Finite Group Databasebeta

← C127×#12699 #44576372

Subgroups — C127×#12699 #44576372

Order 32512.

1146 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124×C127C211yesmaximal, characteristic
2C2×#355×C127C214yesmaximal, complemented
3C127×#1784C212yesmaximal, complemented
4C23×Q8×C127C2211yescharacteristic
5C23×D4×C127C411yescharacteristic
6C2×#124×C127—22no—
7C2×#124×C127—28no—
8C2×#124×C127C412yes—
9C2×#124×C127C2212yes—
10C22×C22⋊C4×C127—22no—
11C22×C42×C127C411yescharacteristic
12C2×#101×C127—22no—
13C2×#101×C127C2214yescomplemented
14#407×C127—22no—
15#355×C127—28no—
16C25×C127—21no—
17C22×Q8×C127—24no—
18C22×Q8×C127—42no—
19C22×Q8×C127—42no—
20C22×Q8×C127C2×C411yescharacteristic
21C22×Q8×C127D412yes—
22C22×Q8×C127C2311yescharacteristic
23C22×D4×C127—24no—
24C22×D4×C127—21no—
25C22×D4×C127—42no—
26C22×D4×C127—42no—
27C22×D4×C127C2×C412yes—
28C23×C4×C127—21no—
29C23×C4×C127C2×C411yescharacteristic
30C23×C4×C127D412yes—
31C2×M16×C127—24no—
32#124×C127—216no—
33#124×C127—24no—
34#124×C127—24no—
35#124×C127—24no—
36#124×C127D418yes—
37C2×C22⋊C4×C127—216no—
38C2×C22⋊C4×C127—24no—
39C2×C22⋊C4×C127—24no—
40C2×C42×C127—21no—
41C2×C42×C127—24no—
42C2×C42×C127C2×C412yes—
43#101×C127—28no—
44#101×C127—44no—
45C22⋊C8×C127—44no—
46C24×C127—24no—
47C24×C127—22no—
48C24×C127—42no—
49C24×C127—48no—
50C24×C127—42no—
51C24×C127C22⋊C411yescharacteristic
52C2×Q8×C127—24no—
53C2×Q8×C127—22no—
54C2×Q8×C127—24no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×Q8×C127—42no—
58C2×Q8×C127—41no—
59C2×Q8×C127—41no—
60C2×Q8×C127—416no—
61C2×Q8×C127C2×D414yes—
62C2×D4×C127—24no—
63C2×D4×C127—22no—
64C2×D4×C127—42no—
65C2×D4×C127—42no—
66C2×D4×C127—416no—
67C2×D4×C127—42no—
68C2×D4×C127—42no—
69C2×D4×C127C22⋊C414yes—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—22no—
76C22×C4×C127—24no—
77C22×C4×C127—42no—
78C22×C4×C127—44no—
79C22×C4×C127C22⋊C412yes—
80C22×C4×C127C22⋊C411yescharacteristic
81C22×C4×C127C22×C411yescharacteristic
82C22×C4×C127C2×D412yes—
83M16×C127—44no—
84M16×C127—44no—
85C2×C8×C127—44no—
86C22⋊C4×C127—216no—
87C22⋊C4×C127—216no—
88C42×C127—24no—
89C42×C127—22no—
90C42×C127C22⋊C414yes—
91C23×C127—24no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—24no—
95C23×C127—48no—
96C23×C127—416no—
97C23×C127—48no—
98C23×C127—42no—
99C23×C127—42no—
100C23×C127—88no—
101C23×C127—84no—
102C23×C127—84no—
103C23×C127#9912yes—
104C23×C127C2×C22⋊C411yescharacteristic
105Q8×C127—44no—
106Q8×C127—48no—
107Q8×C127—44no—
108D4×C127—44no—
109D4×C127—48no—
110D4×C127—44no—
111C2×C4×C127—28no—
112C2×C4×C127—21no—
113C2×C4×C127—22no—
114C2×C4×C127—21no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—24no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—48no—
123C2×C4×C127—48no—
124C2×C4×C127—44no—
125C2×C4×C127#9912yes—
126C2×C4×C127C2×C22⋊C412yes—
127C2×C4×C127#12014yes—
128C8×C127—84no—
129C22×C127—24no—
130C22×C127—24no—
131C22×C127—22no—
132C22×C127—22no—
133C22×C127—22no—
134C22×C127—48no—
135C22×C127—44no—
136C22×C127—816no—
137C22×C127—84no—
138C22×C127—84no—
139C22×C127#35514yescomplemented
140C22×C127#37911yescharacteristic
141C22×C127C2×#9911yescharacteristic
142C22×C127C2×#9911yescharacteristic
143C4×C127—24no—
144C4×C127—22no—
145C4×C127—22no—
146C4×C127—44no—
147C4×C127—42no—
