Finite Group Databasebeta

← C127×#2697 #30635649

Subgroups — C127×#2697 #30635649

Order 16256.

680 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#123×C127C211yesmaximal, characteristic, complemented
2C2×#120×C127C211yesmaximal, characteristic
3#397×C127C212yesmaximal, complemented
4#396×C127C212yesmaximal, complemented
5#394×C127C212yesmaximal, complemented
6#390×C127C211yesmaximal, characteristic, complemented
7#388×C127C212yesmaximal, complemented
8#386×C127C212yesmaximal, complemented
9#379×C127C212yesmaximal, complemented
10C25×C127C2211yescharacteristic
11C22×D4×C127C2212yes—
12C22×D4×C127C2211yescharacteristic
13C23×C4×C127C2211yescharacteristic, complemented
14#123×C127—24no—
15#120×C127—24no—
16C2×C4⋊C4×C127C2212yescomplemented
17C2×C4⋊C4×C127C2211yescharacteristic, complemented
18C2×C4⋊C4×C127C2212yescomplemented
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2212yes—
24C2×C22⋊C4×C127C2212yes—
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C22⋊C4×C127C2211yescharacteristic, complemented
28C2×C22⋊C4×C127C2212yescomplemented
29C2×C42×C127C2211yescharacteristic, complemented
30#95×C127C2212yescomplemented
31#95×C127C2212yescomplemented
32#95×C127C2212yescomplemented
33C24×C127—22no—
34C24×C127—24no—
35C24×C127—42no—
36C24×C127—41no—
37C24×C127—41no—
38C24×C127C2311yescharacteristic
39C24×C127C2311yescharacteristic
40C24×C127C2312yes—
41C2×D4×C127—24no—
42C2×D4×C127—24no—
43C2×D4×C127—41no—
44C2×D4×C127—41no—
45C2×D4×C127—42no—
46C2×D4×C127D414yes—
47C22×C4×C127—22no—
48C22×C4×C127—41no—
49C22×C4×C127—41no—
50C22×C4×C127C2312yes—
51C22×C4×C127C2312yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2312yes—
54C22×C4×C127C2312yes—
55C22×C4×C127C2311yescharacteristic
56C4⋊C4×C127—24no—
57C4⋊C4×C127—41no—
58C4⋊C4×C127—42no—
59C22⋊C4×C127—22no—
60C22⋊C4×C127—24no—
61C22⋊C4×C127—22no—
62C22⋊C4×C127—24no—
63C22⋊C4×C127—24no—
64C22⋊C4×C127—42no—
65C22⋊C4×C127—42no—
66C22⋊C4×C127—42no—
67C22⋊C4×C127—42no—
68C22⋊C4×C127—42no—
69C42×C127—41no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—22no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127—22no—
84C23×C127—48no—
85C23×C127—44no—
86C23×C127—44no—
87C23×C127—48no—
88C23×C127—44no—
89C23×C127C2×D412yes—
90C23×C127C2×D412yes—
91C23×C127#4114yes—
92C23×C127C2411yescharacteristic
93D4×C127—44no—
94D4×C127—48no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—44no—
113C2×C4×C127C2×D412yes—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—24no—
117C22×C127—24no—
118C22×C127—24no—
119C22×C127—24no—
120C22×C127—24no—
121C22×C127—416no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127—42no—
127C22×C127—41no—
128C22×C127—41no—
129C22×C127—41no—
130C22×C127—41no—
131C22×C127—42no—
132C22×C127C22×D411yescharacteristic
133C22×C127C2×#4112yes—
134C22×C127H(5,𝔽2)11yescharacteristic
135C22×C127H(5,𝔽2)11yescharacteristic
136C22×C127H(5,𝔽2)12yes—
137C4×C127—22no—
138C4×C127—42no—
139C4×C127—42no—
140C4×C127—42no—
141C4×C127—42no—
142C4×C127—42no—
143C2×C127—42no—
144C2×C127—42no—
145C2×C127—44no—
146C2×C127#53411yescharacteristic
147C2×C127#53812yes—
148C2×C127#54612yes—
149C2×C127#55111yescharacteristic
150C2×C127#56011yescharacteristic
151#2697C12711yesmaximal, characteristic, direct product factor, Sylow
152C127#269711yescharacteristic, direct product factor, Sylow
153C2×#123C2×C12711yescharacteristic, complemented
154C2×#120C2×C12711yescharacteristic
155#397C2×C12712yescomplemented
156#396C2×C12712yescomplemented
157#394C2×C12712yescomplemented
158#390C2×C12711yescharacteristic, complemented
159#388C2×C12712yescomplemented
