Finite Group Databasebeta

← C127×#2454 #30635406

Subgroups — C127×#2454 #30635406

Order 16256.

548 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×C127C211yesmaximal, characteristic, complemented
2C2×C4×D4×C127C211yesmaximal, characteristic
3#399×C127C212yesmaximal, complemented
4#396×C127C212yesmaximal, complemented
5#390×C127C211yesmaximal, characteristic
6#390×C127C211yesmaximal, characteristic
7#389×C127C211yesmaximal, characteristic, complemented
8#389×C127C211yesmaximal, characteristic, complemented
9#385×C127C212yesmaximal, complemented
10#380×C127C212yesmaximal, complemented
11C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
12C22×D4×C127C2211yescharacteristic
13C23×C4×C127C2212yes—
14#125×C127—24no—
15C4×D4×C127—24no—
16C2×C4⋊C4×C127C2212yes—
17C2×C4⋊C4×C127C2212yes—
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2212yes—
23C2×C4⋊C4×C127C2212yes—
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C22⋊C4×C127C2212yes—
28C2×C22⋊C4×C127C2212yes—
29C2×C22⋊C4×C127C2212yes—
30C2×C42×C127C2211yescharacteristic
31C2×C42×C127C2211yescharacteristic
32C2×C42×C127C2211yescharacteristic
33#95×C127C2211yescharacteristic
34#95×C127C2211yescharacteristic
35#95×C127C2212yes—
36#95×C127C2212yes—
37C24×C127C2312yes—
38C2×D4×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127Q814yes—
41C22×C4×C127—24no—
42C22×C4×C127—24no—
43C22×C4×C127—42no—
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2312yes—
46C22×C4×C127C2311yescharacteristic
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2311yescharacteristic
49C22×C4×C127C2312yes—
50C22×C4×C127C2312yes—
51C22×C4×C127C2311yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—24no—
55C4⋊C4×C127—24no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127—22no—
58C4⋊C4×C127—22no—
59C4⋊C4×C127—22no—
60C4⋊C4×C127—42no—
61C4⋊C4×C127—42no—
62C4⋊C4×C127D414yes—
63C22⋊C4×C127—24no—
64C22⋊C4×C127—24no—
65C22⋊C4×C127—42no—
66C22⋊C4×C127—42no—
67C22⋊C4×C127—42no—
68C42×C127—22no—
69C42×C127—22no—
70C42×C127—22no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127C2×Q814yes—
78C23×C127C2411yescharacteristic
79D4×C127—28no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—22no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—48no—
105C2×C4×C127C2×D414yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×Q812yes—
108C2×C4×C127#4112yes—
109C2×C4×C127#4112yes—
110C2×C4×C127#4112yes—
111C22×C127—24no—
112C22×C127—24no—
113C22×C127—24no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—42no—
117C22×C127—42no—
118C22×C127C22×D411yescharacteristic
119C22×C127C22×Q811yescharacteristic
120C22×C127C2×#4111yescharacteristic
121C22×C127C2×#4111yescharacteristic
122C22×C127C2×#4111yescharacteristic
123C22×C127H(5,𝔽2)11yescharacteristic
124C22×C127H(5,𝔽2)11yescharacteristic
125C4×C127—22no—
126C4×C127—22no—
127C4×C127—24no—
128C4×C127—41no—
129C4×C127—42no—
130C4×C127—41no—
131C4×C127—41no—
132C4×C127—41no—
133C4×C127—42no—
134C4×C127C22⋊Q814yes—
135C2×C127—44no—
136C2×C127C2×C22⋊Q811yescharacteristic
137C2×C127#52911yescharacteristic
138C2×C127#54011yescharacteristic
139C2×C127#54611yescharacteristic
140C2×C127#55011yescharacteristic
141C2×C127#55411yescharacteristic
142C2×C127#55411yescharacteristic
143#2454C12711yesmaximal, characteristic, direct product factor, Sylow
144C127#245411yescharacteristic, direct product factor, Sylow
145C2×#125C2×C12711yescharacteristic, complemented
146C2×C4×D4C2×C12711yescharacteristic
147#399C2×C12712yescomplemented
