Finite Group Databasebeta

← C127×#20688 #44584361

Subgroups — C127×#20688 #44584361

Order 32512.

1510 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
1C22×#117×C127C211yesmaximal, characteristic
2C127×#2280C212yesmaximal
3C127×#1820C214yesmaximal, complemented
4C127×#1819C214yesmaximal
5C127×#1740C214yesmaximal, complemented
6C24×C4×C127C2211yescharacteristic
7C2×#117×C127—24no—
8C2×#117×C127—24no—
9C2×#117×C127C412yescomplemented
10C2×#117×C127C412yescomplemented
11C2×#117×C127C2214yes—
12C2×#117×C127C2212yes—
13C2×#117×C127C2212yes—
14C22×C4⋊C4×C127C412yescomplemented
15C22×C22⋊C4×C127C2212yes—
16C22×C42×C127C412yescomplemented
17#381×C127C2214yes—
18#380×C127—24no—
19#379×C127C2214yes—
20C4×C22⋊C4×C127—24no—
21#342×C127C22116yes—
22C25×C127C2311yescharacteristic
23C23×C4×C127—28no—
24C23×C4×C127C2×C411yescharacteristic
25C23×C4×C127C2×C411yescharacteristic
26C23×C4×C127C2×C411yescharacteristic
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127C2×C414yes—
29C23×C4×C127D414yes—
30C23×C4×C127D414yes—
31C23×C4×C127C2311yescharacteristic
32C23×C4×C127C2311yescharacteristic
33#117×C127—28no—
34#117×C127—24no—
35#117×C127—24no—
36#117×C127—28no—
37#117×C127—24no—
38#117×C127—24no—
39#117×C127—216no—
40C2×C4⋊C4×C127—24no—
41C2×C4⋊C4×C127—24no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127C2×C414yes—
44C2×C4⋊C4×C127C2×C414yes—
45C2×C22⋊C4×C127—28no—
46C2×C22⋊C4×C127—28no—
47C2×C22⋊C4×C127C2×C414yes—
48C2×C22⋊C4×C127C2318yes—
49C2×C22⋊C4×C127C2314yes—
50C2×C42×C127—24no—
51C2×C42×C127—24no—
52C2×C42×C127—24no—
53C2×C42×C127C2×C414yes—
54C2×C42×C127C2×C414yes—
55#95×C127—28no—
56C24×C127—24no—
57C24×C127—44no—
58C24×C127C22×C411yescharacteristic
59C24×C127C22×C411yescharacteristic
60C24×C127C2×D414yes—
61C24×C127C2411yescharacteristic
62C22×C4×C127—24no—
63C22×C4×C127—24no—
64C22×C4×C127—24no—
65C22×C4×C127—24no—
66C22×C4×C127—24no—
67C22×C4×C127—216no—
68C22×C4×C127—216no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—28no—
72C22×C4×C127—48no—
73C22×C4×C127C22⋊C412yes—
74C22×C4×C127C22⋊C414yes—
75C22×C4×C127C22⋊C414yes—
76C22×C4×C127C22⋊C412yes—
77C22×C4×C127C22⋊C412yes—
78C22×C4×C127C22⋊C412yes—
79C22×C4×C127C22×C411yescharacteristic
80C22×C4×C127C22×C411yescharacteristic
81C22×C4×C127C22×C418yes—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C22×C411yescharacteristic
84C22×C4×C127C2×D414yes—
85C22×C4×C127C2×D412yes—
86C22×C4×C127C2×D412yes—
87C22×C4×C127#4118yes—
88C4⋊C4×C127—28no—
89C4⋊C4×C127—28no—
90C4⋊C4×C127—48no—
91C22⋊C4×C127—28no—
92C22⋊C4×C127—28no—
93C22⋊C4×C127—216no—
94C22⋊C4×C127—416no—
95C42×C127—28no—
96C42×C127—24no—
97C42×C127—24no—
98C42×C127—48no—
99C23×C127—22no—
100C23×C127—24no—
101C23×C127—22no—
102C23×C127—28no—
103C23×C127—28no—
104C23×C127—42no—
105C23×C127—44no—
