Finite Group Databasebeta

← C127×#17676 #44581349

Subgroups — C127×#17676 #44581349

Order 32512.

954 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533×C127C211yesmaximal, characteristic
2C2×#432×C127C212yesmaximal
3C127×#1534C214yesmaximal, complemented
4C127×#1531C218yesmaximal, complemented
5#533×C127—22no—
6#533×C127—22no—
7#533×C127C414yes—
8#533×C127C2214yes—
9C2×C4⋊Q8×C127C412yes—
10C2×#125×C127C2212yes—
11C2×C22⋊Q8×C127—22no—
12C2×C4×Q8×C127—22no—
13C2×#117×C127C412yes—
14C22×C42×C127C2211yescharacteristic
15#432×C127—22no—
16#432×C127—22no—
17#432×C127C2214yes—
18#432×C127C2214yes—
19C2×C4⋊C8×C127—22no—
20C2×C22⋊C8×C127—22no—
21C2×#97×C127C2214yes—
22#330×C127C22116yescomplemented
23C22×Q8×C127—22no—
24C23×C4×C127C2×C411yescharacteristic
25C23×C4×C127D412yes—
26C22×C8×C127—24no—
27C4⋊Q8×C127—22no—
28C4⋊Q8×C127—22no—
29C4⋊Q8×C127C2×C414yes—
30C4⋊Q8×C127C2×C414yes—
31#125×C127—24no—
32#125×C127C2×C414yes—
33#125×C127C2314yes—
34C22⋊Q8×C127—24no—
35C22⋊Q8×C127—24no—
36C22⋊Q8×C127—24no—
37C22⋊Q8×C127—24no—
38C4×Q8×C127—24no—
39C4×Q8×C127—24no—
40C4×Q8×C127—24no—
41C4×Q8×C127—24no—
42#117×C127—24no—
43#117×C127C2×C418yes—
44C2×C4⋊C4×C127—22no—
45C2×C4⋊C4×C127—24no—
46C2×C4⋊C4×C127C2×C414yes—
47C2×C22⋊C4×C127—22no—
48C2×C42×C127—22no—
49C2×C42×C127C2×C411yescharacteristic
50C2×C42×C127C2×C412yes—
51C2×C42×C127D412yes—
52C2×C42×C127D412yes—
53C2×C42×C127D412yes—
54C2×C42×C127C2311yescharacteristic
55C2×C42×C127C2312yes—
56C4⋊C8×C127—24no—
57C4⋊C8×C127—24no—
58C22⋊C8×C127—24no—
59C22⋊C8×C127—24no—
60#97×C127—24no—
61#97×C127C2318yes—
62C24×C127C22⋊C411yescharacteristic
63C2×Q8×C127—24no—
64C2×Q8×C127—24no—
65C2×Q8×C127—48no—
66C22×C4×C127—24no—
67C22×C4×C127—24no—
68C22×C4×C127—44no—
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C412yes—
71C22×C4×C127C22⋊C412yes—
72C22×C4×C127C22⋊C412yes—
73C22×C4×C127C22⋊C412yes—
74C22×C4×C127C22×C411yescharacteristic
75C22×C4×C127C22×C412yes—
76C22×C4×C127C2×D412yes—
77C22×C4×C127C2×D412yes—
78C22×C4×C127C2×D412yes—
79C2×C8×C127—28no—
80C2×C8×C127—48no—
81C4⋊C4×C127—28no—
82C4⋊C4×C127—24no—
83C4⋊C4×C127—24no—
84C4⋊C4×C127—24no—
85C4⋊C4×C127—28no—
86C4⋊C4×C127C22×C418yes—
87C22⋊C4×C127—28no—
88C42×C127—28no—
89C42×C127C22⋊C412yes—
90C42×C127C22⋊C412yes—
91C42×C127C22⋊C411yescharacteristic
92C42×C127C22⋊C411yescharacteristic
93C42×C127C22×C411yescharacteristic
94C42×C127C22×C412yes—
95C42×C127C2×D412yes—
96C42×C127C2×D412yes—
97C42×C127C2×D412yes—
98C42×C127C2411yescharacteristic
99C23×C127—42no—
100C23×C127#10114yes—
101C23×C127C2×C22⋊C412yes—
102C23×C127C2×C22⋊C411yescharacteristic
103Q8×C127—48no—
104Q8×C127—48no—
105C2×C4×C127—22no—
106C2×C4×C127—24no—
107C2×C4×C127—22no—
108C2×C4×C127—24no—
109C2×C4×C127—24no—
110C2×C4×C127—24no—
111C2×C4×C127—24no—
112C2×C4×C127—28no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—48no—
117C2×C4×C127C2×C22⋊C412yes—
118C2×C4×C127C2×C22⋊C412yes—
119C2×C4×C127C2×C22⋊C412yes—
120C2×C4×C127C2×C22⋊C411yescharacteristic
121C2×C4×C127C2×C22⋊C412yes—
122C2×C4×C127C23×C411yescharacteristic
123C2×C4×C127C22×D412yes—
124C8×C127—48no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127—22no—
128C22×C127—44no—
129C22×C127C2×#10114yes—
130C22×C127C2×#10112yes—
131C22×C127C22×C22⋊C411yescharacteristic
132C4×C127—22no—
133C4×C127—21no—
134C4×C127—21no—
135C4×C127—22no—
136C4×C127—22no—
137C4×C127—22no—
138C4×C127—48no—
139C4×C127#42114yes—
140#17676C12711yesmaximal, characteristic, direct product factor, Sylow
141C2×C127—22no—
142C2×C127—24no—
