Finite Group Databasebeta

← C127×#2361 #30635313

Subgroups — C127×#2361 #30635313

Order 16256.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×C127C211yesmaximal, characteristic
2C2×C4×D4×C127C211yesmaximal, characteristic, complemented
3C2×C4×D4×C127C212yesmaximal, complemented
4C22×C4⋊C4×C127C211yesmaximal, characteristic, complemented
5#390×C127C211yesmaximal, characteristic, complemented
6#389×C127C211yesmaximal, characteristic, complemented
7#388×C127C212yesmaximal, complemented
8#386×C127C212yesmaximal, complemented
9#385×C127C212yesmaximal, complemented
10C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
11C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
12C22×D4×C127C2212yes—
13C22×D4×C127C2211yescharacteristic
14C23×C4×C127C2212yes—
15C23×C4×C127C2212yes—
16C23×C4×C127C2211yescharacteristic
17#121×C127C418yescomplemented
18C4×D4×C127—24no—
19C4×D4×C127—28no—
20C2×C4⋊C4×C127—24no—
21C2×C4⋊C4×C127C2212yes—
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2212yes—
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C4⋊C4×C127C2211yescharacteristic
26C2×C22⋊C4×C127C2212yes—
27C2×C22⋊C4×C127C2212yes—
28C2×C22⋊C4×C127C2211yescharacteristic
29C2×C22⋊C4×C127C2212yes—
30C2×C22⋊C4×C127C2212yes—
31C2×C22⋊C4×C127C2211yescharacteristic
32C2×C42×C127C2211yescharacteristic
33C2×C42×C127C2212yes—
34C2×C42×C127C2211yescharacteristic
35#95×C127C2211yescharacteristic
36#95×C127C2211yescharacteristic
37#95×C127C2212yes—
38#95×C127C2212yes—
39C24×C127C2312yes—
40C24×C127C2311yescharacteristic
41C2×D4×C127—24no—
42C2×D4×C127—24no—
43C2×D4×C127—22no—
44C2×D4×C127—22no—
45C2×D4×C127C2×C418yescomplemented
46C2×D4×C127C2×C414yescomplemented
47C22×C4×C127—22no—
48C22×C4×C127—22no—
49C22×C4×C127—24no—
50C22×C4×C127—24no—
51C22×C4×C127—24no—
52C22×C4×C127—24no—
53C22×C4×C127—42no—
54C22×C4×C127—42no—
55C22×C4×C127C2×C414yescomplemented
56C22×C4×C127C2311yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C22×C4×C127C2312yes—
59C22×C4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61C22×C4×C127C2311yescharacteristic
62C22×C4×C127C2312yes—
63C22×C4×C127C2311yescharacteristic
64C22×C4×C127C2311yescharacteristic
65C4⋊C4×C127—24no—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—24no—
68C4⋊C4×C127C2×C414yescomplemented
69C4⋊C4×C127D414yescomplemented
70C22⋊C4×C127—22no—
71C22⋊C4×C127—24no—
72C22⋊C4×C127—24no—
73C22⋊C4×C127—42no—
74C22⋊C4×C127C2×C418yescomplemented
75C22⋊C4×C127D414yescomplemented
76C42×C127—24no—
77C42×C127—22no—
78C42×C127—22no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127C22×C414yes—
90C23×C127C22×C412yes—
91C23×C127C2×D412yes—
92C23×C127#4112yes—
93C23×C127C2411yescharacteristic
94D4×C127—216no—
95D4×C127—28no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—22no—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—28no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—22no—
115C2×C4×C127—21no—
116C2×C4×C127—48no—
117C2×C4×C127—48no—
118C2×C4×C127C22×C412yes—
119C2×C4×C127C22×C414yes—
120C2×C4×C127C22×C412yes—
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D414yes—
