Finite Group Databasebeta

← C127×#12685 #44576358

Subgroups — C127×#12685 #44576358

Order 32512.

890 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3573C211yesmaximal, characteristic
2C127×#2117C212yesmaximal, complemented
3C2×#350×C127C212yesmaximal, complemented
4C127×#2115C211yesmaximal, characteristic, complemented
5C127×#2113C211yesmaximal, characteristic, complemented
6#585×C127C411yescharacteristic
7#568×C127—24no—
8#568×C127—22no—
9#568×C127C411yescharacteristic
10#568×C127C412yes—
11C2×#131×C127—24no—
12C2×#131×C127—22no—
13C2×#131×C127C2212yes—
14C2×#131×C127C2211yescharacteristic
15C2×M32×C127C2211yescharacteristic, complemented
16C2×M32×C127C2212yescomplemented
17C22×C16×C127C2211yescharacteristic, complemented
18#350×C127—24no—
19#350×C127—24no—
20#349×C127—22no—
21C22⋊C16×C127—22no—
222−1+4×C127—22no—
232−1+4×C127C812yes—
24H(5,𝔽2)×C127—22no—
25H(5,𝔽2)×C127—22no—
26H(5,𝔽2)×C127C812yes—
27C2×#41×C127—24no—
28C2×#41×C127—22no—
29C2×#41×C127C2×C412yes—
30C2×#41×C127C2×C411yescharacteristic
31#131×C127—24no—
32#131×C127—28no—
33#131×C127—24no—
34#131×C127—24no—
35#131×C127—24no—
36#131×C127—24no—
37#131×C127—22no—
38#131×C127—22no—
39#131×C127—22no—
40#131×C127—22no—
41#131×C127D414yes—
42#131×C127D414yes—
43C2×M16×C127—28no—
44C2×M16×C127—24no—
45C2×M16×C127—22no—
46C2×M16×C127—24no—
47C2×M16×C127—22no—
48C2×M16×C127C2×C412yes—
49C2×M16×C127C2×C411yescharacteristic
50C2×M16×C127D412yes—
51C22×C8×C127—24no—
52C22×C8×C127—22no—
53C22×C8×C127D412yes—
54C22×C8×C127C2311yescharacteristic
55M32×C127—22no—
56M32×C127—22no—
57M32×C127—22no—
58C2×C16×C127—21no—
59C2×C16×C127—22no—
60C2×C16×C127—21no—
61C2×C16×C127—22no—
62#41×C127—24no—
63#41×C127—28no—
64#41×C127—24no—
65#41×C127—24no—
66#41×C127—24no—
67#41×C127—24no—
68#41×C127—22no—
69#41×C127—22no—
70#41×C127—22no—
71#41×C127—22no—
72#41×C127C22⋊C414yes—
73#41×C127C2×C814yes—
74C2×Q8×C127—22no—
75C2×Q8×C127—24no—
76C2×Q8×C127C2×C811yescharacteristic
77C2×Q8×C127M1612yes—
78C2×D4×C127—28no—
79C2×D4×C127—22no—
80C2×D4×C127—24no—
81C2×D4×C127—22no—
82C2×D4×C127—24no—
83C2×D4×C127C2×C811yescharacteristic
84C2×D4×C127M1612yes—
85C2×D4×C127M1612yes—
86C22×C4×C127—24no—
87C22×C4×C127—22no—
88C22×C4×C127C22⋊C412yes—
89C22×C4×C127C22×C411yescharacteristic
90M16×C127—24no—
91M16×C127—28no—
92M16×C127—22no—
93M16×C127—22no—
94M16×C127—24no—
95M16×C127—24no—
96M16×C127—24no—
97M16×C127C22⋊C414yes—
98C2×C8×C127—24no—
99C2×C8×C127—28no—
100C2×C8×C127—22no—
101C2×C8×C127—24no—
102C2×C8×C127—22no—
103C2×C8×C127—24no—
104C2×C8×C127—22no—
105C2×C8×C127C22⋊C412yes—
106C2×C8×C127C2×D412yes—
107C2×C8×C127C2×D411yescharacteristic
108C2×C8×C127C2×D411yescharacteristic
