Finite Group Databasebeta

← C127×#12623 #44576296

Subgroups — C127×#12623 #44576296

Order 32512.

470 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2929C211yesmaximal, characteristic
2C127×#2120C212yesmaximal
3C127×#1757C212yesmaximal, complemented
4C127×#1392C211yesmaximal, characteristic, complemented
5C127×#1391C211yesmaximal, characteristic, complemented
6C2×#131×C127C2211yescharacteristic
7C23×C8×C127C411yescharacteristic, complemented
8#525×C127C2211yescharacteristic
9#435×C127—22no—
10C8×D4×C127—22no—
11#424×C127C411yescharacteristic, complemented
12#413×C127C2212yes—
13#410×C127C2212yes—
14#408×C127C2211yescharacteristic
15#407×C127C411yescharacteristic, complemented
16#405×C127—22no—
17#405×C127C411yescharacteristic, complemented
18#352×C127C414yescomplemented
19AΣL(1,𝔽16)×C127C414yescomplemented
20#344×C127—22no—
21C2×#41×C127C2311yescharacteristic
22C23×C4×C127C2×C411yescharacteristic
23#131×C127—41no—
24#131×C127—41no—
25C2×M16×C127—22no—
26C2×M16×C127C2×C411yescharacteristic
27C2×M16×C127C2×C411yescharacteristic
28C2×M16×C127D412yes—
29C22×C8×C127—21no—
30C22×C8×C127—21no—
31C22×C8×C127—41no—
32C22×C8×C127—41no—
33C22×C8×C127C2×C411yescharacteristic
34C22×C8×C127C2×C411yescharacteristic
35C22×C8×C127D412yes—
36#123×C127C2×C412yes—
37C22⋊Q8×C127—21no—
38#121×C127—21no—
39#120×C127C2×C412yes—
40C4×D4×C127—22no—
41#117×C127—22no—
42#117×C127C2×C411yescharacteristic
43C4⋊C8×C127—22no—
44#101×C127—22no—
45#100×C127C2×C414yes—
46#99×C127—22no—
47#99×C127C2×C414yes—
48C22⋊C8×C127—22no—
49C22⋊C8×C127—22no—
50#97×C127—21no—
51#97×C127—42no—
52C4×C8×C127—21no—
53C4×C8×C127—42no—
54C24×C127C4211yescharacteristic
55#41×C127—41no—
56#41×C127—41no—
57C2×Q8×C127#4111yescharacteristic
58C2×D4×C127—22no—
59C2×D4×C127C22×C412yes—
60C2×D4×C127#4111yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—41no—
64C22×C4×C127—41no—
65C22×C4×C127C4211yescharacteristic
66C22×C4×C127C22×C411yescharacteristic
67C22×C4×C127C2×D412yes—
68M16×C127—24no—
69M16×C127—41no—
70M16×C127—42no—
71M16×C127—42no—
72M16×C127—41no—
73C2×C8×C127—22no—
74C2×C8×C127—21no—
75C2×C8×C127—22no—
76C2×C8×C127—21no—
77C2×C8×C127—21no—
78C2×C8×C127—21no—
79C2×C8×C127—42no—
80C2×C8×C127—42no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127—41no—
84C2×C8×C127C22⋊C411yescharacteristic
85C2×C8×C127C22⋊C412yes—
86C2×C8×C127C22⋊C411yescharacteristic
87C4⋊C4×C127—21no—
88C4⋊C4×C127—22no—
89C4⋊C4×C127—42no—
90C22⋊C4×C127—22no—
91C22⋊C4×C127—22no—
92C22⋊C4×C127—24no—
93C22⋊C4×C127C4212yes—
94C42×C127—21no—
95C42×C127—42no—
96C23×C127—21no—
97C23×C127—41no—
98C23×C127—42no—
99C23×C127C2×C4211yescharacteristic
100C23×C127C4×D412yes—
101Q8×C127—41no—
102D4×C127—42no—
103D4×C127—41no—
104D4×C127—44no—
105C2×C4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—42no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—41no—
115C2×C4×C127—44no—
116C2×C4×C127C2×C22⋊C411yescharacteristic
117C2×C4×C127#11711yescharacteristic
118C2×C4×C127C4×D412yes—
119C8×C127—21no—
120C8×C127—22no—
121C8×C127—44no—
122C8×C127—41no—
123C8×C127—41no—
124C8×C127#9911yescharacteristic
125C8×C127#9911yescharacteristic
126C22×C127—22no—
127C22×C127—22no—
128C22×C127—21no—
129C22×C127—42no—
130C22×C127—41no—
131C22×C127—41no—
132C22×C127—42no—
133C22×C127—42no—
134C22×C127—41no—
135C22×C127C4×C22⋊C411yescharacteristic
136C4×C127—22no—
137C4×C127—21no—
138C4×C127—42no—
139C4×C127—41no—
140C4×C127—84no—
141C4×C127C2×#9911yescharacteristic
142C4×C127#41011yescharacteristic
143#12623C12711yesmaximal, characteristic, direct product factor, Sylow
144C2×C127—21no—
145C2×C127—42no—
146C2×C127—42no—
147C2×C127—41no—
148C2×C127C4×#9911yescharacteristic
149#2929C2×C12711yescharacteristic
