Finite Group Databasebeta

← C127×#3056 #30636008

Subgroups — C127×#3056 #30636008

Order 16256.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#137×C127C211yesmaximal, characteristic, complemented
2C22×M16×C127C211yesmaximal, characteristic, complemented
3#525×C127C211yesmaximal, characteristic
4C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
5#470×C127C212yesmaximal, complemented
6#468×C127C212yesmaximal, complemented
7#467×C127C212yesmaximal, complemented
8#465×C127C212yesmaximal, complemented
9#428×C127C211yesmaximal, characteristic, complemented
10#419×C127C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#41×C127C2211yescharacteristic
13C22×Q8×C127C2211yescharacteristic
14C23×C4×C127C2211yescharacteristic
15#137×C127—24no—
16C2×Q16×C127C2212yescomplemented
17C2×SD16×C127C2212yescomplemented
18C2×M16×C127—24no—
19C2×M16×C127C2211yescharacteristic
20C2×M16×C127C2212yes—
21C2×M16×C127C2211yescharacteristic
22C22×C8×C127C2212yes—
23#123×C127—21no—
24C22⋊Q8×C127—21no—
25C22⋊Q8×C127—21no—
26C22⋊Q8×C127C2212yes—
27C22⋊Q8×C127C2212yes—
28C22⋊Q8×C127C2211yescharacteristic
29C22⋊Q8×C127C2211yescharacteristic
30#121×C127C2211yescharacteristic
31#121×C127C2211yescharacteristic
32#120×C127—21no—
33C4×D4×C127—21no—
34C4×D4×C127—21no—
35#117×C127C2211yescharacteristic
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127C2211yescharacteristic
38C2×C22⋊C4×C127—21no—
39#107×C127C2212yescomplemented
40#106×C127C2212yescomplemented
41#103×C127C2212yescomplemented
42#103×C127C2212yescomplemented
43#103×C127C2212yescomplemented
44#102×C127C2212yescomplemented
45C24×C127D411yescharacteristic
46#41×C127—22no—
47#41×C127—22no—
48C2×Q8×C127—21no—
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127C2312yes—
52C2×Q8×C127C2311yescharacteristic
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127C2311yescharacteristic
57C22×C4×C127—22no—
58C22×C4×C127—22no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127D411yescharacteristic
63C22×C4×C127D411yescharacteristic
64C22×C4×C127D411yescharacteristic
65C22×C4×C127C2311yescharacteristic
66C22×C4×C127C2311yescharacteristic
67C22×C4×C127C2311yescharacteristic
68Q16×C127—24no—
69SD16×C127—24no—
70M16×C127—22no—
71M16×C127—24no—
72M16×C127D414yescomplemented
73C2×C8×C127—24no—
74C2×C8×C127C2312yes—
75C2×C8×C127C2312yes—
76C4⋊C4×C127—22no—
77C4⋊C4×C127—22no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127—21no—
80C4⋊C4×C127—21no—
81C4⋊C4×C127C2312yes—
82C4⋊C4×C127C2311yescharacteristic
83C4⋊C4×C127C2311yescharacteristic
84C22⋊C4×C127—22no—
85C22⋊C4×C127—22no—
86C22⋊C4×C127—21no—
87C22⋊C4×C127—21no—
88C22⋊C4×C127—21no—
89C22⋊C4×C127—21no—
90C22⋊C4×C127—21no—
91C42×C127—21no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—21no—
95C23×C127—41no—
96C23×C127C2×D411yescharacteristic
97C23×C127C2×D411yescharacteristic
98C23×C127C2×D411yescharacteristic
99Q8×C127—22no—
100Q8×C127—21no—
101Q8×C127—22no—
102Q8×C127—21no—
103Q8×C127—42no—
104D4×C127—21no—
105D4×C127—21no—
106D4×C127—42no—
107D4×C127—42no—
108D4×C127—42no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—22no—
119C2×C4×C127—21no—
120C2×C4×C127—22no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127C2×D411yescharacteristic
125C2×C4×C127C2×D411yescharacteristic
126C2×C4×C127C2×D411yescharacteristic
127C2×C4×C127C2×D412yes—
128C2×C4×C127#4112yes—
129C2×C4×C127C2411yescharacteristic
130C8×C127—22no—
131C8×C127C2×D414yes—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—44no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127#12112yes—
144C22×C127#13712yescomplemented
145C22×C127C22×D411yescharacteristic
146C4×C127—21no—
147C4×C127—21no—
148C4×C127—41no—
149C4×C127—42no—
150C4×C127—41no—
151C4×C127—42no—
152C4×C127—41no—
153C4×C127#12112yes—
154C4×C127C22×D411yescharacteristic
155C4×C127C2×#4111yescharacteristic
156C2×C127—22no—
157C2×C127—22no—
158C2×C127—41no—
159C2×C127—81no—
160C2×C127C2×#12111yescharacteristic
161C2×C127C2×#13711yescharacteristic
162C2×C127#57511yescharacteristic
163#3056C12711yesmaximal, characteristic, direct product factor, Sylow
164C127#305611yescharacteristic, direct product factor, Sylow
165C2×#137C2×C12711yescharacteristic, complemented
166C22×M16C2×C12711yescharacteristic, complemented
167#525C2×C12711yescharacteristic
168C2×C22⋊Q8C2×C12711yescharacteristic, complemented
169#470C2×C12712yescomplemented
170#468C2×C12712yescomplemented
171#467C2×C12712yescomplemented
