Finite Group Databasebeta

← C127×#14489 #44578162

Subgroups — C127×#14489 #44578162

Order 32512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#391×C127C211yesmaximal, characteristic
2C127×#1446C212yesmaximal
3C127×#1443C214yesmaximal
4C127×#1436C212yesmaximal
5C127×#1434C214yesmaximal
6C127×#1434C212yesmaximal
7C23×Q8×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2211yescharacteristic
9C22×C42×C127C2212yes—
10C22×C42×C127C2211yescharacteristic
11#391×C127C418yescomplemented
12C2×#95×C127C2214yes—
13C2×#95×C127C2214yes—
14C2×#95×C127C2214yes—
15C2×#95×C127C2212yes—
16C2×#95×C127C2212yes—
17C2×#95×C127C2212yes—
18C2×#95×C127C2214yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2214yes—
22C22×Q8×C127—24no—
23C22×Q8×C127—42no—
24C22×Q8×C127—41no—
25C22×Q8×C127—41no—
26C22×Q8×C127C2×C414yes—
27C23×C4×C127C2314yes—
28C23×C4×C127C2312yes—
29C23×C4×C127C2312yes—
30C23×C4×C127C2312yes—
31C23×C4×C127C2314yes—
32C23×C4×C127C2311yescharacteristic
33C2×C4⋊C4×C127—41no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127C2×C414yes—
36C2×C42×C127—28no—
37C2×C42×C127—42no—
38C2×C42×C127—42no—
39C2×C42×C127C2×C414yes—
40#95×C127—28no—
41#95×C127—28no—
42#95×C127—28no—
43#95×C127—42no—
44#95×C127—44no—
45#95×C127—42no—
46#95×C127—42no—
47#95×C127—42no—
48#95×C127—42no—
49#95×C127C2×C4116yes—
50C24×C127C2411yescharacteristic
51C2×Q8×C127—28no—
52C2×Q8×C127—44no—
53C2×Q8×C127—44no—
54C2×Q8×C127—48no—
55C2×Q8×C127—48no—
56C22×C4×C127—22no—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—24no—
60C22×C4×C127—24no—
61C22×C4×C127—24no—
62C22×C4×C127—24no—
63C22×C4×C127—24no—
64C22×C4×C127—24no—
65C22×C4×C127—24no—
66C22×C4×C127—22no—
67C22×C4×C127—22no—
68C22×C4×C127—24no—
69C22×C4×C127—22no—
70C22×C4×C127—24no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—24no—
83C22×C4×C127—24no—
84C22×C4×C127C22×C414yes—
85C22×C4×C127C22×C414yes—
86C22×C4×C127C22×C414yes—
87C22×C4×C127C22×C412yes—
88C22×C4×C127#4118yes—
89C22×C4×C127#4114yes—
90C4⋊C4×C127—44no—
91C42×C127—44no—
92C42×C127—48no—
93C23×C127C23×C411yescharacteristic
94C23×C127C2×#4112yes—
95C23×C127C2×#4114yes—
96C23×C127H(5,𝔽2)12yes—
97C23×C127H(5,𝔽2)11yescharacteristic
98C23×C127H(5,𝔽2)11yescharacteristic
99C23×C1272−1+414yes—
100Q8×C127—416no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—24no—
104C2×C4×C127—24no—
105C2×C4×C127—24no—
106C2×C4×C127—24no—
107C2×C4×C127—28no—
108C2×C4×C127—24no—
109C2×C4×C127—24no—
110C2×C4×C127—24no—
111C2×C4×C127—24no—
112C2×C4×C127—24no—
113C2×C4×C127—28no—
114C2×C4×C127—28no—
115C2×C4×C127—28no—
116C2×C4×C127—44no—
117C2×C4×C127—44no—
118C2×C4×C127—44no—
119C2×C4×C127—44no—
120C2×C4×C127—44no—
121C2×C4×C127—44no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—44no—
127C2×C4×C127—44no—
128C2×C4×C127—44no—
129C2×C4×C127—44no—
130C2×C4×C127—44no—
131C2×C4×C127—44no—
132C2×C4×C127—44no—
133C2×C4×C127—44no—
134C2×C4×C127#11714yes—
135C22×C127C2×#11711yescharacteristic
136C22×C127C4×#4112yes—
137C22×C127#51811yescharacteristic
138C22×C127#51811yescharacteristic
139C22×C127#51912yes—
140C22×C127#53812yes—
141C22×C127#53812yes—
142C22×C127#53914yes—
143C22×C127#54214yes—
144C22×C127#55112yes—
145C22×C127#55214yes—
146C22×C127#55512yes—
147C22×C127#55614yes—
148C22×C127#56312yes—