109C2×C8×C127C2×D412yes—
110C16×C127—41no—
111C16×C127—42no—
112C16×C127—41no—
113C23×C127—24no—
114C23×C127—22no—
115C23×C127C22⋊C812yes—
116C23×C127C2×M1611yescharacteristic
117Q8×C127—24no—
118Q8×C127—22no—
119Q8×C127—22no—
120Q8×C127C22⋊C814yes—
121D4×C127—28no—
122D4×C127—24no—
123D4×C127—24no—
124D4×C127—24no—
125D4×C127—24no—
126D4×C127—22no—
127D4×C127—22no—
128D4×C127C22⋊C814yes—
129C2×C4×C127—24no—
130C2×C4×C127—28no—
131C2×C4×C127—22no—
132C2×C4×C127—24no—
133C2×C4×C127—22no—
134C2×C4×C127—24no—
135C2×C4×C127—22no—
136C2×C4×C127C22⋊C812yes—
137C2×C4×C127C2×C22⋊C412yes—
138C2×C4×C127C2×C22⋊C411yescharacteristic
139C2×C4×C127C22×C811yescharacteristic
140C2×C4×C127C2×M1612yes—
141C8×C127—24no—
142C8×C127—22no—
143C8×C127#12011yescharacteristic
144C8×C127#12012yes—
145C8×C127#12011yescharacteristic
146C22×C127—22no—
147C22×C127—24no—
148C22×C127—41no—
149C22×C127—42no—
150C22×C127—82no—
151C22×C127—84no—
152C22×C127C2×C22⋊C811yescharacteristic
153C22×C127#40712yes—
154C4×C127—24no—
155C4×C127—22no—
156C4×C127#37911yescharacteristic
157C4×C127C2×C22⋊C812yes—
158C4×C127#40711yescharacteristic
159#12685C12711yesmaximal, characteristic, direct product factor, Sylow
160C2×C127—21no—
161C2×C127—22no—
162C2×C127—42no—
163C2×C127—44no—
164C2×C127#177911yescharacteristic
165#3573C2×C12711yescharacteristic
166#2117C2×C12712yescomplemented
167C2×#350C2×C12712yescomplemented
168#2115C2×C12711yescharacteristic, complemented
169#2113C2×C12711yescharacteristic, complemented
170C127#1268511yescharacteristic, direct product factor, Sylow
171#585C4×C12711yescharacteristic
172#568—24no—
173#568—22no—
174#568C4×C12711yescharacteristic
175#568C4×C12712yes—
176C2×#131—24no—
177C2×#131—22no—
178C2×#131C22×C12712yes—
179C2×#131C22×C12711yescharacteristic
180C2×M32C22×C12711yescharacteristic, complemented
181C2×M32C22×C12712yescomplemented
182C22×C16C22×C12711yescharacteristic, complemented
183#350—24no—
184#350—24no—
185#349—22no—
186C22⋊C16—22no—
1872−1+4—22no—
1882−1+4C8×C12712yes—
189H(5,𝔽2)—22no—
190H(5,𝔽2)—22no—
191H(5,𝔽2)C8×C12712yes—
192C2×#41—24no—
193C2×#41—22no—
194C2×#41C2×C4×C12712yes—
195C2×#41C2×C4×C12711yescharacteristic
196#131—24no—
197#131—28no—
198#131—24no—
199#131—24no—
200#131—24no—
201#131—24no—
202#131—22no—
203#131—22no—
204#131—22no—
205#131—22no—
206#131D4×C12714yes—
207#131D4×C12714yes—
208C2×M16—28no—
209C2×M16—24no—
210C2×M16—22no—
211C2×M16—24no—
212C2×M16—22no—
213C2×M16C2×C4×C12712yes—
214C2×M16C2×C4×C12711yescharacteristic
215C2×M16D4×C12712yes—
216C22×C8—24no—
217C22×C8—22no—
218C22×C8D4×C12712yes—
219C22×C8C23×C12711yescharacteristic
220M32—22no—
221M32—22no—
