Finite Group Databasebeta

← C2×C127×#1620 #44582151

Subgroups — C2×C127×#1620 #44582151

Order 32512.

1190 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
2C2×#493×C127C211yesmaximal, characteristic
3C2×#460×C127C212yesmaximal, complemented
4C127×#1620C218yesmaximal, direct product factor
5C2×#331×C127C211yesmaximal, characteristic, complemented
6C2×Q8⋊C8×C127C212yesmaximal, complemented
7C22×SD16×C127—22no—
8C22×D8×C127—22no—
9#559×C127—22no—
10#559×C127—24no—
11#559×C127C2214yescomplemented
12C2×#127×C127C2211yescharacteristic
13C2×#127×C127C2211yescharacteristic
14C2×#124×C127—22no—
15C2×#123×C127—22no—
16C2×C4×Q8×C127C2212yes—
17C2×C4×D4×C127—22no—
18#493×C127—22no—
19#493×C127C2214yes—
20#460×C127—24no—
21#460×C127C2218yescomplemented
22C2×C4⋊C8×C127C2212yes—
23C2×#102×C127—22no—
24C2×C4×C8×C127C2211yescharacteristic
25#331×C127C2214yescomplemented
26Q8⋊C8×C127C2218yescomplemented
27C22×Q8×C127D412yes—
28C22×D4×C127—22no—
29C22×D4×C127—21no—
30C22×D4×C127—22no—
31C22×D4×C127D412yes—
32C23×C4×C127—22no—
33C2×SD16×C127—28no—
34C2×SD16×C127—48no—
35C2×D8×C127—28no—
36C2×D8×C127—48no—
37C22×C8×C127—21no—
38C22×C8×C127—22no—
39#127×C127—22no—
40#127×C127—22no—
41#127×C127—21no—
42#127×C127D412yes—
43#127×C127C2312yes—
44#127×C127C2312yes—
45#124×C127—28no—
46#124×C127—28no—
47#123×C127—28no—
48#123×C127—28no—
49C4×Q8×C127—22no—
50C4×Q8×C127—22no—
51C4×Q8×C127D414yes—
52C4×Q8×C127C2314yes—
53C4×D4×C127—28no—
54C4×D4×C127—28no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127D412yes—
57C2×C22⋊C4×C127—24no—
58C2×C22⋊C4×C127—22no—
59C2×C42×C127—22no—
60C2×C42×C127C2311yescharacteristic
61C4⋊C8×C127—22no—
62C4⋊C8×C127C2314yes—
63#102×C127—28no—
64C4×C8×C127—21no—
65C4×C8×C127C2312yes—
66C24×C127—22no—
67C24×C127—41no—
68C2×Q8×C127—22no—
69C2×Q8×C127—24no—
70C2×Q8×C127—24no—
71C2×Q8×C127C2×D414yes—
72C2×D4×C127—24no—
73C2×D4×C127—22no—
74C2×D4×C127—24no—
75C2×D4×C127—24no—
76C2×D4×C127—22no—
77C2×D4×C127—44no—
78C2×D4×C127—44no—
79C2×D4×C127—41no—
80C2×D4×C127—42no—
81C2×D4×C127—44no—
82C2×D4×C127—44no—
83C2×D4×C127—44no—
84C2×D4×C127—44no—
85C2×D4×C127C2×D414yes—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—28no—
89C22×C4×C127—84no—
90C22×C4×C127C2×D411yescharacteristic
91C22×C4×C127C2×D412yes—
92SD16×C127—416no—
93D8×C127—416no—
94C2×C8×C127—24no—
95C2×C8×C127—22no—
96C2×C8×C127—24no—
97C2×C8×C127—44no—
98C4⋊C4×C127—24no—
99C4⋊C4×C127—22no—
100C4⋊C4×C127—24no—
101C4⋊C4×C127C2×D414yes—
102C22⋊C4×C127—216no—
103C22⋊C4×C127—28no—
104C42×C127—24no—
105C42×C127—24no—
106C42×C127C2×D411yescharacteristic
107C42×C127C2×D412yes—
108C42×C127C2411yescharacteristic
109C23×C127—24no—
110C23×C127—48no—
111C23×C127—44no—
112C23×C127—42no—
113C23×C127—84no—
114C23×C127—82no—
115C23×C127#12011yescharacteristic
116Q8×C127—24no—
117Q8×C127—24no—
118Q8×C127—44no—
119D4×C127—44no—
120D4×C127—44no—
121D4×C127—44no—
122D4×C127—44no—
123D4×C127—44no—
124D4×C127—44no—
125D4×C127—48no—
126D4×C127—82no—
127D4×C127—82no—
128D4×C127—84no—
129C2×C4×C127—22no—
130C2×C4×C127—24no—
131C2×C4×C127—24no—
132C2×C4×C127—44no—
133C2×C4×C127—44no—
134C2×C4×C127—88no—
135C2×C4×C127#12011yescharacteristic
136C2×C4×C127#12012yes—
137C2×C4×C127#13514yes—
138C2×C4×C127AGL(1,ℤ/8)14yes—
139C2×C4×C127C22×D411yescharacteristic
140C2×C4×C127C22×D412yes—
141C8×C127—24no—
142C8×C127—44no—
143C22×C127—48no—
144C22×C127—44no—
145C22×C127—88no—
146C22×C127—84no—
147C22×C127—82no—
148C22×C127—164no—
149C22×C127#44914yes—
150C22×C127#45312yes—
151C22×C127C2×#12011yescharacteristic
152C4×C127—21no—
153C4×C127—21no—
154C4×C127—44no—
155C4×C127—44no—
156C4×C127#44914yes—
157C4×C127C2×#13512yes—
158C4×C127C2×AGL(1,ℤ/8)12yes—
159C2×#1620C12711yesmaximal, characteristic, direct product factor, Sylow
160C2×C127—84no—
161C2×C127—162no—
162C2×C127#162014yesdirect product factor
163C2×C127C2×#44912yes—
164C2×C127C2×#45311yescharacteristic
165C2×#559C2×C12711yescharacteristic, complemented