149C22×C127#56312yes—
150C4×C127—24no—
151C4×C127—44no—
152C4×C127—44no—
153C4×C127—44no—
154C4×C127—48no—
155C4×C127—84no—
156#14489C12711yesmaximal, characteristic, direct product factor, Sylow
157C2×C127#231211yescharacteristic
158C2×C127#233212yes—
159C2×C127#235612yes—
160C2×C127#237211yescharacteristic
161C2×C127#238111yescharacteristic
162C2×C127#274514yes—
163C2×C127#277514yes—
164C2×#391C2×C12711yescharacteristic
165#1446C2×C12712yes—
166#1443C2×C12714yes—
167#1436C2×C12712yes—
168#1434C2×C12712yes—
169#1434C2×C12714yes—
170C127#1448911yescharacteristic, direct product factor, Sylow
171C23×Q8C22×C12711yescharacteristic
172C22×C4⋊C4C22×C12711yescharacteristic
173C22×C42C22×C12712yes—
174C22×C42C22×C12711yescharacteristic
175#391C4×C12718yescomplemented
176C2×#95C22×C12714yes—
177C2×#95C22×C12714yes—
178C2×#95C22×C12714yes—
179C2×#95C22×C12712yes—
180C2×#95C22×C12712yes—
181C2×#95C22×C12712yes—
182C2×#95C22×C12714yes—
183C2×#95C22×C12712yes—
184C2×#95C22×C12712yes—
185C2×#95C22×C12714yes—
186C22×Q8—24no—
187C22×Q8—42no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8C2×C4×C12714yes—
191C23×C4C23×C12714yes—
192C23×C4C23×C12712yes—
193C23×C4C23×C12712yes—
194C23×C4C23×C12712yes—
195C23×C4C23×C12714yes—
196C23×C4C23×C12711yescharacteristic
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C2×C4×C12714yes—
200C2×C42—28no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42C2×C4×C12714yes—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—42no—
208#95—44no—
209#95—42no—
210#95—42no—
211#95—42no—
212#95—42no—
213#95C2×C4×C127116yes—
214C24C24×C12711yescharacteristic
215C2×Q8—28no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—48no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4C22×C4×C12714yes—
249C22×C4C22×C4×C12714yes—
250C22×C4C22×C4×C12714yes—
251C22×C4C22×C4×C12712yes—
252C22×C4#41×C12718yes—
253C22×C4#41×C12714yes—
254C4⋊C4—44no—
255C42—44no—
256C42—48no—
257C23C23×C4×C12711yescharacteristic
258C23C2×#41×C12712yes—
259C23C2×#41×C12714yes—
260C23H(5,𝔽2)×C12712yes—
261C23H(5,𝔽2)×C12711yescharacteristic
262C23H(5,𝔽2)×C12711yescharacteristic
263C232−1+4×C12714yes—
264Q8—416no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—28no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4#117×C12714yes—
299C22C2×#117×C12711yescharacteristic
300C22C4×#41×C12712yes—
301C22#518×C12711yescharacteristic
302C22#518×C12711yescharacteristic
303C22#519×C12712yes—
304C22#538×C12712yes—
305C22#538×C12712yes—
306C22#539×C12714yes—
307C22#542×C12714yes—
308C22#551×C12712yes—
309C22#552×C12714yes—
310C22#555×C12712yes—
311C22#556×C12714yes—
312C22#563×C12712yes—
313C22#563×C12712yes—
314C4—24no—
315C4—44no—
316C4—44no—
317C4—44no—
318C4—48no—
319C4—84no—
320C2C127×#231211yescharacteristic
321C2C127×#233212yes—
322C2C127×#235612yes—
323C2C127×#237211yescharacteristic
324C2C127×#238111yescharacteristic
325C2C127×#274514yes—
326C2C127×#277514yes—
subgroupG/H
center, socle, prime radicalC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
11. #391×C127C4semidirect
156. #14489C127direct (N × H)
170. C127#14489direct (N × H)
175. #391C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#391×C1272
C127×#14462
C127×#14432
C127×#14362
C127×#14342
#14489127