172#465C2×C12712yescomplemented
173#428C2×C12711yescharacteristic, complemented
174#419C2×C12711yescharacteristic, complemented
175#417C2×C12711yescharacteristic, complemented
176C2×#41C22×C12711yescharacteristic
177C22×Q8C22×C12711yescharacteristic
178C23×C4C22×C12711yescharacteristic
179#137—24no—
180C2×Q16C22×C12712yescomplemented
181C2×SD16C22×C12712yescomplemented
182C2×M16—24no—
183C2×M16C22×C12711yescharacteristic
184C2×M16C22×C12712yes—
185C2×M16C22×C12711yescharacteristic
186C22×C8C22×C12712yes—
187#123—21no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8C22×C12712yes—
191C22⋊Q8C22×C12712yes—
192C22⋊Q8C22×C12711yescharacteristic
193C22⋊Q8C22×C12711yescharacteristic
194#121C22×C12711yescharacteristic
195#121C22×C12711yescharacteristic
196#120—21no—
197C4×D4—21no—
198C4×D4—21no—
199#117C22×C12711yescharacteristic
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22×C12711yescharacteristic
202C2×C22⋊C4—21no—
203#107C22×C12712yescomplemented
204#106C22×C12712yescomplemented
205#103C22×C12712yescomplemented
206#103C22×C12712yescomplemented
207#103C22×C12712yescomplemented
208#102C22×C12712yescomplemented
209C24D4×C12711yescharacteristic
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C23×C12712yes—
216C2×Q8C23×C12711yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4C23×C12711yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4D4×C12711yescharacteristic
227C22×C4D4×C12711yescharacteristic
228C22×C4D4×C12711yescharacteristic
229C22×C4C23×C12711yescharacteristic
230C22×C4C23×C12711yescharacteristic
231C22×C4C23×C12711yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—22no—
235M16—24no—
236M16D4×C12714yescomplemented
237C2×C8—24no—
238C2×C8C23×C12712yes—
239C2×C8C23×C12712yes—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4C23×C12712yes—
246C4⋊C4C23×C12711yescharacteristic
247C4⋊C4C23×C12711yescharacteristic
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C42—21no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—41no—
260C23C2×D4×C12711yescharacteristic
261C23C2×D4×C12711yescharacteristic
262C23C2×D4×C12711yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—42no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4C2×D4×C12711yescharacteristic
289C2×C4C2×D4×C12711yescharacteristic
290C2×C4C2×D4×C12711yescharacteristic
291C2×C4C2×D4×C12712yes—
292C2×C4#41×C12712yes—
293C2×C4C24×C12711yescharacteristic
294C8—22no—
295C8C2×D4×C12714yes—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22#121×C12712yes—
308C22#137×C12712yescomplemented
309C22C22×D4×C12711yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—42no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4#121×C12712yes—
318C4C22×D4×C12711yescharacteristic
319C4C2×#41×C12711yescharacteristic
320C2—22no—
321C2—22no—
322C2—41no—
323C2—81no—
324C2C2×#121×C12711yescharacteristic
325C2C2×#137×C12711yescharacteristic
326C2#575×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C22×D4
commutator, Frattini, cosocleC2×C4C24×C127
prime radical#525×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#137×C127C2semidirect
2. C22×M16×C127C2semidirect
4. C2×C22⋊Q8×C127C2semidirect
5. #470×C127C2semidirect
6. #468×C127C2semidirect
7. #467×C127C2semidirect
8. #465×C127C2semidirect
9. #428×C127C2semidirect
10. #419×C127C2semidirect
11. #417×C127C2semidirect
16. C2×Q16×C127C22semidirect
17. C2×SD16×C127C22semidirect
39. #107×C127C22semidirect
40. #106×C127C22semidirect
41. #103×C127C22semidirect
42. #103×C127C22semidirect
43. #103×C127C22semidirect
44. #102×C127C22semidirect
72. M16×C127D4semidirect
144. C22×C127#137semidirect
163. #3056C127direct (N × H)
164. C127#3056direct (N × H)
165. C2×#137C2×C127semidirect
166. C22×M16C2×C127semidirect
168. C2×C22⋊Q8C2×C127semidirect
169. #470C2×C127semidirect
170. #468C2×C127semidirect
171. #467C2×C127semidirect
172. #465C2×C127semidirect
173. #428C2×C127semidirect
174. #419C2×C127semidirect
175. #417C2×C127semidirect
180. C2×Q16C22×C127semidirect
181. C2×SD16C22×C127semidirect
203. #107C22×C127semidirect
204. #106C22×C127semidirect
205. #103C22×C127semidirect
206. #103C22×C127semidirect
207. #103C22×C127semidirect
208. #102C22×C127semidirect
236. M16D4×C127semidirect
308. C22#137×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#137×C1272
C22×M16×C1272
#525×C1272
C2×C22⋊Q8×C1272
#470×C1272
#468×C1272
#467×C1272
#465×C1272
#428×C1272
#419×C1272
#417×C1272
#3056127