166C2×#493C2×C12711yescharacteristic
167C2×#460C2×C12712yescomplemented
168#1620C2×C12718yesdirect product factor
169C2×#331C2×C12711yescharacteristic, complemented
170C2×Q8⋊C8C2×C12712yescomplemented
171C127C2×#162011yescharacteristic, direct product factor, Sylow
172C22×SD16—22no—
173C22×D8—22no—
174#559—22no—
175#559—24no—
176#559C22×C12714yescomplemented
177C2×#127C22×C12711yescharacteristic
178C2×#127C22×C12711yescharacteristic
179C2×#124—22no—
180C2×#123—22no—
181C2×C4×Q8C22×C12712yes—
182C2×C4×D4—22no—
183#493—22no—
184#493C22×C12714yes—
185#460—24no—
186#460C22×C12718yescomplemented
187C2×C4⋊C8C22×C12712yes—
188C2×#102—22no—
189C2×C4×C8C22×C12711yescharacteristic
190#331C22×C12714yescomplemented
191Q8⋊C8C22×C12718yescomplemented
192C22×Q8D4×C12712yes—
193C22×D4—22no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4D4×C12712yes—
197C23×C4—22no—
198C2×SD16—28no—
199C2×SD16—48no—
200C2×D8—28no—
201C2×D8—48no—
202C22×C8—21no—
203C22×C8—22no—
204#127—22no—
205#127—22no—
206#127—21no—
207#127D4×C12712yes—
208#127C23×C12712yes—
209#127C23×C12712yes—
210#124—28no—
211#124—28no—
212#123—28no—
213#123—28no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8D4×C12714yes—
217C4×Q8C23×C12714yes—
218C4×D4—28no—
219C4×D4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4D4×C12712yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42C23×C12711yescharacteristic
226C4⋊C8—22no—
227C4⋊C8C23×C12714yes—
228#102—28no—
229C4×C8—21no—
230C4×C8C23×C12712yes—
231C24—22no—
232C24—41no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8C2×D4×C12714yes—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4C2×D4×C12714yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—84no—
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4C2×D4×C12712yes—
257SD16—416no—
258D8—416no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—44no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4C2×D4×C12714yes—
267C22⋊C4—216no—
268C22⋊C4—28no—
269C42—24no—
270C42—24no—
271C42C2×D4×C12711yescharacteristic
272C42C2×D4×C12712yes—
273C42C24×C12711yescharacteristic
274C23—24no—
275C23—48no—
276C23—44no—
277C23—42no—
278C23—84no—
279C23—82no—
280C23#120×C12711yescharacteristic
281Q8—24no—
282Q8—24no—
283Q8—44no—
284D4—44no—
285D4—44no—
286D4—44no—
287D4—44no—
288D4—44no—
289D4—44no—
290D4—48no—
291D4—82no—
292D4—82no—
293D4—84no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—88no—
300C2×C4#120×C12711yescharacteristic
301C2×C4#120×C12712yes—
302C2×C4#135×C12714yes—
303C2×C4AGL(1,ℤ/8)×C12714yes—
304C2×C4C22×D4×C12711yescharacteristic
305C2×C4C22×D4×C12712yes—
306C8—24no—
307C8—44no—
308C22—48no—
309C22—44no—
310C22—88no—
311C22—84no—
312C22—82no—
313C22—164no—
314C22#449×C12714yes—
315C22#453×C12712yes—
316C22C2×#120×C12711yescharacteristic
317C4—21no—
318C4—21no—
319C4—44no—
320C4—44no—
321C4#449×C12714yes—
322C4C2×#135×C12712yes—
323C4C2×AGL(1,ℤ/8)×C12712yes—
324C2—84no—
325C2—162no—
326C2C127×#162014yesdirect product factor
327C2C2×#449×C12712yes—
328C2C2×#453×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC42C24×C127
prime radicalC2×#493×C127C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#559×C127C2semidirect
3. C2×#460×C127C2semidirect
4. C127×#1620C2direct (N × H)
5. C2×#331×C127C2semidirect
6. C2×Q8⋊C8×C127C2semidirect
11. #559×C127C22semidirect
21. #460×C127C22semidirect
25. #331×C127C22semidirect
26. Q8⋊C8×C127C22semidirect
159. C2×#1620C127direct (N × H)
162. C2×C127#1620direct (N × H)
165. C2×#559C2×C127semidirect
167. C2×#460C2×C127semidirect
168. #1620C2×C127direct (N × H)
169. C2×#331C2×C127semidirect
170. C2×Q8⋊C8C2×C127semidirect
171. C127C2×#1620direct (N × H)
176. #559C22×C127semidirect
186. #460C22×C127semidirect
190. #331C22×C127semidirect
191. Q8⋊C8C22×C127semidirect
326. C2C127×#1620direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#559×C1272
C2×#493×C1272
C2×#460×C1272
C127×#16202
C2×#331×C1272
C2×Q8⋊C8×C1272
C2×#1620127