Finite Group Databasebeta

← C2×C127×#1533 #44581329

Subgroups — C2×C127×#1533 #44581329

Order 32512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124×C127C211yesmaximal, characteristic
2C2×#432×C127C212yesmaximal, complemented
3C127×#1533C218yesmaximal, direct product factor
4C2×#329×C127C214yesmaximal, complemented
5C23×Q8×C127C411yescharacteristic
6C23×D4×C127C411yescharacteristic
7C2×#124×C127—22no—
8C2×#124×C127—24no—
9C2×#124×C127C412yes—
10C2×#124×C127C414yes—
11C2×#124×C127C2212yes—
12C2×#124×C127C2214yes—
13C22×C22⋊C4×C127—22no—
14C22×C42×C127C2211yescharacteristic
15#432×C127—24no—
16#432×C127C2218yescomplemented
17C2×C4⋊C8×C127—22no—
18C2×C22⋊C8×C127—22no—
19C2×#97×C127C2214yescomplemented
20#329×C127C22116yescomplemented
21C25×C127—21no—
22C22×Q8×C127—22no—
23C22×Q8×C127—21no—
24C22×Q8×C127—44no—
25C22×Q8×C127C2×C414yes—
26C22×Q8×C127C2×C412yes—
27C22×D4×C127—22no—
28C22×D4×C127—21no—
29C22×D4×C127—44no—
30C22×D4×C127C2×C414yes—
31C22×D4×C127C2×C412yes—
32C23×C4×C127—21no—
33C23×C4×C127C2×C411yescharacteristic
34C23×C4×C127D412yes—
35C22×C8×C127—24no—
36#124×C127—28no—
37#124×C127—24no—
38#124×C127—24no—
39#124×C127—28no—
40#124×C127C2×C414yes—
41#124×C127C2×C418yes—
42#124×C127C2314yes—
43C2×C22⋊C4×C127—28no—
44C2×C22⋊C4×C127—24no—
45C2×C22⋊C4×C127—28no—
46C2×C22⋊C4×C127—24no—
47C2×C42×C127C2×C411yescharacteristic
48C2×C42×C127C2×C412yes—
49C2×C42×C127D412yes—
50C2×C42×C127D414yes—
51C2×C42×C127C2311yescharacteristic
52C2×C42×C127C2312yes—
53C4⋊C8×C127—28no—
54C22⋊C8×C127—28no—
55#97×C127—24no—
56#97×C127C2318yescomplemented
57C24×C127—24no—
58C24×C127—22no—
59C24×C127—48no—
60C24×C127—44no—
61C24×C127C22⋊C411yescharacteristic
62C2×Q8×C127—22no—
63C2×Q8×C127—24no—
64C2×Q8×C127—48no—
65C2×Q8×C127—44no—
66C2×Q8×C127—48no—
67C2×Q8×C127—44no—
68C2×Q8×C127C22×C414yes—
69C2×D4×C127—22no—
70C2×D4×C127—24no—
71C2×D4×C127—48no—
72C2×D4×C127—44no—
73C2×D4×C127—48no—
74C2×D4×C127—44no—
75C2×D4×C127C22×C414yes—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—22no—
79C22×C4×C127—44no—
80C22×C4×C127—44no—
81C22×C4×C127C22⋊C412yes—
82C22×C4×C127C22⋊C412yes—
83C22×C4×C127C22⋊C411yescharacteristic
84C22×C4×C127C22⋊C414yes—
85C22×C4×C127C22×C412yes—
86C22×C4×C127C22×C411yescharacteristic
87C22×C4×C127C2×D412yes—
88C22×C4×C127C2×D414yes—
89C2×C8×C127—28no—
90C2×C8×C127—48no—
91C22⋊C4×C127—28no—
92C22⋊C4×C127—216no—
93C22⋊C4×C127—28no—
94C42×C127C22⋊C414yes—
95C42×C127C22⋊C412yes—
96C42×C127C22×C411yescharacteristic
97C42×C127C22×C412yes—
98C42×C127C2×D412yes—
99C42×C127C2×D414yes—
100C42×C127C2411yescharacteristic
101C23×C127—24no—
102C23×C127—24no—