148#396C2×C12712yescomplemented
149#390C2×C12711yescharacteristic
150#390C2×C12711yescharacteristic
151#389C2×C12711yescharacteristic, complemented
152#389C2×C12711yescharacteristic, complemented
153#385C2×C12712yescomplemented
154#380C2×C12712yescomplemented
155C4×C4⋊C4C2×C12711yescharacteristic, complemented
156C22×D4C22×C12711yescharacteristic
157C23×C4C22×C12712yes—
158#125—24no—
159C4×D4—24no—
160C2×C4⋊C4C22×C12712yes—
161C2×C4⋊C4C22×C12712yes—
162C2×C4⋊C4C22×C12711yescharacteristic
163C2×C4⋊C4C22×C12711yescharacteristic
164C2×C4⋊C4C22×C12711yescharacteristic
165C2×C4⋊C4C22×C12711yescharacteristic
166C2×C4⋊C4C22×C12712yes—
167C2×C4⋊C4C22×C12712yes—
168C2×C4⋊C4C22×C12711yescharacteristic
169C2×C22⋊C4C22×C12712yes—
170C2×C22⋊C4C22×C12712yes—
171C2×C22⋊C4C22×C12712yes—
172C2×C22⋊C4C22×C12712yes—
173C2×C22⋊C4C22×C12712yes—
174C2×C42C22×C12711yescharacteristic
175C2×C42C22×C12711yescharacteristic
176C2×C42C22×C12711yescharacteristic
177#95C22×C12711yescharacteristic
178#95C22×C12711yescharacteristic
179#95C22×C12712yes—
180#95C22×C12712yes—
181C24C23×C12712yes—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4Q8×C12714yes—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—42no—
188C22×C4C23×C12711yescharacteristic
189C22×C4C23×C12712yes—
190C22×C4C23×C12711yescharacteristic
191C22×C4C23×C12711yescharacteristic
192C22×C4C23×C12711yescharacteristic
193C22×C4C23×C12712yes—
194C22×C4C23×C12712yes—
195C22×C4C23×C12711yescharacteristic
196C22×C4C23×C12711yescharacteristic
197C22×C4C23×C12711yescharacteristic
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4D4×C12714yes—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—22no—
213C42—22no—
214C42—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23C2×Q8×C12714yes—
222C23C24×C12711yescharacteristic
223D4—28no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—48no—
249C2×C4C2×D4×C12714yes—
250C2×C4C2×D4×C12712yes—
251C2×C4C2×Q8×C12712yes—
252C2×C4#41×C12712yes—
253C2×C4#41×C12712yes—
254C2×C4#41×C12712yes—
255C22—24no—
256C22—24no—
257C22—24no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22C22×D4×C12711yescharacteristic
263C22C22×Q8×C12711yescharacteristic
264C22C2×#41×C12711yescharacteristic
265C22C2×#41×C12711yescharacteristic
266C22C2×#41×C12711yescharacteristic
267C22H(5,𝔽2)×C12711yescharacteristic
268C22H(5,𝔽2)×C12711yescharacteristic
269C4—22no—
270C4—22no—
271C4—24no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—42no—
278C4C22⋊Q8×C12714yes—
279C2—44no—
280C2C2×C22⋊Q8×C12711yescharacteristic
281C2#529×C12711yescharacteristic
282C2#540×C12711yescharacteristic
283C2#546×C12711yescharacteristic
284C2#550×C12711yescharacteristic
285C2#554×C12711yescharacteristic
286C2#554×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC22×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#125×C127C2semidirect
3. #399×C127C2semidirect
4. #396×C127C2semidirect
7. #389×C127C2semidirect
8. #389×C127C2semidirect
9. #385×C127C2semidirect
10. #380×C127C2semidirect
11. C4×C4⋊C4×C127C2semidirect
143. #2454C127direct (N × H)
144. C127#2454direct (N × H)
145. C2×#125C2×C127semidirect
147. #399C2×C127semidirect
148. #396C2×C127semidirect
151. #389C2×C127semidirect
152. #389C2×C127semidirect
153. #385C2×C127semidirect
154. #380C2×C127semidirect
155. C4×C4⋊C4C2×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#125×C1272
C2×C4×D4×C1272
#399×C1272
#396×C1272
#390×C1272
#389×C1272
#385×C1272
#380×C1272
C4×C4⋊C4×C1272
#2454127