123C2×C4×C127C2×D412yes—
124C2×C4×C127C2×D412yes—
125C2×C4×C127#4112yes—
126C22×C127—22no—
127C22×C127—22no—
128C22×C127—22no—
129C22×C127—22no—
130C22×C127—22no—
131C22×C127—22no—
132C22×C127—24no—
133C22×C127—24no—
134C22×C127—24no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—42no—
139C22×C127C4×D414yescomplemented
140C22×C127C23×C411yescharacteristic
141C22×C127C22×D411yescharacteristic
142C22×C127C22×D411yescharacteristic
143C22×C127C2×#4111yescharacteristic
144C22×C127C2×#4111yescharacteristic
145C22×C127H(5,𝔽2)11yescharacteristic
146C22×C1272−1+411yescharacteristic
147C4×C127—22no—
148C4×C127—24no—
149C4×C127—22no—
150C4×C127—24no—
151C4×C127—22no—
152C4×C127—42no—
153C4×C127—41no—
154C4×C127—41no—
155C4×C127C4×D414yescomplemented
156C2×C127—24no—
157C2×C127—44no—
158C2×C127C2×C4×D411yescharacteristic
159C2×C127C2×C4×D411yescharacteristic
160C2×C127#52011yescharacteristic
161C2×C127D4211yescharacteristic
162C2×C127#54711yescharacteristic
163C2×C127#54811yescharacteristic
164C2×C127#55311yescharacteristic
165#2361C12711yesmaximal, characteristic, direct product factor, Sylow
166C127#236111yescharacteristic, direct product factor, Sylow
167C2×#121C2×C12711yescharacteristic
168C2×C4×D4C2×C12711yescharacteristic, complemented
169C2×C4×D4C2×C12712yescomplemented
170C22×C4⋊C4C2×C12711yescharacteristic, complemented
171#390C2×C12711yescharacteristic, complemented
172#389C2×C12711yescharacteristic, complemented
173#388C2×C12712yescomplemented
174#386C2×C12712yescomplemented
175#385C2×C12712yescomplemented
176C4×C4⋊C4C2×C12711yescharacteristic, complemented
177C4×C22⋊C4C2×C12711yescharacteristic, complemented
178C22×D4C22×C12712yes—
179C22×D4C22×C12711yescharacteristic
180C23×C4C22×C12712yes—
181C23×C4C22×C12712yes—
182C23×C4C22×C12711yescharacteristic
183#121C4×C12718yescomplemented
184C4×D4—24no—
185C4×D4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4C22×C12712yes—
188C2×C4⋊C4C22×C12711yescharacteristic
189C2×C4⋊C4C22×C12712yes—
190C2×C4⋊C4C22×C12711yescharacteristic
191C2×C4⋊C4C22×C12711yescharacteristic
192C2×C22⋊C4C22×C12712yes—
193C2×C22⋊C4C22×C12712yes—
194C2×C22⋊C4C22×C12711yescharacteristic
195C2×C22⋊C4C22×C12712yes—
196C2×C22⋊C4C22×C12712yes—
197C2×C22⋊C4C22×C12711yescharacteristic
198C2×C42C22×C12711yescharacteristic
199C2×C42C22×C12712yes—
200C2×C42C22×C12711yescharacteristic
201#95C22×C12711yescharacteristic
202#95C22×C12711yescharacteristic
203#95C22×C12712yes—
204#95C22×C12712yes—
205C24C23×C12712yes—
206C24C23×C12711yescharacteristic
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C2×C4×C12718yescomplemented
212C2×D4C2×C4×C12714yescomplemented
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C2×C4×C12714yescomplemented
222C22×C4C23×C12711yescharacteristic
223C22×C4C23×C12711yescharacteristic
224C22×C4C23×C12712yes—
225C22×C4C23×C12712yes—
226C22×C4C23×C12711yescharacteristic
227C22×C4C23×C12711yescharacteristic
228C22×C4C23×C12712yes—
229C22×C4C23×C12711yescharacteristic
230C22×C4C23×C12711yescharacteristic
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—24no—
234C4⋊C4C2×C4×C12714yescomplemented