103C23×C127—416no—
104C23×C127—48no—
105C23×C127—44no—
106C23×C127—48no—
107C23×C127—816no—
108C23×C127#10014yes—
109C23×C127C2×C22⋊C412yes—
110C23×C127C2×C22⋊C411yescharacteristic
111Q8×C127—44no—
112Q8×C127—44no—
113Q8×C127—48no—
114D4×C127—44no—
115D4×C127—44no—
116D4×C127—48no—
117C2×C4×C127—24no—
118C2×C4×C127—22no—
119C2×C4×C127—24no—
120C2×C4×C127—22no—
121C2×C4×C127—24no—
122C2×C4×C127—24no—
123C2×C4×C127—24no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—48no—
127C2×C4×C127—44no—
128C2×C4×C127C2×C22⋊C412yes—
129C2×C4×C127C2×C22⋊C414yes—
130C2×C4×C127C2×C22⋊C411yescharacteristic
131C2×C4×C127C2×C22⋊C412yes—
132C2×C4×C127C23×C411yescharacteristic
133C2×C4×C127C22×D412yes—
134C8×C127—48no—
135C22×C127—24no—
136C22×C127—28no—
137C22×C127—44no—
138C22×C127—48no—
139C22×C127—816no—
140C22×C127—88no—
141C22×C127C2×#10014yes—
142C22×C127C2×#10012yes—
143C22×C127#42114yes—
144C22×C127C22×C22⋊C411yescharacteristic
145C4×C127—21no—
146C4×C127—22no—
147C4×C127—21no—
148C4×C127—22no—
149C4×C127—22no—
150C4×C127—24no—
151C4×C127—44no—
152C2×#1533C12711yesmaximal, characteristic, direct product factor, Sylow
153C2×C127—24no—
154C2×C127—84no—
155C2×C127#153314yesdirect product factor
156C2×C127C22×#10011yescharacteristic
157C2×C127C2×#42112yes—
158C22×#124C2×C12711yescharacteristic
159C2×#432C2×C12712yescomplemented
160#1533C2×C12718yesdirect product factor
161C2×#329C2×C12714yescomplemented
162C127C2×#153311yescharacteristic, direct product factor, Sylow
163C23×Q8C4×C12711yescharacteristic
164C23×D4C4×C12711yescharacteristic
165C2×#124—22no—
166C2×#124—24no—
167C2×#124C4×C12712yes—
168C2×#124C4×C12714yes—
169C2×#124C22×C12712yes—
170C2×#124C22×C12714yes—
171C22×C22⋊C4—22no—
172C22×C42C22×C12711yescharacteristic
173#432—24no—
174#432C22×C12718yescomplemented
175C2×C4⋊C8—22no—
176C2×C22⋊C8—22no—
177C2×#97C22×C12714yescomplemented
178#329C22×C127116yescomplemented
179C25—21no—
180C22×Q8—22no—
181C22×Q8—21no—
182C22×Q8—44no—
183C22×Q8C2×C4×C12712yes—
184C22×Q8C2×C4×C12714yes—
185C22×D4—22no—
186C22×D4—21no—
187C22×D4—44no—
188C22×D4C2×C4×C12712yes—
189C22×D4C2×C4×C12714yes—
190C23×C4—21no—
191C23×C4C2×C4×C12711yescharacteristic
192C23×C4D4×C12712yes—
193C22×C8—24no—
194#124—28no—
195#124—24no—
196#124—24no—
197#124—28no—
198#124C2×C4×C12714yes—
199#124C2×C4×C12718yes—
200#124C23×C12714yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C42C2×C4×C12711yescharacteristic
206C2×C42C2×C4×C12712yes—
207C2×C42D4×C12712yes—
208C2×C42D4×C12714yes—
209C2×C42C23×C12711yescharacteristic
210C2×C42C23×C12712yes—
211C4⋊C8—28no—
212C22⋊C8—28no—
213#97—24no—