106C23×C127—42no—
107C23×C127—416no—
108C23×C127C2×C22⋊C412yes—
109C23×C127C2×C22⋊C412yes—
110C23×C127C23×C411yescharacteristic
111C23×C127C22×D412yes—
112C23×C127C2×#4114yes—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—28no—
116C2×C4×C127—24no—
117C2×C4×C127—28no—
118C2×C4×C127—28no—
119C2×C4×C127—24no—
120C2×C4×C127—24no—
121C2×C4×C127—24no—
122C2×C4×C127—28no—
123C2×C4×C127—48no—
124C2×C4×C127—48no—
125C2×C4×C127—48no—
126C2×C4×C127—416no—
127C2×C4×C127—48no—
128C2×C4×C127C2×C22⋊C412yes—
129C2×C4×C127C2×C22⋊C412yes—
130C2×C4×C127C2×C22⋊C411yescharacteristic
131C2×C4×C127C2×C22⋊C412yes—
132C2×C4×C127C2×C22⋊C411yescharacteristic
133C2×C4×C127#11714yes—
134C2×C4×C127#11714yes—
135C2×C4×C127#12318yes—
136C2×C4×C127#12318yes—
137C22×C127—28no—
138C22×C127—22no—
139C22×C127—22no—
140C22×C127—24no—
141C22×C127—22no—
142C22×C127—24no—
143C22×C127—22no—
144C22×C127—44no—
145C22×C127—416no—
146C22×C127—44no—
147C22×C127#41014yes—
148C22×C127C22×C22⋊C411yescharacteristic
149C22×C127C2×#11712yes—
150C22×C127C2×#12314yes—
151C4×C127—22no—
152C4×C127—22no—
153C4×C127—48no—
154C4×C127—88no—
155C4×C127#38112yes—
156C4×C127#38114yes—
157C4×C127#38112yes—
158#20688C12711yesmaximal, characteristic, direct product factor, Sylow
159C2×C127—24no—
160C2×C127—22no—
161C2×C127—24no—
162C2×C127—42no—
163C2×C127C2×#38111yescharacteristic
164C2×C127C2×#41012yes—
165C22×#117C2×C12711yescharacteristic
166#2280C2×C12712yes—
167#1820C2×C12714yescomplemented
168#1819C2×C12714yes—
169#1740C2×C12714yescomplemented
170C127#2068811yescharacteristic, direct product factor, Sylow
171C24×C4C22×C12711yescharacteristic
172C2×#117—24no—
173C2×#117—24no—
174C2×#117C4×C12712yescomplemented
175C2×#117C4×C12712yescomplemented
176C2×#117C22×C12714yes—
177C2×#117C22×C12712yes—
178C2×#117C22×C12712yes—
179C22×C4⋊C4C4×C12712yescomplemented
180C22×C22⋊C4C22×C12712yes—
181C22×C42C4×C12712yescomplemented
182#381C22×C12714yes—
183#380—24no—
184#379C22×C12714yes—
185C4×C22⋊C4—24no—
186#342C22×C127116yes—
187C25C23×C12711yescharacteristic
188C23×C4—28no—
189C23×C4C2×C4×C12711yescharacteristic
190C23×C4C2×C4×C12711yescharacteristic
191C23×C4C2×C4×C12711yescharacteristic
192C23×C4C2×C4×C12711yescharacteristic
193C23×C4C2×C4×C12714yes—
194C23×C4D4×C12714yes—
195C23×C4D4×C12714yes—
196C23×C4C23×C12711yescharacteristic
197C23×C4C23×C12711yescharacteristic
198#117—28no—
199#117—24no—
200#117—24no—
201#117—28no—
202#117—24no—
203#117—24no—
204#117—216no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C2×C4×C12714yes—
209C2×C4⋊C4C2×C4×C12714yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4C2×C4×C12714yes—
213C2×C22⋊C4C23×C12718yes—
214C2×C22⋊C4C23×C12714yes—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42C2×C4×C12714yes—
219C2×C42C2×C4×C12714yes—