143C2×C127C22×#10111yescharacteristic
144C2×C127C2×#42112yes—
145C2×#533C2×C12711yescharacteristic
146C2×#432C2×C12712yes—
147#1534C2×C12714yescomplemented
148#1531C2×C12718yescomplemented
149C127#1767611yescharacteristic, direct product factor, Sylow
150#533—22no—
151#533—22no—
152#533C4×C12714yes—
153#533C22×C12714yes—
154C2×C4⋊Q8C4×C12712yes—
155C2×#125C22×C12712yes—
156C2×C22⋊Q8—22no—
157C2×C4×Q8—22no—
158C2×#117C4×C12712yes—
159C22×C42C22×C12711yescharacteristic
160#432—22no—
161#432—22no—
162#432C22×C12714yes—
163#432C22×C12714yes—
164C2×C4⋊C8—22no—
165C2×C22⋊C8—22no—
166C2×#97C22×C12714yes—
167#330C22×C127116yescomplemented
168C22×Q8—22no—
169C23×C4C2×C4×C12711yescharacteristic
170C23×C4D4×C12712yes—
171C22×C8—24no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8C2×C4×C12714yes—
175C4⋊Q8C2×C4×C12714yes—
176#125—24no—
177#125C2×C4×C12714yes—
178#125C23×C12714yes—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187#117—24no—
188#117C2×C4×C12718yes—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4C2×C4×C12714yes—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42C2×C4×C12711yescharacteristic
195C2×C42C2×C4×C12712yes—
196C2×C42D4×C12712yes—
197C2×C42D4×C12712yes—
198C2×C42D4×C12712yes—
199C2×C42C23×C12711yescharacteristic
200C2×C42C23×C12712yes—
201C4⋊C8—24no—
202C4⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205#97—24no—
206#97C23×C12718yes—
207C24C22⋊C4×C12711yescharacteristic
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—48no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—44no—
214C22×C4C22⋊C4×C12711yescharacteristic
215C22×C4C22⋊C4×C12712yes—
216C22×C4C22⋊C4×C12712yes—
217C22×C4C22⋊C4×C12712yes—
218C22×C4C22⋊C4×C12712yes—
219C22×C4C22×C4×C12711yescharacteristic
220C22×C4C22×C4×C12712yes—
221C22×C4C2×D4×C12712yes—
222C22×C4C2×D4×C12712yes—
223C22×C4C2×D4×C12712yes—
224C2×C8—28no—
225C2×C8—48no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—28no—
231C4⋊C4C22×C4×C12718yes—
232C22⋊C4—28no—
233C42—28no—
234C42C22⋊C4×C12712yes—
235C42C22⋊C4×C12712yes—
236C42C22⋊C4×C12711yescharacteristic
237C42C22⋊C4×C12711yescharacteristic
238C42C22×C4×C12711yescharacteristic
239C42C22×C4×C12712yes—
240C42C2×D4×C12712yes—
241C42C2×D4×C12712yes—
242C42C2×D4×C12712yes—
243C42C24×C12711yescharacteristic
244C23—42no—
245C23#101×C12714yes—
246C23C2×C22⋊C4×C12712yes—
247C23C2×C22⋊C4×C12711yescharacteristic
248Q8—48no—
249Q8—48no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—28no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—48no—
262C2×C4C2×C22⋊C4×C12712yes—
263C2×C4C2×C22⋊C4×C12712yes—
264C2×C4C2×C22⋊C4×C12712yes—
265C2×C4C2×C22⋊C4×C12711yescharacteristic
266C2×C4C2×C22⋊C4×C12712yes—
267C2×C4C23×C4×C12711yescharacteristic
268C2×C4C22×D4×C12712yes—
269C8—48no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—44no—
274C22C2×#101×C12714yes—
275C22C2×#101×C12712yes—
276C22C22×C22⋊C4×C12711yescharacteristic
277C4—22no—
278C4—21no—
279C4—21no—
280C4—22no—
281C4—22no—
282C4—22no—
283C4—48no—
284C4#421×C12714yes—
285C2—22no—
286C2—24no—
287C2C22×#101×C12711yescharacteristic
288C2C2×#421×C12712yes—
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC42C24×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1534C2semidirect
4. C127×#1531C2semidirect
22. #330×C127C22semidirect
140. #17676C127direct (N × H)
147. #1534C2×C127semidirect
148. #1531C2×C127semidirect
149. C127#17676direct (N × H)
167. #330C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#533×C1272
C2×#432×C1272
C127×#15342
C127×#15312
#17676127