Finite Group Databasebeta

← C127×#8339 #44572012

Subgroups — C127×#8339 #44572012

Order 32512.

1718 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 238 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×#127×C127C211yesmaximal, characteristic
2C127×#1797C212yesmaximal, complemented
3C2×#331×C127C214yesmaximal, complemented
4C23×D4×C127—22no—
5C23×D4×C127C412yes—
6C2×#127×C127—28no—
7C2×#127×C127—22no—
8C2×#127×C127C412yes—
9C2×#127×C127C2212yes—
10C22×C42×C127C2211yescharacteristic
11C2×C4⋊C8×C127C2214yescomplemented
12C2×C22⋊C8×C127—22no—
13#336×C127—22no—
14#331×C127—28no—
15C25×C127—22no—
16C22×D4×C127—216no—
17C22×D4×C127—28no—
18C22×D4×C127—22no—
19C22×D4×C127—24no—
20C22×D4×C127—24no—
21C22×D4×C127—48no—
22C22×D4×C127—44no—
23C22×D4×C127—44no—
24C22×D4×C127—44no—
25C22×D4×C127—44no—
26C22×D4×C127C2×C414yes—
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D412yes—
29C22×C8×C127—24no—
30#127×C127—216no—
31#127×C127—28no—
32#127×C127—24no—
33#127×C127D418yes—
34C2×C42×C127—24no—
35C2×C42×C127C2×C411yescharacteristic
36C2×C42×C127D412yes—
37C2×C42×C127C2311yescharacteristic
38C4⋊C8×C127—28no—
39C22⋊C8×C127—44no—
40C24×C127—28no—
41C24×C127—24no—
42C24×C127—44no—
43C24×C127—48no—
44C24×C127—44no—
45C24×C127—48no—
46C24×C127C22⋊C411yescharacteristic
47C2×D4×C127—28no—
48C2×D4×C127—216no—
49C2×D4×C127—24no—
50C2×D4×C127—216no—
51C2×D4×C127—44no—
52C2×D4×C127—44no—
53C2×D4×C127—432no—
54C2×D4×C127—44no—
55C2×D4×C127—48no—
56C2×D4×C127—432no—
57C2×D4×C127—432no—
58C2×D4×C127—44no—
59C2×D4×C127—48no—
60C2×D4×C127—44no—
61C2×D4×C127—44no—
62C2×D4×C127—48no—
63C2×D4×C127C22⋊C418yes—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—22no—
67C22×C4×C127—24no—
68C22×C4×C127—24no—
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C412yes—
71C22×C4×C127D814yes—
72C22×C4×C127SD1614yes—
73C22×C4×C127C22×C411yescharacteristic
74C22×C4×C127C2×D412yes—
75C2×C8×C127—44no—
76C2×C8×C127—44no—
77C2×C8×C127—44no—
78C42×C127—22no—
79C42×C127—24no—
80C42×C127C2×D414yes—
81C23×C127—21no—
82C23×C127—21no—
83C23×C127—22no—
84C23×C127—28no—
85C23×C127—416no—
86C23×C127—432no—
87C23×C127—44no—
88C23×C127—44no—
89C23×C127—48no—
90C23×C127—48no—
91C23×C127—816no—
92C23×C127—88no—
93C23×C127—88no—
94C23×C127#10012yes—
95C23×C127#10214yes—
96C23×C127C2×C22⋊C411yescharacteristic
97D4×C127—48no—
98D4×C127—416no—
99D4×C127—48no—
100D4×C127—432no—
101D4×C127—416no—
102D4×C127—416no—
103C2×C4×C127—28no—
104C2×C4×C127—21no—
105C2×C4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—22no—
110C2×C4×C127—28no—
111C2×C4×C127—28no—
112C2×C4×C127#10012yes—
113C2×C4×C127#10214yes—
114C2×C4×C127C2×C22⋊C412yes—
115C2×C4×C127#12014yes—
116C2×C4×C127C2×D812yes—
117C2×C4×C127C2×SD1612yes—
118C2×C4×C127AGL(1,ℤ/8)14yes—
119C8×C127—84no—
120C22×C127—24no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—24no—
124C22×C127—48no—
125C22×C127—416no—
126C22×C127—832no—
127C22×C127—84no—
128C22×C127—88no—
129C22×C127—84no—
130C22×C127#33114yescomplemented
131C22×C127#37911yescharacteristic
132C22×C127C2×#10011yescharacteristic
133C22×C127C2×#10011yescharacteristic
134C22×C127C2×#10212yes—
135C22×C127#41812yes—
136C4×C127—22no—
137C4×C127—22no—
138C4×C127—24no—
139C4×C127#44714yes—
140C4×C127#45014yes—
141#8339C12711yesmaximal, characteristic, direct product factor, Sylow
142C2×C127—24no—
143C2×C127—88no—
144C2×C127C2×#33111yescharacteristic
145C2×C127#154512yes—
