Finite Group Databasebeta

← C127×#12587 #44576260

Subgroups — C127×#12587 #44576260

Order 32512.

794 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#413×C127C212yesmaximal
2C127×#2311C211yesmaximal, characteristic
3C127×#1394C214yesmaximal, complemented
4C22×#41×C127C2211yescharacteristic
5C22×M16×C127—22no—
6C2×#117×C127C411yescharacteristic
7C22×C4⋊C4×C127C411yescharacteristic
8#413×C127—24no—
9#413×C127C418yescomplemented
10C2×#101×C127C2214yes—
11C2×#100×C127—22no—
12#391×C127C2212yes—
13#390×C127C412yes—
14#386×C127—22no—
15#380×C127—22no—
16C2×#41×C127—42no—
17C2×#41×C127—42no—
18C2×#41×C127C2×C414yes—
19C2×#41×C127D412yes—
20C2×#41×C127Q812yes—
21C22×Q8×C127C2311yescharacteristic
22C22×D4×C127C2×C411yescharacteristic
23C22×D4×C127D412yes—
24C23×C4×C127—21no—
25C23×C4×C127C2×C411yescharacteristic
26C23×C4×C127D412yes—
27C2×M16×C127—28no—
28C2×M16×C127—24no—
29C2×M16×C127—42no—
30C2×M16×C127—44no—
31C22×C8×C127—42no—
32#117×C127—42no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—42no—
36C2×C4⋊C4×C127C2×C412yes—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—24no—
39C2×C42×C127C2×C412yes—
40#101×C127—24no—
41#101×C127C2×C418yescomplemented
42#100×C127—28no—
43#95×C127—24no—
44C24×C127C22⋊C412yes—
45C24×C127C22⋊C411yescharacteristic
46#41×C127—44no—
47#41×C127—44no—
48#41×C127—44no—
49#41×C127—44no—
50C2×Q8×C127—41no—
51C2×Q8×C127—41no—
52C2×Q8×C127C22×C412yes—
53C2×Q8×C127C2×D411yescharacteristic
54C2×Q8×C127C2×Q811yescharacteristic
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—42no—
58C2×D4×C127—42no—
59C2×D4×C127C4212yes—
60C2×D4×C127C22⋊C414yes—
61C2×D4×C127C4⋊C412yes—
62C2×D4×C127C4⋊C414yes—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—24no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—22no—
70C22×C4×C127—42no—
71C22×C4×C127—82no—
72C22×C4×C127C4212yes—
73C22×C4×C127C22⋊C414yes—
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C4⋊C412yes—
76C22×C4×C127C4⋊C414yes—
77C22×C4×C127C22×C411yescharacteristic
78C22×C4×C127C2×D412yes—
79M16×C127—28no—
80M16×C127—44no—
81M16×C127—48no—
82C2×C8×C127—44no—
83C2×C8×C127—48no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—42no—
86C4⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C22⋊C4×C127—84no—
89C42×C127—42no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—41no—
95C23×C127—41no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127#9512yes—
99C23×C127#9514yes—
100C23×C127C2×C22⋊C411yescharacteristic
101Q8×C127—42no—
102Q8×C127—41no—
103Q8×C127—41no—
104D4×C127—42no—
105D4×C127—44no—
106D4×C127—42no—
107D4×C127—44no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—24no—
113C2×C4×C127—24no—
114C2×C4×C127—24no—
115C2×C4×C127—24no—
116C2×C4×C127—24no—
117C2×C4×C127—24no—
118C2×C4×C127—42no—
119C2×C4×C127—44no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—84no—
124C2×C4×C127#9512yes—
125C2×C4×C127#9912yes—
126C2×C4×C127#9912yes—
127C2×C4×C127C2×C4211yescharacteristic
128C2×C4×C127C2×C22⋊C412yes—
129C2×C4×C127C2×C4⋊C411yescharacteristic
130C2×C4×C127C2×C4⋊C412yes—
131C8×C127—48no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—22no—
135C22×C127—24no—
136C22×C127—42no—
137C22×C127—44no—
138C22×C127—42no—
139C22×C127—41no—
140C22×C127—42no—
141C22×C127—41no—
142C22×C127—42no—
143C22×C127—82no—
144C22×C127C2×#9511yescharacteristic
145C22×C127C2×#9911yescharacteristic
146C22×C127C2×#9911yescharacteristic
147C4×C127—22no—
148C4×C127—21no—
149C4×C127—21no—
150C4×C127—22no—
151C4×C127—82no—
152C4×C127—82no—
153C4×C127#34214yes—
