Finite Group Databasebeta

← C127×#2534 #30635486

Subgroups — C127×#2534 #30635486

Order 16256.

800 conjugacy classes of subgroups.

The subgroup lattice has 340 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
3C2×C4×D4×C127C211yesmaximal, characteristic, complemented
4C22×C22⋊C4×C127C211yesmaximal, characteristic, complemented
5#396×C127C212yesmaximal, complemented
6#394×C127C212yesmaximal, complemented
7#388×C127C212yesmaximal, complemented
8#386×C127C212yesmaximal, complemented
9#381×C127C211yesmaximal, characteristic, complemented
10#380×C127C211yesmaximal, characteristic, complemented
11#379×C127C211yesmaximal, characteristic, complemented
12C25×C127C2211yescharacteristic
13C22×D4×C127C2212yes—
14C22×D4×C127C2211yescharacteristic
15C23×C4×C127C2211yescharacteristic
16C23×C4×C127C2211yescharacteristic
17C23×C4×C127C2211yescharacteristic, complemented
18#123×C127—24no—
19#120×C127—24no—
20C4×D4×C127—24no—
21C2×C4⋊C4×C127C2211yescharacteristic, complemented
22C2×C4⋊C4×C127C2212yescomplemented
23C2×C4⋊C4×C127C2212yescomplemented
24C2×C22⋊C4×C127—24no—
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C22⋊C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2212yes—
30C2×C22⋊C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127C2212yes—
32C2×C22⋊C4×C127C2212yes—
33C2×C22⋊C4×C127C2211yescharacteristic, complemented
34C2×C22⋊C4×C127C2212yescomplemented
35C2×C42×C127C2211yescharacteristic, complemented
36#95×C127C2212yescomplemented
37#95×C127C2212yescomplemented
38#95×C127C2212yescomplemented
39C24×C127—22no—
40C24×C127—24no—
41C24×C127—22no—
42C24×C127—22no—
43C24×C127—42no—
44C24×C127C2311yescharacteristic
45C24×C127C2312yes—
46C24×C127C2311yescharacteristic
47C2×D4×C127—22no—
48C2×D4×C127—22no—
49C2×D4×C127—24no—
50C2×D4×C127—24no—
51C2×D4×C127—42no—
52C2×D4×C127D414yes—
53C22×C4×C127—22no—
54C22×C4×C127—22no—
55C22×C4×C127—22no—
56C22×C4×C127—22no—
57C22×C4×C127—42no—
58C22×C4×C127—42no—
59C22×C4×C127D414yescomplemented
60C22×C4×C127C2312yes—
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2311yescharacteristic
63C22×C4×C127C2312yes—
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C22×C4×C127C2311yescharacteristic
67C22×C4×C127C2311yescharacteristic
68C4⋊C4×C127—22no—
69C4⋊C4×C127—24no—
70C4⋊C4×C127—42no—
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—22no—
80C22⋊C4×C127—42no—
81C42×C127—22no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127—28no—
85C23×C127—22no—
86C23×C127—21no—
87C23×C127—22no—
88C23×C127—21no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—22no—
95C23×C127—48no—
96C23×C127—48no—
97C23×C127—44no—
98C23×C127—44no—
99C23×C127C2×D412yes—
100C23×C127C2×D412yes—
101C23×C127C2×D412yes—
102C23×C127#4114yes—
103C23×C127#4112yes—
104C23×C127C2411yescharacteristic
105D4×C127—28no—
106D4×C127—48no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—21no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—44no—
127C2×C4×C127C2×D412yes—
128C2×C4×C127C2×D412yes—
129C2×C4×C127C2×D412yes—
130C22×C127—22no—
131C22×C127—22no—
132C22×C127—24no—
133C22×C127—24no—
134C22×C127—22no—
135C22×C127—24no—
136C22×C127—24no—
137C22×C127—24no—
138C22×C127—24no—
139C22×C127—416no—
140C22×C127—42no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—42no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—42no—
147C22×C127#12114yes—
148C22×C127C22×D411yescharacteristic
149C22×C127C22×D411yescharacteristic
150C22×C127C2×#4111yescharacteristic
151C22×C127C2×#4112yes—
152C22×C127H(5,𝔽2)12yes—
153C4×C127—22no—
154C4×C127—22no—
155C4×C127—22no—
156C4×C127—42no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—42no—
160C4×C127—42no—
161C2×C127—24no—
162C2×C127—42no—
163C2×C127—44no—
164C2×C127C2×#12111yescharacteristic
165C2×C127#52511yescharacteristic
166C2×C127#53411yescharacteristic
167C2×C127#54612yes—
168C2×C127#55112yes—
169#2534C12711yesmaximal, characteristic, direct product factor, Sylow
170C127#253411yescharacteristic, direct product factor, Sylow
171C2×#123C2×C12711yescharacteristic, complemented
172C2×#120C2×C12711yescharacteristic
173C2×C4×D4C2×C12711yescharacteristic, complemented
174C22×C22⋊C4C2×C12711yescharacteristic, complemented
175#396C2×C12712yescomplemented
