Finite Group Databasebeta

← C127×#11706 #44575379

Subgroups — C127×#11706 #44575379

Order 32512.

456 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467×C127C211yesmaximal, characteristic
2C127×#2112C211yesmaximal, characteristic
3C127×#1735C211yesmaximal, characteristic
4C127×#1348C214yesmaximal, complemented
5C22×Q16×C127C411yescharacteristic, complemented
6C23×C8×C127C2211yescharacteristic
7C2×C22⋊Q8×C127—21no—
8C2×#117×C127—21no—
9C2×M32×C127—21no—
10#467×C127—21no—
11#467×C127—21no—
12#467×C127C412yescomplemented
13#467×C127C2212yes—
14C2×#107×C127C411yescharacteristic, complemented
15C2×#103×C127—21no—
16C22⋊C16×C127—21no—
17C22⋊C16×C127C2212yes—
18#340×C127—21no—
19#340×C127C2212yes—
20C22×Q8×C127—21no—
21C23×C4×C127D411yescharacteristic
22C2×Q16×C127—21no—
23C2×Q16×C127—42no—
24C2×Q16×C127C2×C412yes—
25C22×C8×C127—42no—
26C22×C8×C127C2×C411yescharacteristic
27C22×C8×C127C2×C411yescharacteristic
28C22×C8×C127D411yescharacteristic
29C22×C8×C127D411yescharacteristic
30C22×C8×C127D411yescharacteristic
31C22×C8×C127C2311yescharacteristic
32C22⋊Q8×C127—22no—
33C22⋊Q8×C127—22no—
34C22⋊Q8×C127—41no—
35C22⋊Q8×C127—41no—
36#117×C127—24no—
37#117×C127—42no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—22no—
40C2×C4⋊C4×C127—41no—
41C2×C22⋊C4×C127—41no—
42C2×C22⋊C4×C127—41no—
43C2×C42×C127—41no—
44M32×C127—42no—
45C2×C16×C127—22no—
46#107×C127—21no—
47#107×C127C2×C412yes—
48#103×C127—22no—
49#103×C127—22no—
50C24×C127D811yescharacteristic
51C2×Q8×C127—22no—
52C2×Q8×C127—41no—
53C2×Q8×C127—42no—
54C2×Q8×C127—42no—
55C22×C4×C127—22no—
56C22×C4×C127—22no—
57C22×C4×C127—41no—
58C22×C4×C127—41no—
59C22×C4×C127—42no—
60C22×C4×C127C22⋊C411yescharacteristic
61C22×C4×C127C22⋊C411yescharacteristic
62C22×C4×C127D811yescharacteristic
63C22×C4×C127SD1611yescharacteristic
64C22×C4×C127SD1611yescharacteristic
65C22×C4×C127C2×D411yescharacteristic
66Q16×C127—42no—
67Q16×C127—42no—
68C2×C8×C127—22no—
69C2×C8×C127—22no—
70C2×C8×C127—41no—
71C2×C8×C127—41no—
72C2×C8×C127—41no—
73C2×C8×C127—41no—
74C2×C8×C127C22⋊C411yescharacteristic
75C2×C8×C127C22⋊C411yescharacteristic
76C2×C8×C127C22×C411yescharacteristic
77C2×C8×C127C2×D411yescharacteristic
78C4⋊C4×C127—24no—
79C4⋊C4×C127—22no—
80C4⋊C4×C127—42no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—42no—
84C22⋊C4×C127—44no—
85C22⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C42×C127—44no—
88C16×C127—42no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—41no—
92C23×C127#10211yescharacteristic
93C23×C127#10211yescharacteristic
94C23×C127C2×D811yescharacteristic
95Q8×C127—42no—
96Q8×C127—42no—
97Q8×C127—82no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—44no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—82no—
109C2×C4×C127—82no—
110C2×C4×C127—82no—
111C2×C4×C127—82no—
112C2×C4×C127#9912yes—
113C2×C4×C127#10211yescharacteristic
114C2×C4×C127#10211yescharacteristic
115C2×C4×C127#10412yes—
116C2×C4×C127C2×C22⋊C411yescharacteristic
117C2×C4×C127C2×SD1611yescharacteristic
118C8×C127—21no—
119C8×C127—21no—
120C8×C127—21no—
121C8×C127—21no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—21no—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—44no—
130C22×C127#32712yes—
131C22×C127C2×#10211yescharacteristic
132C22×C127#51012yes—
133C4×C127—21no—
134C4×C127—21no—
135C4×C127—84no—
136C4×C127—82no—
137C4×C127—82no—
138C4×C127#32712yes—
139C4×C127C2×#9911yescharacteristic
