Finite Group Databasebeta

← C127×#2342 #30635294

Subgroups — C127×#2342 #30635294

Order 16256.

792 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×C127C211yesmaximal, characteristic, complemented
2C22×C22⋊C4×C127C211yesmaximal, characteristic
3C22×C22⋊C4×C127C211yesmaximal, characteristic
4#388×C127C212yesmaximal, complemented
5#388×C127C212yesmaximal, complemented
6#385×C127C212yesmaximal, complemented
7#381×C127C211yesmaximal, characteristic, complemented
8#380×C127C211yesmaximal, characteristic, complemented
9#379×C127C211yesmaximal, characteristic
10C4×C22⋊C4×C127C212yesmaximal, complemented
11C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
12C25×C127C2211yescharacteristic
13C23×C4×C127C2212yes—
14C23×C4×C127C2211yescharacteristic
15C23×C4×C127C2211yescharacteristic
16#117×C127—24no—
17C2×C4⋊C4×C127C2212yescomplemented
18C2×C4⋊C4×C127C2212yescomplemented
19C2×C22⋊C4×C127—24no—
20C2×C22⋊C4×C127C418yescomplemented
21C2×C22⋊C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C22⋊C4×C127C2211yescharacteristic
28C2×C22⋊C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127C2211yescharacteristic
31C2×C22⋊C4×C127C2211yescharacteristic
32C2×C42×C127C2212yescomplemented
33C2×C42×C127C2212yescomplemented
34#95×C127C2212yescomplemented
35#95×C127C2212yescomplemented
36#95×C127C2212yescomplemented
37#95×C127C2212yescomplemented
38C24×C127—22no—
39C24×C127—24no—
40C24×C127—22no—
41C24×C127—42no—
42C24×C127C2×C414yes—
43C24×C127C2311yescharacteristic
44C24×C127C2311yescharacteristic
45C24×C127C2311yescharacteristic
46C22×C4×C127—22no—
47C22×C4×C127—22no—
48C22×C4×C127—24no—
49C22×C4×C127—24no—
50C22×C4×C127—42no—
51C22×C4×C127—42no—
52C22×C4×C127C2×C414yescomplemented
53C22×C4×C127C2×C414yescomplemented
54C22×C4×C127C2312yes—
55C22×C4×C127C2312yes—
56C22×C4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C22×C4×C127C2311yescharacteristic
61C22×C4×C127C2312yes—
62C4⋊C4×C127—24no—
63C4⋊C4×C127—42no—
64C22⋊C4×C127—22no—
65C22⋊C4×C127—22no—
66C22⋊C4×C127—24no—
67C22⋊C4×C127—22no—
68C22⋊C4×C127—24no—
69C22⋊C4×C127—42no—
70C22⋊C4×C127C2×C414yes—
71C22⋊C4×C127C2×C414yes—
72C22⋊C4×C127C2×C414yescomplemented
73C22⋊C4×C127C2×C414yescomplemented
74C22⋊C4×C127D414yes—
75C42×C127—24no—
76C42×C127—42no—
77C23×C127—21no—
78C23×C127—22no—
79C23×C127—21no—
80C23×C127—22no—
81C23×C127—22no—
82C23×C127—28no—
83C23×C127—28no—
84C23×C127—21no—
85C23×C127—22no—
86C23×C127—21no—
87C23×C127—22no—
88C23×C127—44no—
89C23×C127—48no—
90C23×C127—48no—
91C23×C127C22×C412yes—
92C23×C127C22×C412yes—
93C23×C127C22×C412yes—
94C23×C127C2×D412yes—
95C23×C127#4114yes—
96C23×C127#4112yes—
97C23×C127C2411yescharacteristic
98C2×C4×C127—28no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
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—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—44no—
117C2×C4×C127—48no—
118C2×C4×C127C22×C412yes—
119C2×C4×C127C22×C412yes—
120C2×C4×C127C22×C412yes—