214#97C23×C12718yescomplemented
215C24—24no—
216C24—22no—
217C24—48no—
218C24—44no—
219C24C22⋊C4×C12711yescharacteristic
220C2×Q8—22no—
221C2×Q8—24no—
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—48no—
225C2×Q8—44no—
226C2×Q8C22×C4×C12714yes—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—48no—
230C2×D4—44no—
231C2×D4—48no—
232C2×D4—44no—
233C2×D4C22×C4×C12714yes—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4C22⋊C4×C12712yes—
240C22×C4C22⋊C4×C12712yes—
241C22×C4C22⋊C4×C12711yescharacteristic
242C22×C4C22⋊C4×C12714yes—
243C22×C4C22×C4×C12712yes—
244C22×C4C22×C4×C12711yescharacteristic
245C22×C4C2×D4×C12712yes—
246C22×C4C2×D4×C12714yes—
247C2×C8—28no—
248C2×C8—48no—
249C22⋊C4—28no—
250C22⋊C4—216no—
251C22⋊C4—28no—
252C42C22⋊C4×C12714yes—
253C42C22⋊C4×C12712yes—
254C42C22×C4×C12711yescharacteristic
255C42C22×C4×C12712yes—
256C42C2×D4×C12712yes—
257C42C2×D4×C12714yes—
258C42C24×C12711yescharacteristic
259C23—24no—
260C23—24no—
261C23—416no—
262C23—48no—
263C23—44no—
264C23—48no—
265C23—816no—
266C23#100×C12714yes—
267C23C2×C22⋊C4×C12712yes—
268C23C2×C22⋊C4×C12711yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—48no—
272D4—44no—
273D4—44no—
274D4—48no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4C2×C22⋊C4×C12712yes—
287C2×C4C2×C22⋊C4×C12714yes—
288C2×C4C2×C22⋊C4×C12711yescharacteristic
289C2×C4C2×C22⋊C4×C12712yes—
290C2×C4C23×C4×C12711yescharacteristic
291C2×C4C22×D4×C12712yes—
292C8—48no—
293C22—24no—
294C22—28no—
295C22—44no—
296C22—48no—
297C22—816no—
298C22—88no—
299C22C2×#100×C12714yes—
300C22C2×#100×C12712yes—
301C22#421×C12714yes—
302C22C22×C22⋊C4×C12711yescharacteristic
303C4—21no—
304C4—22no—
305C4—21no—
306C4—22no—
307C4—22no—
308C4—24no—
309C4—44no—
310C2—24no—
311C2—84no—
312C2C127×#153314yesdirect product factor
313C2C22×#100×C12711yescharacteristic
314C2C2×#421×C12712yes—
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC42C24×C127
prime radicalC23×D4×C127C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#432×C127C2semidirect
3. C127×#1533C2direct (N × H)
4. C2×#329×C127C2semidirect
16. #432×C127C22semidirect
19. C2×#97×C127C22semidirect
20. #329×C127C22semidirect
56. #97×C127C23semidirect
152. C2×#1533C127direct (N × H)
155. C2×C127#1533direct (N × H)
159. C2×#432C2×C127semidirect
160. #1533C2×C127direct (N × H)
161. C2×#329C2×C127semidirect
162. C127C2×#1533direct (N × H)
174. #432C22×C127semidirect
177. C2×#97C22×C127semidirect
178. #329C22×C127semidirect
214. #97C23×C127semidirect
312. C2C127×#1533direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#124×C1272
C2×#432×C1272
C127×#15332
C2×#329×C1272
C2×#1533127