235C4⋊C4D4×C12714yescomplemented
236C22⋊C4—22no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4C2×C4×C12718yescomplemented
241C22⋊C4D4×C12714yescomplemented
242C42—24no—
243C42—22no—
244C42—22no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23C22×C4×C12714yes—
256C23C22×C4×C12712yes—
257C23C2×D4×C12712yes—
258C23#41×C12712yes—
259C23C24×C12711yescharacteristic
260D4—216no—
261D4—28no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—28no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—48no—
283C2×C4—48no—
284C2×C4C22×C4×C12712yes—
285C2×C4C22×C4×C12714yes—
286C2×C4C22×C4×C12712yes—
287C2×C4C2×D4×C12712yes—
288C2×C4C2×D4×C12714yes—
289C2×C4C2×D4×C12712yes—
290C2×C4C2×D4×C12712yes—
291C2×C4#41×C12712yes—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—24no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22C4×D4×C12714yescomplemented
306C22C23×C4×C12711yescharacteristic
307C22C22×D4×C12711yescharacteristic
308C22C22×D4×C12711yescharacteristic
309C22C2×#41×C12711yescharacteristic
310C22C2×#41×C12711yescharacteristic
311C22H(5,𝔽2)×C12711yescharacteristic
312C222−1+4×C12711yescharacteristic
313C4—22no—
314C4—24no—
315C4—22no—
316C4—24no—
317C4—22no—
318C4—42no—
319C4—41no—
320C4—41no—
321C4C4×D4×C12714yescomplemented
322C2—24no—
323C2—44no—
324C2C2×C4×D4×C12711yescharacteristic
325C2C2×C4×D4×C12711yescharacteristic
326C2#520×C12711yescharacteristic
327C2D42×C12711yescharacteristic
328C2#547×C12711yescharacteristic
329C2#548×C12711yescharacteristic
330C2#553×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutatorC22C23×C4×C127
Frattini, cosocleC23C24×C127
prime radicalC2×#121×C127C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×D4×C127C2semidirect
3. C2×C4×D4×C127C2semidirect
4. C22×C4⋊C4×C127C2semidirect
5. #390×C127C2semidirect
6. #389×C127C2semidirect
7. #388×C127C2semidirect
8. #386×C127C2semidirect
9. #385×C127C2semidirect
10. C4×C4⋊C4×C127C2semidirect
11. C4×C22⋊C4×C127C2semidirect
17. #121×C127C4semidirect
45. C2×D4×C127C2×C4semidirect
46. C2×D4×C127C2×C4semidirect
55. C22×C4×C127C2×C4semidirect
68. C4⋊C4×C127C2×C4semidirect
69. C4⋊C4×C127D4semidirect
74. C22⋊C4×C127C2×C4semidirect
75. C22⋊C4×C127D4semidirect
139. C22×C127C4×D4semidirect
155. C4×C127C4×D4semidirect
165. #2361C127direct (N × H)
166. C127#2361direct (N × H)
168. C2×C4×D4C2×C127semidirect
169. C2×C4×D4C2×C127semidirect
170. C22×C4⋊C4C2×C127semidirect
171. #390C2×C127semidirect
172. #389C2×C127semidirect
173. #388C2×C127semidirect
174. #386C2×C127semidirect
175. #385C2×C127semidirect
176. C4×C4⋊C4C2×C127semidirect
177. C4×C22⋊C4C2×C127semidirect
183. #121C4×C127semidirect
211. C2×D4C2×C4×C127semidirect
212. C2×D4C2×C4×C127semidirect
221. C22×C4C2×C4×C127semidirect
234. C4⋊C4C2×C4×C127semidirect
235. C4⋊C4D4×C127semidirect
240. C22⋊C4C2×C4×C127semidirect
241. C22⋊C4D4×C127semidirect
305. C22C4×D4×C127semidirect
321. C4C4×D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C1272
C2×C4×D4×C1272
C22×C4⋊C4×C1272
#390×C1272
#389×C1272
#388×C1272
#386×C1272
#385×C1272
C4×C4⋊C4×C1272
C4×C22⋊C4×C1272
#2361127