148#12699C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—21no—
150C2×C127—24no—
151C2×C127—21no—
152C2×C127—84no—
153C2×C127#178111yescharacteristic
154C2×C127C2×#35512yes—
155C22×#124C2×C12711yescharacteristic
156C2×#355C2×C12714yescomplemented
157#1784C2×C12712yescomplemented
158C127#1269911yescharacteristic, direct product factor, Sylow
159C23×Q8C22×C12711yescharacteristic
160C23×D4C4×C12711yescharacteristic
161C2×#124—22no—
162C2×#124—28no—
163C2×#124C4×C12712yes—
164C2×#124C22×C12712yes—
165C22×C22⋊C4—22no—
166C22×C42C4×C12711yescharacteristic
167C2×#101—22no—
168C2×#101C22×C12714yescomplemented
169#407—22no—
170#355—28no—
171C25—21no—
172C22×Q8—24no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8C2×C4×C12711yescharacteristic
176C22×Q8D4×C12712yes—
177C22×Q8C23×C12711yescharacteristic
178C22×D4—24no—
179C22×D4—21no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4C2×C4×C12712yes—
183C23×C4—21no—
184C23×C4C2×C4×C12711yescharacteristic
185C23×C4D4×C12712yes—
186C2×M16—24no—
187#124—216no—
188#124—24no—
189#124—24no—
190#124—24no—
191#124D4×C12718yes—
192C2×C22⋊C4—216no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C42—21no—
196C2×C42—24no—
197C2×C42C2×C4×C12712yes—
198#101—28no—
199#101—44no—
200C22⋊C8—44no—
201C24—24no—
202C24—22no—
203C24—42no—
204C24—48no—
205C24—42no—
206C24C22⋊C4×C12711yescharacteristic
207C2×Q8—24no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—416no—
216C2×Q8C2×D4×C12714yes—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—416no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4C22⋊C4×C12714yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4C22⋊C4×C12712yes—
235C22×C4C22⋊C4×C12711yescharacteristic
236C22×C4C22×C4×C12711yescharacteristic
237C22×C4C2×D4×C12712yes—
238M16—44no—
239M16—44no—
240C2×C8—44no—
241C22⋊C4—216no—
242C22⋊C4—216no—
243C42—24no—
244C42—22no—
245C42C22⋊C4×C12714yes—
246C23—24no—
247C23—21no—
248C23—21no—
249C23—24no—
250C23—48no—
251C23—416no—
252C23—48no—
253C23—42no—
254C23—42no—
255C23—88no—
256C23—84no—
257C23—84no—
258C23#99×C12712yes—
259C23C2×C22⋊C4×C12711yescharacteristic
260Q8—44no—
261Q8—48no—
262Q8—44no—
263D4—44no—
264D4—48no—
265D4—44no—
266C2×C4—28no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—44no—
280C2×C4#99×C12712yes—
281C2×C4C2×C22⋊C4×C12712yes—
282C2×C4#120×C12714yes—
283C8—84no—
284C22—24no—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—48no—
290C22—44no—
291C22—816no—
292C22—84no—
293C22—84no—
294C22#355×C12714yescomplemented
295C22#379×C12711yescharacteristic
296C22C2×#99×C12711yescharacteristic
297C22C2×#99×C12711yescharacteristic
298C4—24no—
299C4—22no—
300C4—22no—
301C4—44no—
302C4—42no—
303C2—21no—
304C2—24no—
305C2—21no—
306C2—84no—
307C2C127×#178111yescharacteristic
308C2C2×#355×C12712yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC23×D4×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C23×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#355×C127C2semidirect
3. C127×#1784C2semidirect
13. C2×#101×C127C22semidirect
139. C22×C127#355semidirect
148. #12699C127direct (N × H)
156. C2×#355C2×C127semidirect
157. #1784C2×C127semidirect
158. C127#12699direct (N × H)
168. C2×#101C22×C127semidirect
294. C22#355×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#124×C1272
C2×#355×C1272
C127×#17842
#12699127