Finite Group Databasebeta

← C127×#18526 #44582199

Subgroups — C127×#18526 #44582199

Order 32512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559×C127C211yesmaximal, characteristic, complemented
2C127×#3151C211yesmaximal, characteristic
3C127×#3112C212yesmaximal, complemented
4C127×#1630C214yesmaximal, complemented
5C127×#1620C214yesmaximal, complemented
6C127×#1555C211yesmaximal, characteristic, complemented
7C127×#1491C212yesmaximal, complemented
8C2×AGL(1,ℤ/8)×C127—22no—
9#559×C127—22no—
10#559×C127—24no—
11#559×C127C2214yes—
12#532×C127C2211yescharacteristic
13C2×#127×C127C2211yescharacteristic
14C2×#124×C127—22no—
15C2×#123×C127—22no—
16C2×C4×Q8×C127C2212yes—
17C2×C4×D4×C127—22no—
18#496×C127—22no—
19#493×C127C2212yes—
20#492×C127C2212yes—
21#480×C127—22no—
22#477×C127C2214yescomplemented
23#460×C127C2214yescomplemented
24#448×C127—22no—
25#440×C127—24no—
26#432×C127C2212yes—
27C4×M16×C127C2211yescharacteristic
28#331×C127C2214yes—
29Q8⋊C8×C127C2218yescomplemented
30C2×#41×C127—21no—
31C22×Q8×C127D412yes—
32C22×D4×C127—22no—
33C22×D4×C127D412yes—
34C23×C4×C127—22no—
35AGL(1,ℤ/8)×C127—48no—
36C2×SD16×C127—24no—
37C2×SD16×C127—24no—
38C2×D8×C127—24no—
39C2×M16×C127—21no—
40C4⋊Q8×C127C2311yescharacteristic
41#127×C127—22no—
42#127×C127D412yes—
43#127×C127C2311yescharacteristic
44#127×C127C2312yes—
45#124×C127—28no—
46#124×C127—28no—
47#124×C127—21no—
48#123×C127—28no—
49#123×C127—28no—
50#121×C127—22no—
51C4×Q8×C127—22no—
52C4×Q8×C127—22no—
53C4×Q8×C127D414yes—
54C4×Q8×C127C2314yes—
55C4×D4×C127—28no—
56C4×D4×C127—28no—
57C4×D4×C127—22no—
58C2×C4⋊C4×C127—22no—
59C2×C4⋊C4×C127D412yes—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—22no—
62C2×C42×C127—22no—
63C2×C42×C127C2311yescharacteristic
64#107×C127—24no—
65C4⋊C8×C127C2314yes—
66#103×C127—24no—
67#102×C127—24no—
68C22⋊C8×C127—22no—
69#97×C127—21no—
70#97×C127—22no—
71C4×C8×C127C2312yes—
72C24×C127—22no—
73#41×C127—44no—
74C2×Q8×C127—22no—
75C2×Q8×C127—21no—
76C2×Q8×C127—44no—
77C2×Q8×C127C2×D414yes—
78C2×D4×C127—24no—
79C2×D4×C127—22no—
80C2×D4×C127—24no—
81C2×D4×C127—22no—
82C2×D4×C127—21no—
83C2×D4×C127—48no—
84C2×D4×C127—44no—
85C2×D4×C127—41no—
86C2×D4×C127C2×D414yes—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—28no—
90C22×C4×C127—41no—
91C22×C4×C127—84no—
92C22×C4×C127C2×D412yes—
93C22×C4×C127C2×D411yescharacteristic
94SD16×C127—48no—
95SD16×C127—48no—
96D8×C127—48no—
97M16×C127—24no—
98C2×C8×C127—22no—
99C2×C8×C127—24no—
100C4⋊C4×C127—24no—
101C4⋊C4×C127—22no—
102C4⋊C4×C127—24no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127C2×D414yes—
105C22⋊C4×C127—216no—
106C22⋊C4×C127—28no—
107C22⋊C4×C127—42no—
108C42×C127—24no—
109C42×C127—24no—
110C42×C127C2×D412yes—
111C42×C127C2×D411yescharacteristic
112C42×C127C2411yescharacteristic
113C23×C127—24no—
114C23×C127—48no—
115C23×C127—44no—
116C23×C127—41no—
117C23×C127#12011yescharacteristic
118Q8×C127—24no—
119Q8×C127—24no—
120Q8×C127—44no—
121Q8×C127—42no—
122D4×C127—44no—
123D4×C127—48no—
124D4×C127—42no—
125D4×C127—48no—
126D4×C127—44no—
127D4×C127—82no—
128D4×C127—82no—
129C2×C4×C127—22no—
130C2×C4×C127—24no—
131C2×C4×C127—24no—
132C2×C4×C127—24no—
133C2×C4×C127—44no—
134C2×C4×C127—44no—
135C2×C4×C127—41no—
136C2×C4×C127—82no—
137C2×C4×C127—88no—
138C2×C4×C127#12012yes—
139C2×C4×C127#12011yescharacteristic
140C2×C4×C127C22×D411yescharacteristic
141C2×C4×C127C22×D412yes—
142C8×C127—24no—
143C8×C127—44no—
144C22×C127—22no—
145C22×C127—48no—
146C22×C127—44no—
147C22×C127—88no—
148C22×C127—82no—
149C22×C127—81no—
150C22×C127#45312yes—
151C22×C127C2×#12011yescharacteristic
152C4×C127—22no—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—44no—
156C4×C127—44no—
157C4×C127—81no—
158C4×C127#57514yes—
159#18526C12711yesmaximal, characteristic, direct product factor, Sylow
160C2×C127—22no—
161C2×C127—84no—
162C2×C127—161no—
163C2×C127#300512yes—