220#95—28no—
221C24—24no—
222C24—44no—
223C24C22×C4×C12711yescharacteristic
224C24C22×C4×C12711yescharacteristic
225C24C2×D4×C12714yes—
226C24C24×C12711yescharacteristic
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—216no—
233C22×C4—216no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—48no—
238C22×C4C22⋊C4×C12712yes—
239C22×C4C22⋊C4×C12712yes—
240C22×C4C22⋊C4×C12714yes—
241C22×C4C22⋊C4×C12714yes—
242C22×C4C22⋊C4×C12712yes—
243C22×C4C22⋊C4×C12712yes—
244C22×C4C22×C4×C12711yescharacteristic
245C22×C4C22×C4×C12711yescharacteristic
246C22×C4C22×C4×C12718yes—
247C22×C4C22×C4×C12711yescharacteristic
248C22×C4C22×C4×C12711yescharacteristic
249C22×C4C2×D4×C12714yes—
250C22×C4C2×D4×C12712yes—
251C22×C4C2×D4×C12712yes—
252C22×C4#41×C12718yes—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C4⋊C4—48no—
256C22⋊C4—28no—
257C22⋊C4—28no—
258C22⋊C4—216no—
259C22⋊C4—416no—
260C42—28no—
261C42—24no—
262C42—24no—
263C42—48no—
264C23—22no—
265C23—24no—
266C23—22no—
267C23—28no—
268C23—28no—
269C23—42no—
270C23—44no—
271C23—42no—
272C23—416no—
273C23C2×C22⋊C4×C12712yes—
274C23C2×C22⋊C4×C12712yes—
275C23C23×C4×C12711yescharacteristic
276C23C22×D4×C12712yes—
277C23C2×#41×C12714yes—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—28no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—28no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—416no—
292C2×C4—48no—
293C2×C4C2×C22⋊C4×C12712yes—
294C2×C4C2×C22⋊C4×C12712yes—
295C2×C4C2×C22⋊C4×C12711yescharacteristic
296C2×C4C2×C22⋊C4×C12711yescharacteristic
297C2×C4C2×C22⋊C4×C12712yes—
298C2×C4#117×C12714yes—
299C2×C4#117×C12714yes—
300C2×C4#123×C12718yes—
301C2×C4#123×C12718yes—
302C22—28no—
303C22—22no—
304C22—22no—
305C22—24no—
306C22—22no—
307C22—24no—
308C22—22no—
309C22—44no—
310C22—416no—
311C22—44no—
312C22#410×C12714yes—
313C22C22×C22⋊C4×C12711yescharacteristic
314C22C2×#117×C12712yes—
315C22C2×#123×C12714yes—
316C4—22no—
317C4—22no—
318C4—48no—
319C4—88no—
320C4#381×C12712yes—
321C4#381×C12712yes—
322C4#381×C12714yes—
323C2—24no—
324C2—22no—
325C2—24no—
326C2—42no—
327C2C2×#381×C12711yescharacteristic
328C2C2×#410×C12712yes—
subgroupG/H
centerC2×C4×C127C2×C22⋊C4
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC25×C127C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1820C2semidirect
5. C127×#1740C2semidirect
9. C2×#117×C127C4semidirect
10. C2×#117×C127C4semidirect
14. C22×C4⋊C4×C127C4semidirect
16. C22×C42×C127C4semidirect
158. #20688C127direct (N × H)
167. #1820C2×C127semidirect
169. #1740C2×C127semidirect
170. C127#20688direct (N × H)
174. C2×#117C4×C127semidirect
175. C2×#117C4×C127semidirect
179. C22×C4⋊C4C4×C127semidirect
181. C22×C42C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C1272
C127×#22802
C127×#18202
C127×#18192
C127×#17402
#20688127