160#386C2×C12712yescomplemented
161#379C2×C12712yescomplemented
162C25C22×C12711yescharacteristic
163C22×D4C22×C12712yes—
164C22×D4C22×C12711yescharacteristic
165C23×C4C22×C12711yescharacteristic, complemented
166#123—24no—
167#120—24no—
168C2×C4⋊C4C22×C12712yescomplemented
169C2×C4⋊C4C22×C12711yescharacteristic, complemented
170C2×C4⋊C4C22×C12712yescomplemented
171C2×C22⋊C4C22×C12712yes—
172C2×C22⋊C4C22×C12711yescharacteristic
173C2×C22⋊C4C22×C12712yes—
174C2×C22⋊C4C22×C12712yes—
175C2×C22⋊C4C22×C12712yes—
176C2×C22⋊C4C22×C12712yes—
177C2×C22⋊C4C22×C12712yes—
178C2×C22⋊C4C22×C12712yes—
179C2×C22⋊C4C22×C12711yescharacteristic, complemented
180C2×C22⋊C4C22×C12712yescomplemented
181C2×C42C22×C12711yescharacteristic, complemented
182#95C22×C12712yescomplemented
183#95C22×C12712yescomplemented
184#95C22×C12712yescomplemented
185C24—22no—
186C24—24no—
187C24—42no—
188C24—41no—
189C24—41no—
190C24C23×C12711yescharacteristic
191C24C23×C12711yescharacteristic
192C24C23×C12712yes—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4D4×C12714yes—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4C23×C12712yes—
203C22×C4C23×C12712yes—
204C22×C4C23×C12712yes—
205C22×C4C23×C12712yes—
206C22×C4C23×C12712yes—
207C22×C4C23×C12711yescharacteristic
208C4⋊C4—24no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C22⋊C4—22no—
212C22⋊C4—24no—
213C22⋊C4—22no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—41no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—48no—
237C23—44no—
238C23—44no—
239C23—48no—
240C23—44no—
241C23C2×D4×C12712yes—
242C23C2×D4×C12712yes—
243C23#41×C12714yes—
244C23C24×C12711yescharacteristic
245D4—44no—
246D4—48no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—44no—
265C2×C4C2×D4×C12712yes—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—416no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—41no—
284C22C22×D4×C12711yescharacteristic
285C22C2×#41×C12712yes—
286C22H(5,𝔽2)×C12711yescharacteristic
287C22H(5,𝔽2)×C12711yescharacteristic
288C22H(5,𝔽2)×C12712yes—
289C4—22no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—42no—
294C4—42no—
295C2—42no—
296C2—44no—
297C2—42no—
298C2#534×C12711yescharacteristic
299C2#538×C12712yes—
300C2#546×C12712yes—
301C2#551×C12711yescharacteristic
302C2#560×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC2×#120×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#123×C127C2semidirect
3. #397×C127C2semidirect
4. #396×C127C2semidirect
5. #394×C127C2semidirect
6. #390×C127C2semidirect
7. #388×C127C2semidirect
8. #386×C127C2semidirect
9. #379×C127C2semidirect
13. C23×C4×C127C22semidirect
16. C2×C4⋊C4×C127C22semidirect
17. C2×C4⋊C4×C127C22semidirect
18. C2×C4⋊C4×C127C22semidirect
27. C2×C22⋊C4×C127C22semidirect
28. C2×C22⋊C4×C127C22semidirect
29. C2×C42×C127C22semidirect
30. #95×C127C22semidirect
31. #95×C127C22semidirect
32. #95×C127C22semidirect
151. #2697C127direct (N × H)
152. C127#2697direct (N × H)
153. C2×#123C2×C127semidirect
155. #397C2×C127semidirect
156. #396C2×C127semidirect
157. #394C2×C127semidirect
158. #390C2×C127semidirect
159. #388C2×C127semidirect
160. #386C2×C127semidirect
161. #379C2×C127semidirect
165. C23×C4C22×C127semidirect
168. C2×C4⋊C4C22×C127semidirect
169. C2×C4⋊C4C22×C127semidirect
170. C2×C4⋊C4C22×C127semidirect
179. C2×C22⋊C4C22×C127semidirect
180. C2×C22⋊C4C22×C127semidirect
181. C2×C42C22×C127semidirect
182. #95C22×C127semidirect
183. #95C22×C127semidirect
184. #95C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C1272
C2×#120×C1272
#397×C1272
#396×C1272
#394×C1272
#390×C1272
#388×C1272
#386×C1272
#379×C1272
#2697127