146C2×C127#155511yescharacteristic
147C2×C127#178311yescharacteristic
148C2×C127#178612yes—
149C22×#127C2×C12711yescharacteristic
150#1797C2×C12712yescomplemented
151C2×#331C2×C12714yescomplemented
152C127#833911yescharacteristic, direct product factor, Sylow
153C23×D4—22no—
154C23×D4C4×C12712yes—
155C2×#127—28no—
156C2×#127—22no—
157C2×#127C4×C12712yes—
158C2×#127C22×C12712yes—
159C22×C42C22×C12711yescharacteristic
160C2×C4⋊C8C22×C12714yescomplemented
161C2×C22⋊C8—22no—
162#336—22no—
163#331—28no—
164C25—22no—
165C22×D4—216no—
166C22×D4—28no—
167C22×D4—22no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—44no—
171C22×D4—48no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4C2×C4×C12714yes—
176C23×C4C2×C4×C12711yescharacteristic
177C23×C4D4×C12712yes—
178C22×C8—24no—
179#127—216no—
180#127—28no—
181#127—24no—
182#127D4×C12718yes—
183C2×C42—24no—
184C2×C42C2×C4×C12711yescharacteristic
185C2×C42D4×C12712yes—
186C2×C42C23×C12711yescharacteristic
187C4⋊C8—28no—
188C22⋊C8—44no—
189C24—28no—
190C24—24no—
191C24—44no—
192C24—48no—
193C24—44no—
194C24—48no—
195C24C22⋊C4×C12711yescharacteristic
196C2×D4—28no—
197C2×D4—216no—
198C2×D4—24no—
199C2×D4—216no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—432no—
203C2×D4—44no—
204C2×D4—48no—
205C2×D4—432no—
206C2×D4—432no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4C22⋊C4×C12718yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4C22⋊C4×C12711yescharacteristic
219C22×C4C22⋊C4×C12712yes—
220C22×C4D8×C12714yes—
221C22×C4SD16×C12714yes—
222C22×C4C22×C4×C12711yescharacteristic
223C22×C4C2×D4×C12712yes—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C42—22no—
228C42—24no—
229C42C2×D4×C12714yes—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—28no—
234C23—416no—
235C23—432no—
236C23—44no—
237C23—44no—
238C23—48no—
239C23—48no—
240C23—816no—
241C23—88no—
242C23—88no—
243C23#100×C12712yes—
244C23#102×C12714yes—
245C23C2×C22⋊C4×C12711yescharacteristic
246D4—48no—
247D4—416no—
248D4—48no—
249D4—432no—
250D4—416no—
251D4—416no—
252C2×C4—28no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—28no—
260C2×C4—28no—
261C2×C4#100×C12712yes—
262C2×C4#102×C12714yes—
263C2×C4C2×C22⋊C4×C12712yes—
264C2×C4#120×C12714yes—
265C2×C4C2×D8×C12712yes—
266C2×C4C2×SD16×C12712yes—
267C2×C4AGL(1,ℤ/8)×C12714yes—
268C8—84no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—48no—
274C22—416no—
275C22—832no—
276C22—88no—
277C22—84no—
278C22—84no—
279C22#331×C12714yescomplemented
280C22#379×C12711yescharacteristic
281C22C2×#100×C12711yescharacteristic
282C22C2×#100×C12711yescharacteristic
283C22C2×#102×C12712yes—
284C22#418×C12712yes—
285C4—22no—
286C4—22no—
287C4—24no—
288C4#447×C12714yes—
289C4#450×C12714yes—
290C2—24no—
291C2—88no—
292C2C2×#331×C12711yescharacteristic
293C2C127×#154512yes—
294C2C127×#155511yescharacteristic
295C2C127×#178311yescharacteristic
296C2C127×#178612yes—
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×#127×C127C2
LCSG ▷ C22×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. C127×#1797C2semidirect
3. C2×#331×C127C2semidirect
11. C2×C4⋊C8×C127C22semidirect
130. C22×C127#331semidirect
141. #8339C127direct (N × H)
150. #1797C2×C127semidirect
151. C2×#331C2×C127semidirect
152. C127#8339direct (N × H)
160. C2×C4⋊C8C22×C127semidirect
279. C22#331×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#127×C1272
C127×#17972
C2×#331×C1272
#8339127