C4—42no—
202C23×C4C2×C4×C12711yescharacteristic
203#117—42no—
204#117—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×C4×C12712yescomplemented
207C2×C22⋊C4—28no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C42—21no—
216C2×C42C2×C4×C12712yescomplemented
217#99—28no—
218#99—24no—
219#99C2×C4×C12718yescomplemented
220C24—22no—
221C24—24no—
222C24—22no—
223C24—84no—
224C24C22⋊C4×C12712yes—
225C24C22⋊C4×C12714yes—
226C24C22⋊C4×C12711yescharacteristic
227C24C4⋊C4×C12714yes—
228C24C2×D4×C12712yes—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—44no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4C42×C12712yes—
240C2×D4C22⋊C4×C12714yes—
241C2×D4C4⋊C4×C12714yes—
242C2×D4C4⋊C4×C12712yes—
243C2×D4C22×C4×C12712yes—
244C2×D4C2×D4×C12711yescharacteristic
245C2×D4C2×Q8×C12711yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—88no—
253C22×C4C42×C12712yes—
254C22×C4C22⋊C4×C12711yescharacteristic
255C22×C4C4⋊C4×C12712yes—
256C22×C4C22×C4×C12711yescharacteristic
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—28no—
260C22⋊C4—48no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—84no—
265C22⋊C4—84no—
266C42—42no—
267C42—42no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—48no—
278C23—44no—
279C23—44no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23—88no—
284C23#95×C12714yes—
285C23#95×C12712yes—
286C23#99×C12712yes—
287C23#99×C12712yes—
288C23C2×C22⋊C4×C12711yescharacteristic
289C23C2×C22⋊C4×C12712yes—
290C23C2×C4⋊C4×C12712yes—
291D4—42no—
292D4—44no—
293D4—44no—
294D4—42no—
295D4—42no—
296D4—41no—
297D4—41no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—48no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—816no—
311C2×C4—88no—
312C2×C4#95×C12712yes—
313C2×C4C2×C42×C12711yescharacteristic
314C2×C4C2×C4⋊C4×C12711yescharacteristic
315C22—22no—
316C22—24no—
317C22—22no—
318C22—24no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—44no—
323C22—42no—
324C22—42no—
325C22—48no—
326C22—48no—
327C22—44no—
328C22—44no—
329C22—82no—
330C22—82no—
331C22—84no—
332C22—84no—
333C22#342×C12714yes—
334C22C2×#95×C12711yescharacteristic
335C22C2×#99×C12711yescharacteristic
336C22C2×#99×C12711yescharacteristic
337C4—22no—
338C4—21no—
339C4—21no—
340C4—82no—
341C4—88no—
342C4—82no—
343C2—21no—
344C2—21no—
345C2—24no—
346C2—42no—
347C2—44no—
348C2—42no—
349C2C2×#342×C12711yescharacteristic
350C2C127×#212712yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC2×C4C2×C42×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC23×D4×C127C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C23×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1393C2semidirect
5. C2×#117×C127C4semidirect
6. C2×#117×C127C4semidirect
10. C2×#99×C127C4semidirect
13. #390×C127C4semidirect
30. C2×C4⋊C4×C127C2×C4semidirect
40. C2×C42×C127C2×C4semidirect
43. #99×C127C2×C4semidirect
167. #12582C127direct (N × H)
178. #1393C2×C127semidirect
179. C127#12582direct (N × H)
181. C2×#117C4×C127semidirect
182. C2×#117C4×C127semidirect
186. C2×#99C4×C127semidirect
189. #390C4×C127semidirect
206. C2×C4⋊C4C2×C4×C127semidirect
216. C2×C42C2×C4×C127semidirect
219. #99C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#99×C1272
C127×#23092
C127×#13932
#12582127