Finite Group Databasebeta

← C127×#2423 #30635375

Subgroups — C127×#2423 #30635375

Order 16256.

500 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×C127C211yesmaximal, characteristic
2C2×C4×Q8×C127C211yesmaximal, characteristic, complemented
3#399×C127C212yesmaximal
4#398×C127C212yesmaximal, complemented
5#391×C127C211yesmaximal, characteristic, complemented
6#389×C127C211yesmaximal, characteristic, complemented
7#388×C127C212yesmaximal
8#387×C127C212yesmaximal, complemented
9#384×C127C211yesmaximal, characteristic, complemented
10#380×C127C211yesmaximal, characteristic
11C4×C22⋊C4×C127C211yesmaximal, characteristic
12C22×Q8×C127C2211yescharacteristic
13C23×C4×C127C2211yescharacteristic
14C22⋊Q8×C127—24no—
15C4×Q8×C127—24no—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2212yes—
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2212yes—
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2212yes—
22C2×C4⋊C4×C127C2212yes—
23C2×C4⋊C4×C127C2212yes—
24C2×C22⋊C4×C127C2212yes—
25C2×C22⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2212yes—
27C2×C22⋊C4×C127C2211yescharacteristic
28C2×C42×C127C2211yescharacteristic
29C2×C42×C127C2212yes—
30C2×C42×C127C2211yescharacteristic
31#95×C127C2211yescharacteristic
32#95×C127C2211yescharacteristic
33#95×C127C2212yes—
34#95×C127C2212yes—
35#95×C127C2212yes—
36#95×C127C2212yes—
37C24×C127C2311yescharacteristic
38C2×Q8×C127—22no—
39C2×Q8×C127—22no—
40C2×Q8×C127D414yes—
41C22×C4×C127—22no—
42C22×C4×C127—22no—
43C22×C4×C127—22no—
44C22×C4×C127C2312yes—
45C22×C4×C127C2311yescharacteristic
46C22×C4×C127C2311yescharacteristic
47C22×C4×C127C2312yes—
48C22×C4×C127C2312yes—
49C22×C4×C127C2312yes—
50C22×C4×C127C2311yescharacteristic
51C22×C4×C127C2311yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—24no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127—24no—
58C4⋊C4×C127—24no—
59C4⋊C4×C127—22no—
60C4⋊C4×C127—42no—
61C4⋊C4×C127—42no—
62C22⋊C4×C127—24no—
63C22⋊C4×C127—22no—
64C22⋊C4×C127—42no—
65C22⋊C4×C127Q814yescomplemented
66C42×C127—24no—
67C42×C127—22no—
68C42×C127—22no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127—21no—
74C23×C127—21no—
75C23×C127C2×Q812yes—
76C23×C127C2411yescharacteristic
77Q8×C127—28no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—44no—
102C2×C4×C127C2×D414yes—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×Q812yes—
105C2×C4×C127C2×Q812yes—
106C2×C4×C127#4112yes—
107C2×C4×C127#4112yes—
108C2×C4×C127#4112yes—
109C22×C127—22no—
110C22×C127—22no—
111C22×C127—22no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127C22×D411yescharacteristic
117C22×C127C22×Q811yescharacteristic
118C22×C127C2×#4111yescharacteristic
119C22×C127C2×#4111yescharacteristic
120C22×C127C2×#4111yescharacteristic
121C22×C127H(5,𝔽2)11yescharacteristic
122C22×C1272−1+411yescharacteristic
123C4×C127—24no—
124C4×C127—22no—
125C4×C127—22no—
126C4×C127—22no—
127C4×C127—41no—
128C4×C127—42no—
129C4×C127—41no—
130C4×C127—42no—
131C4×C127—42no—
132C4×C127#12314yes—
133C2×C127—42no—
134C2×C127C2×#12311yescharacteristic
135C2×C127#53311yescharacteristic
136C2×C127#54211yescharacteristic
137C2×C127D4×Q811yescharacteristic
138C2×C127#55011yescharacteristic
139C2×C127#55211yescharacteristic
140C2×C127#55411yescharacteristic
141#2423C12711yesmaximal, characteristic, direct product factor, Sylow
142C127#242311yescharacteristic, direct product factor, Sylow
143C2×C22⋊Q8C2×C12711yescharacteristic
144C2×C4×Q8C2×C12711yescharacteristic, complemented