222M32—22no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—22no—
226C2×C16—22no—
227#41—24no—
228#41—28no—
229#41—24no—
230#41—24no—
231#41—24no—
232#41—24no—
233#41—22no—
234#41—22no—
235#41—22no—
236#41—22no—
237#41C22⋊C4×C12714yes—
238#41C2×C8×C12714yes—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8C2×C8×C12711yescharacteristic
242C2×Q8M16×C12712yes—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4C2×C8×C12711yescharacteristic
249C2×D4M16×C12712yes—
250C2×D4M16×C12712yes—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4C22⋊C4×C12712yes—
254C22×C4C22×C4×C12711yescharacteristic
255M16—24no—
256M16—28no—
257M16—22no—
258M16—22no—
259M16—24no—
260M16—24no—
261M16—24no—
262M16C22⋊C4×C12714yes—
263C2×C8—24no—
264C2×C8—28no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8C22⋊C4×C12712yes—
271C2×C8C2×D4×C12711yescharacteristic
272C2×C8C2×D4×C12712yes—
273C2×C8C2×D4×C12712yes—
274C2×C8C2×D4×C12711yescharacteristic
275C16—41no—
276C16—41no—
277C16—42no—
278C23—24no—
279C23—22no—
280C23C22⋊C8×C12712yes—
281C23C2×M16×C12711yescharacteristic
282Q8—24no—
283Q8—22no—
284Q8—22no—
285Q8C22⋊C8×C12714yes—
286D4—28no—
287D4—24no—
288D4—24no—
289D4—24no—
290D4—24no—
291D4—22no—
292D4—22no—
293D4C22⋊C8×C12714yes—
294C2×C4—24no—
295C2×C4—28no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4C22⋊C8×C12712yes—
302C2×C4C2×C22⋊C4×C12712yes—
303C2×C4C2×C22⋊C4×C12711yescharacteristic
304C2×C4C22×C8×C12711yescharacteristic
305C2×C4C2×M16×C12712yes—
306C8—24no—
307C8—22no—
308C8#120×C12711yescharacteristic
309C8#120×C12711yescharacteristic
310C8#120×C12712yes—
311C22—22no—
312C22—24no—
313C22—41no—
314C22—42no—
315C22—82no—
316C22—84no—
317C22C2×C22⋊C8×C12711yescharacteristic
318C22#407×C12712yes—
319C4—24no—
320C4—22no—
321C4#379×C12711yescharacteristic
322C4C2×C22⋊C8×C12712yes—
323C4#407×C12711yescharacteristic
324C2—21no—
325C2—22no—
326C2—42no—
327C2—44no—
328C2C127×#177911yescharacteristic
subgroupG/H
centerC8×C127#120
commutatorC2×C4C22×C8×C127
Frattini, cosocleC22×C8C23×C127
socleC2×C127#1779
prime radical#585×C127C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2117C2semidirect
3. C2×#350×C127C2semidirect
4. C127×#2115C2semidirect
5. C127×#2113C2semidirect
15. C2×M32×C127C22semidirect
16. C2×M32×C127C22semidirect
17. C22×C16×C127C22semidirect
159. #12685C127direct (N × H)
166. #2117C2×C127semidirect
167. C2×#350C2×C127semidirect
168. #2115C2×C127semidirect
169. #2113C2×C127semidirect
170. C127#12685direct (N × H)
180. C2×M32C22×C127semidirect
181. C2×M32C22×C127semidirect
182. C22×C16C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35732
C127×#21172
C2×#350×C1272
C127×#21152
C127×#21132
#12685127