121C2×C4×C127C22×C412yes—
122C2×C4×C127C2×D414yes—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127—24no—
128C22×C127—24no—
129C22×C127—24no—
130C22×C127—24no—
131C22×C127—22no—
132C22×C127—22no—
133C22×C127—22no—
134C22×C127—416no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127#11714yescomplemented
138C22×C127C4×D414yes—
139C22×C127C23×C411yescharacteristic
140C22×C127C22×D411yescharacteristic
141C22×C127C2×#4112yes—
142C22×C127C2×#4111yescharacteristic
143C22×C127H(5,𝔽2)12yes—
144C4×C127—22no—
145C4×C127—22no—
146C4×C127—22no—
147C4×C127—22no—
148C4×C127—24no—
149C4×C127—42no—
150C4×C127—42no—
151C4×C127—42no—
152C2×C127—24no—
153C2×C127—24no—
154C2×C127—42no—
155C2×C127C2×#11711yescharacteristic
156C2×C127C2×C4×D411yescharacteristic
157C2×C127#51811yescharacteristic
158C2×C127#54612yes—
159C2×C127#55112yes—
160#2342C12711yesmaximal, characteristic, direct product factor, Sylow
161C127#234211yescharacteristic, direct product factor, Sylow
162C2×#117C2×C12711yescharacteristic, complemented
163C22×C22⋊C4C2×C12711yescharacteristic
164C22×C22⋊C4C2×C12711yescharacteristic
165#388C2×C12712yescomplemented
166#388C2×C12712yescomplemented
167#385C2×C12712yescomplemented
168#381C2×C12711yescharacteristic, complemented
169#380C2×C12711yescharacteristic, complemented
170#379C2×C12711yescharacteristic
171C4×C22⋊C4C2×C12712yescomplemented
172C4×C22⋊C4C2×C12711yescharacteristic, complemented
173C25C22×C12711yescharacteristic
174C23×C4C22×C12712yes—
175C23×C4C22×C12711yescharacteristic
176C23×C4C22×C12711yescharacteristic
177#117—24no—
178C2×C4⋊C4C22×C12712yescomplemented
179C2×C4⋊C4C22×C12712yescomplemented
180C2×C22⋊C4—24no—
181C2×C22⋊C4C4×C12718yescomplemented
182C2×C22⋊C4C22×C12711yescharacteristic
183C2×C22⋊C4C22×C12712yes—
184C2×C22⋊C4C22×C12711yescharacteristic
185C2×C22⋊C4C22×C12711yescharacteristic
186C2×C22⋊C4C22×C12712yes—
187C2×C22⋊C4C22×C12712yes—
188C2×C22⋊C4C22×C12711yescharacteristic
189C2×C22⋊C4C22×C12711yescharacteristic
190C2×C22⋊C4C22×C12711yescharacteristic
191C2×C22⋊C4C22×C12711yescharacteristic
192C2×C22⋊C4C22×C12711yescharacteristic
193C2×C42C22×C12712yescomplemented
194C2×C42C22×C12712yescomplemented
195#95C22×C12712yescomplemented
196#95C22×C12712yescomplemented
197#95C22×C12712yescomplemented
198#95C22×C12712yescomplemented
199C24—22no—
200C24—24no—
201C24—22no—
202C24—42no—
203C24C2×C4×C12714yes—
204C24C23×C12711yescharacteristic
205C24C23×C12711yescharacteristic
206C24C23×C12711yescharacteristic
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C2×C4×C12714yescomplemented
214C22×C4C2×C4×C12714yescomplemented
215C22×C4C23×C12712yes—
216C22×C4C23×C12712yes—
217C22×C4C23×C12712yes—
218C22×C4C23×C12711yescharacteristic
219C22×C4C23×C12711yescharacteristic
220C22×C4C23×C12711yescharacteristic
221C22×C4C23×C12711yescharacteristic
222C22×C4C23×C12712yes—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—24no—
228C22⋊C4—22no—
229C22⋊C4—24no—
230C22⋊C4—42no—