154#12587C12711yesmaximal, characteristic, direct product factor, Sylow
155C2×C127—21no—
156C2×C127—21no—
157C2×C127—42no—
158C2×C127—44no—
159C2×C127C2×#34211yescharacteristic
160C2×C127#213411yescharacteristic
161C2×C127#213511yescharacteristic
162C2×#413C2×C12712yes—
163#2311C2×C12711yescharacteristic
164#1394C2×C12714yescomplemented
165C127#1258711yescharacteristic, direct product factor, Sylow
166C22×#41C22×C12711yescharacteristic
167C22×M16—22no—
168C2×#117C4×C12711yescharacteristic
169C22×C4⋊C4C4×C12711yescharacteristic
170#413—24no—
171#413C4×C12718yescomplemented
172C2×#101C22×C12714yes—
173C2×#100—22no—
174#391C22×C12712yes—
175#390C4×C12712yes—
176#386—22no—
177#380—22no—
178C2×#41—42no—
179C2×#41—42no—
180C2×#41C2×C4×C12714yes—
181C2×#41D4×C12712yes—
182C2×#41Q8×C12712yes—
183C22×Q8C23×C12711yescharacteristic
184C22×D4C2×C4×C12711yescharacteristic
185C22×D4D4×C12712yes—
186C23×C4—21no—
187C23×C4C2×C4×C12711yescharacteristic
188C23×C4D4×C12712yes—
189C2×M16—28no—
190C2×M16—24no—
191C2×M16—42no—
192C2×M16—44no—
193C22×C8—42no—
194#117—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C2×C4×C12712yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C42C2×C4×C12712yes—
202#101—24no—
203#101C2×C4×C12718yescomplemented
204#100—28no—
205#95—24no—
206C24C22⋊C4×C12712yes—
207C24C22⋊C4×C12711yescharacteristic
208#41—44no—
209#41—44no—
210#41—44no—
211#41—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8C22×C4×C12712yes—
215C2×Q8C2×D4×C12711yescharacteristic
216C2×Q8C2×Q8×C12711yescharacteristic
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4C42×C12712yes—
222C2×D4C22⋊C4×C12714yes—
223C2×D4C4⋊C4×C12712yes—
224C2×D4C4⋊C4×C12714yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—82no—
234C22×C4C42×C12712yes—
235C22×C4C22⋊C4×C12714yes—
236C22×C4C22⋊C4×C12711yescharacteristic
237C22×C4C4⋊C4×C12712yes—
238C22×C4C4⋊C4×C12714yes—
239C22×C4C22×C4×C12711yescharacteristic
240C22×C4C2×D4×C12712yes—
241M16—28no—
242M16—44no—
243M16—48no—
244C2×C8—44no—
245C2×C8—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—84no—
251C42—42no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23#95×C12712yes—
261C23#95×C12714yes—
262C23C2×C22⋊C4×C12711yescharacteristic
263Q8—42no—
264Q8—41no—
265Q8—41no—
266D4—42no—
267D4—44no—
268D4—42no—
269D4—44no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—84no—
286C2×C4#95×C12712yes—
287C2×C4#99×C12712yes—
288C2×C4#99×C12712yes—
289C2×C4C2×C42×C12711yescharacteristic
290C2×C4C2×C22⋊C4×C12712yes—
291C2×C4C2×C4⋊C4×C12711yescharacteristic
292C2×C4C2×C4⋊C4×C12712yes—
293C8—48no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—24no—
298C22—42no—
299C22—44no—
300C22—42no—
301C22—41no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—82no—
306C22C2×#95×C12711yescharacteristic
307C22C2×#99×C12711yescharacteristic
308C22C2×#99×C12711yescharacteristic
309C4—22no—
310C4—21no—
311C4—21no—
312C4—22no—
313C4—82no—
314C4—82no—
315C4#342×C12714yes—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—44no—
320C2C2×#342×C12711yescharacteristic
321C2C127×#213411yescharacteristic
322C2C127×#213511yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC2×C4C2×C42×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1394C2semidirect
9. #413×C127C4semidirect
41. #101×C127C2×C4semidirect
154. #12587C127direct (N × H)
164. #1394C2×C127semidirect
165. C127#12587direct (N × H)
171. #413C4×C127semidirect
203. #101C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#413×C1272
C127×#23112
C127×#13942
#12587127