145#399C2×C12712yes—
146#398C2×C12712yescomplemented
147#391C2×C12711yescharacteristic, complemented
148#389C2×C12711yescharacteristic, complemented
149#388C2×C12712yes—
150#387C2×C12712yescomplemented
151#384C2×C12711yescharacteristic, complemented
152#380C2×C12711yescharacteristic
153C4×C22⋊C4C2×C12711yescharacteristic
154C22×Q8C22×C12711yescharacteristic
155C23×C4C22×C12711yescharacteristic
156C22⋊Q8—24no—
157C4×Q8—24no—
158C2×C4⋊C4C22×C12711yescharacteristic
159C2×C4⋊C4C22×C12712yes—
160C2×C4⋊C4C22×C12711yescharacteristic
161C2×C4⋊C4C22×C12712yes—
162C2×C4⋊C4C22×C12711yescharacteristic
163C2×C4⋊C4C22×C12712yes—
164C2×C4⋊C4C22×C12712yes—
165C2×C4⋊C4C22×C12712yes—
166C2×C22⋊C4C22×C12712yes—
167C2×C22⋊C4C22×C12711yescharacteristic
168C2×C22⋊C4C22×C12712yes—
169C2×C22⋊C4C22×C12711yescharacteristic
170C2×C42C22×C12711yescharacteristic
171C2×C42C22×C12712yes—
172C2×C42C22×C12711yescharacteristic
173#95C22×C12711yescharacteristic
174#95C22×C12711yescharacteristic
175#95C22×C12712yes—
176#95C22×C12712yes—
177#95C22×C12712yes—
178#95C22×C12712yes—
179C24C23×C12711yescharacteristic
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8D4×C12714yes—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4C23×C12712yes—
187C22×C4C23×C12711yescharacteristic
188C22×C4C23×C12711yescharacteristic
189C22×C4C23×C12712yes—
190C22×C4C23×C12712yes—
191C22×C4C23×C12712yes—
192C22×C4C23×C12711yescharacteristic
193C22×C4C23×C12711yescharacteristic
194C22×C4C23×C12711yescharacteristic
195C22×C4C23×C12711yescharacteristic
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C22⋊C4—24no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4Q8×C12714yescomplemented
208C42—24no—
209C42—22no—
210C42—22no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23C2×Q8×C12712yes—
218C23C24×C12711yescharacteristic
219Q8—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—44no—
244C2×C4C2×D4×C12714yes—
245C2×C4C2×D4×C12712yes—
246C2×C4C2×Q8×C12712yes—
247C2×C4C2×Q8×C12712yes—
248C2×C4#41×C12712yes—
249C2×C4#41×C12712yes—
250C2×C4#41×C12712yes—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22C22×D4×C12711yescharacteristic
259C22C22×Q8×C12711yescharacteristic
260C22C2×#41×C12711yescharacteristic
261C22C2×#41×C12711yescharacteristic
262C22C2×#41×C12711yescharacteristic
263C22H(5,𝔽2)×C12711yescharacteristic
264C222−1+4×C12711yescharacteristic
265C4—24no—
266C4—22no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—42no—
271C4—41no—
272C4—42no—
273C4—42no—
274C4#123×C12714yes—
275C2—42no—
276C2C2×#123×C12711yescharacteristic
277C2#533×C12711yescharacteristic
278C2#542×C12711yescharacteristic
279C2D4×Q8×C12711yescharacteristic
280C2#550×C12711yescharacteristic
281C2#552×C12711yescharacteristic
282C2#554×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC24×C127C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×Q8×C127C2semidirect
4. #398×C127C2semidirect
5. #391×C127C2semidirect
6. #389×C127C2semidirect
8. #387×C127C2semidirect
9. #384×C127C2semidirect
65. C22⋊C4×C127Q8semidirect
141. #2423C127direct (N × H)
142. C127#2423direct (N × H)
144. C2×C4×Q8C2×C127semidirect
146. #398C2×C127semidirect
147. #391C2×C127semidirect
148. #389C2×C127semidirect
150. #387C2×C127semidirect
151. #384C2×C127semidirect
207. C22⋊C4Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×C1272
C2×C4×Q8×C1272
#399×C1272
#398×C1272
#391×C1272
#389×C1272
#388×C1272
#387×C1272
#384×C1272
#380×C1272
C4×C22⋊C4×C1272
#2423127