140C4×C127C2×#10411yescharacteristic
141#11706C12711yesmaximal, characteristic, direct product factor, Sylow
142C2×C127—22no—
143C2×C127—22no—
144C2×C127—41no—
145C2×C127C2×#32711yescharacteristic
146C2×C127#214111yescharacteristic
147C2×C127#214511yescharacteristic
148C2×#467C2×C12711yescharacteristic
149#2112C2×C12711yescharacteristic
150#1735C2×C12711yescharacteristic
151#1348C2×C12714yescomplemented
152C127#1170611yescharacteristic, direct product factor, Sylow
153C22×Q16C4×C12711yescharacteristic, complemented
154C23×C8C22×C12711yescharacteristic
155C2×C22⋊Q8—21no—
156C2×#117—21no—
157C2×M32—21no—
158#467—21no—
159#467—21no—
160#467C4×C12712yescomplemented
161#467C22×C12712yes—
162C2×#107C4×C12711yescharacteristic, complemented
163C2×#103—21no—
164C22⋊C16—21no—
165C22⋊C16C22×C12712yes—
166#340—21no—
167#340C22×C12712yes—
168C22×Q8—21no—
169C23×C4D4×C12711yescharacteristic
170C2×Q16—21no—
171C2×Q16—42no—
172C2×Q16C2×C4×C12712yes—
173C22×C8—42no—
174C22×C8C2×C4×C12711yescharacteristic
175C22×C8C2×C4×C12711yescharacteristic
176C22×C8D4×C12711yescharacteristic
177C22×C8D4×C12711yescharacteristic
178C22×C8D4×C12711yescharacteristic
179C22×C8C23×C12711yescharacteristic
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184#117—24no—
185#117—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C42—41no—
192M32—42no—
193C2×C16—22no—
194#107—21no—
195#107C2×C4×C12712yes—
196#103—22no—
197#103—22no—
198C24D8×C12711yescharacteristic
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4C22⋊C4×C12711yescharacteristic
209C22×C4C22⋊C4×C12711yescharacteristic
210C22×C4D8×C12711yescharacteristic
211C22×C4SD16×C12711yescharacteristic
212C22×C4SD16×C12711yescharacteristic
213C22×C4C2×D4×C12711yescharacteristic
214Q16—42no—
215Q16—42no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8C22⋊C4×C12711yescharacteristic
223C2×C8C22⋊C4×C12711yescharacteristic
224C2×C8C22×C4×C12711yescharacteristic
225C2×C8C2×D4×C12711yescharacteristic
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C42—44no—
236C16—42no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23#102×C12711yescharacteristic
241C23#102×C12711yescharacteristic
242C23C2×D8×C12711yescharacteristic
243Q8—42no—
244Q8—42no—
245Q8—82no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4#99×C12712yes—
261C2×C4#102×C12711yescharacteristic
262C2×C4#102×C12711yescharacteristic
263C2×C4#104×C12712yes—
264C2×C4C2×C22⋊C4×C12711yescharacteristic
265C2×C4C2×SD16×C12711yescharacteristic
266C8—21no—
267C8—21no—
268C8—21no—
269C8—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22#327×C12712yes—
279C22C2×#102×C12711yescharacteristic
280C22#510×C12712yes—
281C4—21no—
282C4—21no—
283C4—84no—
284C4—82no—
285C4—82no—
286C4#327×C12712yes—
287C4C2×#99×C12711yescharacteristic
288C4C2×#104×C12711yescharacteristic
289C2—22no—
290C2—22no—
291C2—41no—
292C2C2×#327×C12711yescharacteristic
293C2C127×#214111yescharacteristic
294C2C127×#214511yescharacteristic
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC24×C127D8
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1348C2semidirect
5. C22×Q16×C127C4semidirect
12. #467×C127C4semidirect
14. C2×#107×C127C4semidirect
141. #11706C127direct (N × H)
151. #1348C2×C127semidirect
152. C127#11706direct (N × H)
153. C22×Q16C4×C127semidirect
160. #467C4×C127semidirect
162. C2×#107C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#467×C1272
C127×#21122
C127×#17352
C127×#13482
#11706127