164C2×C127C2×#45311yescharacteristic
165C2×#559C2×C12711yescharacteristic, complemented
166#3151C2×C12711yescharacteristic
167#3112C2×C12712yescomplemented
168#1630C2×C12714yescomplemented
169#1620C2×C12714yescomplemented
170#1555C2×C12711yescharacteristic, complemented
171#1491C2×C12712yescomplemented
172C127#1852611yescharacteristic, direct product factor, Sylow
173C2×AGL(1,ℤ/8)—22no—
174#559—22no—
175#559—24no—
176#559C22×C12714yes—
177#532C22×C12711yescharacteristic
178C2×#127C22×C12711yescharacteristic
179C2×#124—22no—
180C2×#123—22no—
181C2×C4×Q8C22×C12712yes—
182C2×C4×D4—22no—
183#496—22no—
184#493C22×C12712yes—
185#492C22×C12712yes—
186#480—22no—
187#477C22×C12714yescomplemented
188#460C22×C12714yescomplemented
189#448—22no—
190#440—24no—
191#432C22×C12712yes—
192C4×M16C22×C12711yescharacteristic
193#331C22×C12714yes—
194Q8⋊C8C22×C12718yescomplemented
195C2×#41—21no—
196C22×Q8D4×C12712yes—
197C22×D4—22no—
198C22×D4D4×C12712yes—
199C23×C4—22no—
200AGL(1,ℤ/8)—48no—
201C2×SD16—24no—
202C2×SD16—24no—
203C2×D8—24no—
204C2×M16—21no—
205C4⋊Q8C23×C12711yescharacteristic
206#127—22no—
207#127D4×C12712yes—
208#127C23×C12711yescharacteristic
209#127C23×C12712yes—
210#124—28no—
211#124—28no—
212#124—21no—
213#123—28no—
214#123—28no—
215#121—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8D4×C12714yes—
219C4×Q8C23×C12714yes—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4D4×C12712yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42C23×C12711yescharacteristic
229#107—24no—
230C4⋊C8C23×C12714yes—
231#103—24no—
232#102—24no—
233C22⋊C8—22no—
234#97—21no—
235#97—22no—
236C4×C8C23×C12712yes—
237C24—22no—
238#41—44no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—44no—
242C2×Q8C2×D4×C12714yes—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—48no—
249C2×D4—44no—
250C2×D4—41no—
251C2×D4C2×D4×C12714yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—41no—
256C22×C4—84no—
257C22×C4C2×D4×C12712yes—
258C22×C4C2×D4×C12711yescharacteristic
259SD16—48no—
260SD16—48no—
261D8—48no—
262M16—24no—
263C2×C8—22no—
264C2×C8—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4C2×D4×C12714yes—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—42no—
273C42—24no—
274C42—24no—
275C42C2×D4×C12712yes—
276C42C2×D4×C12711yescharacteristic
277C42C24×C12711yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—41no—
282C23#120×C12711yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—44no—
286Q8—42no—
287D4—44no—
288D4—48no—
289D4—42no—
290D4—48no—
291D4—44no—
292D4—82no—
293D4—82no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—88no—
303C2×C4#120×C12712yes—
304C2×C4#120×C12711yescharacteristic
305C2×C4C22×D4×C12711yescharacteristic
306C2×C4C22×D4×C12712yes—
307C8—24no—
308C8—44no—
309C22—22no—
310C22—48no—
311C22—44no—
312C22—88no—
313C22—82no—
314C22—81no—
315C22#453×C12712yes—
316C22C2×#120×C12711yescharacteristic
317C4—22no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—44no—
322C4—81no—
323C4#575×C12714yes—
324C2—22no—
325C2—84no—
326C2—161no—
327C2C127×#300512yes—
328C2C2×#453×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC42C24×C127
prime radicalC127×#3151C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#559×C127C2semidirect
3. C127×#3112C2semidirect
4. C127×#1630C2semidirect
5. C127×#1620C2semidirect
6. C127×#1555C2semidirect
7. C127×#1491C2semidirect
22. #477×C127C22semidirect
23. #460×C127C22semidirect
29. Q8⋊C8×C127C22semidirect
159. #18526C127direct (N × H)
165. C2×#559C2×C127semidirect
167. #3112C2×C127semidirect
168. #1630C2×C127semidirect
169. #1620C2×C127semidirect
170. #1555C2×C127semidirect
171. #1491C2×C127semidirect
172. C127#18526direct (N × H)
187. #477C22×C127semidirect
188. #460C22×C127semidirect
194. Q8⋊C8C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#559×C1272
C127×#31512
C127×#31122
C127×#16302
C127×#16202
C127×#15552
C127×#14912
#18526127