150#2120C2×C12712yes—
151#1757C2×C12712yescomplemented
152#1392C2×C12711yescharacteristic, complemented
153#1391C2×C12711yescharacteristic, complemented
154C127#1262311yescharacteristic, direct product factor, Sylow
155C2×#131C22×C12711yescharacteristic
156C23×C8C4×C12711yescharacteristic, complemented
157#525C22×C12711yescharacteristic
158#435—22no—
159C8×D4—22no—
160#424C4×C12711yescharacteristic, complemented
161#413C22×C12712yes—
162#410C22×C12712yes—
163#408C22×C12711yescharacteristic
164#407C4×C12711yescharacteristic, complemented
165#405—22no—
166#405C4×C12711yescharacteristic, complemented
167#352C4×C12714yescomplemented
168AΣL(1,𝔽16)C4×C12714yescomplemented
169#344—22no—
170C2×#41C23×C12711yescharacteristic
171C23×C4C2×C4×C12711yescharacteristic
172#131—41no—
173#131—41no—
174C2×M16—22no—
175C2×M16C2×C4×C12711yescharacteristic
176C2×M16C2×C4×C12711yescharacteristic
177C2×M16D4×C12712yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8C2×C4×C12711yescharacteristic
183C22×C8C2×C4×C12711yescharacteristic
184C22×C8D4×C12712yes—
185#123C2×C4×C12712yes—
186C22⋊Q8—21no—
187#121—21no—
188#120C2×C4×C12712yes—
189C4×D4—22no—
190#117—22no—
191#117C2×C4×C12711yescharacteristic
192C4⋊C8—22no—
193#101—22no—
194#100C2×C4×C12714yes—
195#99—22no—
196#99C2×C4×C12714yes—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#97—21no—
200#97—42no—
201C4×C8—21no—
202C4×C8—42no—
203C24C42×C12711yescharacteristic
204#41—41no—
205#41—41no—
206C2×Q8#41×C12711yescharacteristic
207C2×D4—22no—
208C2×D4C22×C4×C12712yes—
209C2×D4#41×C12711yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4C42×C12711yescharacteristic
215C22×C4C22×C4×C12711yescharacteristic
216C22×C4C2×D4×C12712yes—
217M16—24no—
218M16—41no—
219M16—42no—
220M16—42no—
221M16—41no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8C22⋊C4×C12711yescharacteristic
234C2×C8C22⋊C4×C12712yes—
235C2×C8C22⋊C4×C12711yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4C42×C12712yes—
243C42—21no—
244C42—42no—
245C23—21no—
246C23—41no—
247C23—42no—
248C23C2×C42×C12711yescharacteristic
249C23C4×D4×C12712yes—
250Q8—41no—
251D4—42no—
252D4—41no—
253D4—44no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—44no—
265C2×C4C2×C22⋊C4×C12711yescharacteristic
266C2×C4#117×C12711yescharacteristic
267C2×C4C4×D4×C12712yes—
268C8—21no—
269C8—22no—
270C8—44no—
271C8—41no—
272C8—41no—
273C8#99×C12711yescharacteristic
274C8#99×C12711yescharacteristic
275C22—22no—
276C22—22no—
277C22—21no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22C4×C22⋊C4×C12711yescharacteristic
285C4—22no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—84no—
290C4C2×#99×C12711yescharacteristic
291C4#410×C12711yescharacteristic
292C2—21no—
293C2—42no—
294C2—42no—
295C2—41no—
296C2C4×#99×C12711yescharacteristic
subgroupG/H
centerC8×C127#99
commutatorC23C2×C42×C127
Frattini, cosocleC2×#41C23×C127
socleC2×C127C4×#99
prime radical#525×C127C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8×C127 ◁ C2×C8×C127 ◁ C2×#131×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1757C2semidirect
4. C127×#1392C2semidirect
5. C127×#1391C2semidirect
7. C23×C8×C127C4semidirect
11. #424×C127C4semidirect
15. #407×C127C4semidirect
17. #405×C127C4semidirect
18. #352×C127C4semidirect
19. AΣL(1,𝔽16)×C127C4semidirect
143. #12623C127direct (N × H)
151. #1757C2×C127semidirect
152. #1392C2×C127semidirect
153. #1391C2×C127semidirect
154. C127#12623direct (N × H)
156. C23×C8C4×C127semidirect
160. #424C4×C127semidirect
164. #407C4×C127semidirect
166. #405C4×C127semidirect
167. #352C4×C127semidirect
168. AΣL(1,𝔽16)C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#29292
C127×#21202
C127×#17572
C127×#13922
C127×#13912
#12623127