176#394C2×C12712yescomplemented
177#388C2×C12712yescomplemented
178#386C2×C12712yescomplemented
179#381C2×C12711yescharacteristic, complemented
180#380C2×C12711yescharacteristic, complemented
181#379C2×C12711yescharacteristic, complemented
182C25C22×C12711yescharacteristic
183C22×D4C22×C12712yes—
184C22×D4C22×C12711yescharacteristic
185C23×C4C22×C12711yescharacteristic
186C23×C4C22×C12711yescharacteristic
187C23×C4C22×C12711yescharacteristic, complemented
188#123—24no—
189#120—24no—
190C4×D4—24no—
191C2×C4⋊C4C22×C12711yescharacteristic, complemented
192C2×C4⋊C4C22×C12712yescomplemented
193C2×C4⋊C4C22×C12712yescomplemented
194C2×C22⋊C4—24no—
195C2×C22⋊C4C22×C12712yes—
196C2×C22⋊C4C22×C12712yes—
197C2×C22⋊C4C22×C12711yescharacteristic
198C2×C22⋊C4C22×C12711yescharacteristic
199C2×C22⋊C4C22×C12712yes—
200C2×C22⋊C4C22×C12711yescharacteristic
201C2×C22⋊C4C22×C12712yes—
202C2×C22⋊C4C22×C12712yes—
203C2×C22⋊C4C22×C12711yescharacteristic, complemented
204C2×C22⋊C4C22×C12712yescomplemented
205C2×C42C22×C12711yescharacteristic, complemented
206#95C22×C12712yescomplemented
207#95C22×C12712yescomplemented
208#95C22×C12712yescomplemented
209C24—22no—
210C24—24no—
211C24—22no—
212C24—22no—
213C24—42no—
214C24C23×C12711yescharacteristic
215C24C23×C12712yes—
216C24C23×C12711yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—42no—
222C2×D4D4×C12714yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4D4×C12714yescomplemented
230C22×C4C23×C12712yes—
231C22×C4C23×C12711yescharacteristic
232C22×C4C23×C12711yescharacteristic
233C22×C4C23×C12712yes—
234C22×C4C23×C12712yes—
235C22×C4C23×C12711yescharacteristic
236C22×C4C23×C12711yescharacteristic
237C22×C4C23×C12711yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—42no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—22no—
250C22⋊C4—42no—
251C42—22no—
252C23—22no—
253C23—22no—
254C23—28no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—48no—
266C23—48no—
267C23—44no—
268C23—44no—
269C23C2×D4×C12712yes—
270C23C2×D4×C12712yes—
271C23C2×D4×C12712yes—
272C23#41×C12714yes—
273C23#41×C12712yes—
274C23C24×C12711yescharacteristic
275D4—28no—
276D4—48no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4C2×D4×C12712yes—
298C2×C4C2×D4×C12712yes—
299C2×C4C2×D4×C12712yes—
300C22—22no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22#121×C12714yes—
318C22C22×D4×C12711yescharacteristic
319C22C22×D4×C12711yescharacteristic
320C22C2×#41×C12711yescharacteristic
321C22C2×#41×C12712yes—
322C22H(5,𝔽2)×C12712yes—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—42no—
330C4—42no—
331C2—24no—
332C2—44no—
333C2—42no—
334C2C2×#121×C12711yescharacteristic
335C2#525×C12711yescharacteristic
336C2#534×C12711yescharacteristic
337C2#546×C12712yes—
338C2#551×C12712yes—
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. C2×C4×D4×C127C2semidirect
4. C22×C22⋊C4×C127C2semidirect
5. #396×C127C2semidirect
6. #394×C127C2semidirect
7. #388×C127C2semidirect
8. #386×C127C2semidirect
9. #381×C127C2semidirect
10. #380×C127C2semidirect
11. #379×C127C2semidirect
17. C23×C4×C127C22semidirect
21. C2×C4⋊C4×C127C22semidirect
22. C2×C4⋊C4×C127C22semidirect
23. C2×C4⋊C4×C127C22semidirect
33. C2×C22⋊C4×C127C22semidirect
34. C2×C22⋊C4×C127C22semidirect
35. C2×C42×C127C22semidirect
36. #95×C127C22semidirect
37. #95×C127C22semidirect
38. #95×C127C22semidirect
59. C22×C4×C127D4semidirect
169. #2534C127direct (N × H)
170. C127#2534direct (N × H)
171. C2×#123C2×C127semidirect
173. C2×C4×D4C2×C127semidirect
174. C22×C22⋊C4C2×C127semidirect
175. #396C2×C127semidirect
176. #394C2×C127semidirect
177. #388C2×C127semidirect
178. #386C2×C127semidirect
179. #381C2×C127semidirect
180. #380C2×C127semidirect
181. #379C2×C127semidirect
187. C23×C4C22×C127semidirect
191. C2×C4⋊C4C22×C127semidirect
192. C2×C4⋊C4C22×C127semidirect
193. C2×C4⋊C4C22×C127semidirect
203. C2×C22⋊C4C22×C127semidirect
204. C2×C22⋊C4C22×C127semidirect
205. C2×C42C22×C127semidirect
206. #95C22×C127semidirect
207. #95C22×C127semidirect
208. #95C22×C127semidirect
229. C22×C4D4×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
C2×C4×D4×C1272
C22×C22⋊C4×C1272
#396×C1272
#394×C1272
#388×C1272
#386×C1272
#381×C1272
#380×C1272
#379×C1272
#2534127