231C22⋊C4C2×C4×C12714yes—
232C22⋊C4C2×C4×C12714yes—
233C22⋊C4C2×C4×C12714yescomplemented
234C22⋊C4C2×C4×C12714yescomplemented
235C22⋊C4D4×C12714yes—
236C42—24no—
237C42—42no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—28no—
244C23—28no—
245C23—21no—
246C23—22no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—48no—
251C23—48no—
252C23C22×C4×C12712yes—
253C23C22×C4×C12712yes—
254C23C22×C4×C12712yes—
255C23C2×D4×C12712yes—
256C23#41×C12714yes—
257C23#41×C12712yes—
258C23C24×C12711yescharacteristic
259C2×C4—28no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—44no—
278C2×C4—48no—
279C2×C4C22×C4×C12712yes—
280C2×C4C22×C4×C12712yes—
281C2×C4C22×C4×C12712yes—
282C2×C4C22×C4×C12712yes—
283C2×C4C2×D4×C12714yes—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—416no—
296C22—42no—
297C22—42no—
298C22#117×C12714yescomplemented
299C22C4×D4×C12714yes—
300C22C23×C4×C12711yescharacteristic
301C22C22×D4×C12711yescharacteristic
302C22C2×#41×C12712yes—
303C22C2×#41×C12711yescharacteristic
304C22H(5,𝔽2)×C12712yes—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—24no—
310C4—42no—
311C4—42no—
312C4—42no—
313C2—24no—
314C2—24no—
315C2—42no—
316C2C2×#117×C12711yescharacteristic
317C2C2×C4×D4×C12711yescharacteristic
318C2#518×C12711yescharacteristic
319C2#546×C12712yes—
320C2#551×C12712yes—
subgroupG/H
center, socleC23×C127C24
commutatorC22C23×C4×C127
Frattini, cosocleC23C24×C127
prime radicalC25×C127C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#117×C127C2semidirect
4. #388×C127C2semidirect
5. #388×C127C2semidirect
6. #385×C127C2semidirect
7. #381×C127C2semidirect
8. #380×C127C2semidirect
10. C4×C22⋊C4×C127C2semidirect
11. C4×C22⋊C4×C127C2semidirect
17. C2×C4⋊C4×C127C22semidirect
18. C2×C4⋊C4×C127C22semidirect
20. C2×C22⋊C4×C127C4semidirect
32. C2×C42×C127C22semidirect
33. C2×C42×C127C22semidirect
34. #95×C127C22semidirect
35. #95×C127C22semidirect
36. #95×C127C22semidirect
37. #95×C127C22semidirect
52. C22×C4×C127C2×C4semidirect
53. C22×C4×C127C2×C4semidirect
72. C22⋊C4×C127C2×C4semidirect
73. C22⋊C4×C127C2×C4semidirect
137. C22×C127#117semidirect
160. #2342C127direct (N × H)
161. C127#2342direct (N × H)
162. C2×#117C2×C127semidirect
165. #388C2×C127semidirect
166. #388C2×C127semidirect
167. #385C2×C127semidirect
168. #381C2×C127semidirect
169. #380C2×C127semidirect
171. C4×C22⋊C4C2×C127semidirect
172. C4×C22⋊C4C2×C127semidirect
178. C2×C4⋊C4C22×C127semidirect
179. C2×C4⋊C4C22×C127semidirect
181. C2×C22⋊C4C4×C127semidirect
193. C2×C42C22×C127semidirect
194. C2×C42C22×C127semidirect
195. #95C22×C127semidirect
196. #95C22×C127semidirect
197. #95C22×C127semidirect
198. #95C22×C127semidirect
213. C22×C4C2×C4×C127semidirect
214. C22×C4C2×C4×C127semidirect
233. C22⋊C4C2×C4×C127semidirect
234. C22⋊C4C2×C4×C127semidirect
298. C22#117×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#117×C1272
C22×C22⋊C4×C1272
#388×C1272
#385×C1272
#381×C1272
#380×C1272
#379×C1272
C4×C22⋊C4×C1272
#2342127