Finite Group Databasebeta

← C127×#12736 #44576409

Subgroups — C127×#12736 #44576409

Order 32512.

1162 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 182 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
1C23×D8×C127C211yesmaximal, characteristic
2C2×#361×C127C214yesmaximal, complemented
3C127×#2112C211yesmaximal, characteristic, complemented
4C127×#1810C211yesmaximal, characteristic, complemented
5C23×D4×C127—21no—
6C22×D8×C127—28no—
7C22×D8×C127—21no—
8C22×D8×C127—22no—
9C22×D8×C127C414yes—
10C22×D8×C127C2212yes—
11C23×C8×C127C2211yescharacteristic
12C2×M32×C127C2212yescomplemented
13C2×#108×C127C2212yescomplemented
14#407×C127—21no—
15#361×C127—28no—
16C22⋊C16×C127—22no—
17#341×C127—21no—
18C25×C127—41no—
19C22×D4×C127—28no—
20C22×D4×C127—24no—
21C22×D4×C127—22no—
22C22×D4×C127—44no—
23C22×D4×C127—42no—
24C22×D4×C127—41no—
25C23×C4×C127D411yescharacteristic
26C2×D8×C127—28no—
27C2×D8×C127—216no—
28C2×D8×C127—216no—
29C2×D8×C127—24no—
30C2×D8×C127—24no—
31C2×D8×C127—22no—
32C2×D8×C127C2×C414yes—
33C2×D8×C127D418yes—
34C2×M16×C127—22no—
35C22×C8×C127—24no—
36C22×C8×C127C2×C412yes—
37C22×C8×C127D411yescharacteristic
38C22×C8×C127D412yes—
39C22×C8×C127C2311yescharacteristic
40M32×C127—24no—
41C2×C16×C127—22no—
42#108×C127—24no—
43C22⋊C8×C127—42no—
44C24×C127—48no—
45C24×C127—44no—
46C24×C127—42no—
47C24×C127—88no—
48C24×C127SD1611yescharacteristic
49C2×D4×C127—216no—
50C2×D4×C127—24no—
51C2×D4×C127—28no—
52C2×D4×C127—48no—
53C2×D4×C127—48no—
54C2×D4×C127—416no—
55C2×D4×C127—44no—
56C2×D4×C127—42no—
57C2×D4×C127—44no—
58C22×C4×C127—22no—
59C22×C4×C127—22no—
60C22×C4×C127C22⋊C412yes—
61C22×C4×C127D812yes—
62C22×C4×C127SD1611yescharacteristic
63C22×C4×C127C2×D411yescharacteristic
64D8×C127—216no—
65D8×C127—24no—
66D8×C127—28no—
67D8×C127C22⋊C418yes—
68M16×C127—42no—
69M16×C127—42no—
70C2×C8×C127—22no—
71C2×C8×C127—24no—
72C2×C8×C127—24no—
73C2×C8×C127—42no—
74C2×C8×C127C22⋊C412yes—
75C2×C8×C127C22×C411yescharacteristic
76C2×C8×C127C2×D414yes—
77C2×C8×C127C2×D411yescharacteristic
78C16×C127—42no—
79C23×C127—24no—
80C23×C127—22no—
81C23×C127—416no—
82C23×C127—44no—
83C23×C127—48no—
84C23×C127—832no—
85C23×C127—88no—
86C23×C127—88no—
87C23×C127—88no—
88C23×C127#10212yes—
89C23×C127C2×SD1611yescharacteristic
90D4×C127—28no—
91D4×C127—416no—
92D4×C127—44no—
93D4×C127—48no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—24no—
98C2×C4×C127#10212yes—
99C2×C4×C127C2×C22⋊C411yescharacteristic
100C2×C4×C127#12012yes—
101C2×C4×C127C2×D811yescharacteristic
102C2×C4×C127AGL(1,ℤ/8)12yes—
103C8×C127—22no—
104C8×C127—82no—
105C8×C127C2×C22⋊C412yes—
106C8×C127#12012yes—
107C22×C127—28no—
108C22×C127—22no—
109C22×C127—22no—
110C22×C127—22no—
111C22×C127—48no—
112C22×C127—832no—
113C22×C127—816no—
114C22×C127—88no—
115C22×C127#36114yescomplemented
116C22×C127C2×#10211yescharacteristic
117C22×C127#45012yes—
118C4×C127—22no—
119C4×C127#37911yescharacteristic
120C4×C127#41811yescharacteristic
121C4×C127#44712yes—
122#12736C12711yesmaximal, characteristic, direct product factor, Sylow
123C2×C127—24no—
124C2×C127—22no—
125C2×C127—88no—
126C2×C127#178611yescharacteristic
127C2×C127C2×#36112yes—
128C23×D8C2×C12711yescharacteristic
129C2×#361C2×C12714yescomplemented
130#2112C2×C12711yescharacteristic, complemented
131#1810C2×C12711yescharacteristic, complemented
132C127#1273611yescharacteristic, direct product factor, Sylow
133C23×D4—21no—
134C22×D8—28no—
135C22×D8—21no—
136C22×D8—22no—
137C22×D8C4×C12714yes—
138C22×D8C22×C12712yes—
139C23×C8C22×C12711yescharacteristic
140C2×M32C22×C12712yescomplemented
141C2×#108C22×C12712yescomplemented
142#407—21no—
143#361—28no—
144C22⋊C16—22no—
145#341—21no—
146C25—41no—
147C22×D4—28no—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—44no—
151C22×D4—42no—
152C22×D4—41no—
153C23×C4D4×C12711yescharacteristic
154C2×D8—28no—
155C2×D8—216no—
156C2×D8—216no—
157C2×D8—24no—
158C2×D8—24no—
159C2×D8—22no—
160C2×D8C2×C4×C12714yes—
161C2×D8D4×C12718yes—
162C2×M16—22no—
163C22×C8—24no—
164C22×C8C2×C4×C12712yes—
165C22×C8D4×C12711yescharacteristic
166C22×C8D4×C12712yes—
167C22×C8C23×C12711yescharacteristic
168M32—24no—
169C2×C16—22no—
170#108—24no—
171C22⋊C8—42no—
172C24—48no—
173C24—44no—
174C24—42no—
175C24—88no—
176C24SD16×C12711yescharacteristic
177C2×D4—216no—
178C2×D4—24no—
179C2×D4—28no—
180C2×D4—48no—
181C2×D4—48no—
182C2×D4—416no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—44no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4C22⋊C4×C12712yes—
189C22×C4D8×C12712yes—
190C22×C4SD16×C12711yescharacteristic
191C22×C4C2×D4×C12711yescharacteristic
192D8—216no—
193D8—24no—
194D8—28no—
195D8C22⋊C4×C12718yes—
196M16—42no—
197M16—42no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—42no—
202C2×C8C22⋊C4×C12712yes—
203C2×C8C22×C4×C12711yescharacteristic
204C2×C8C2×D4×C12714yes—
205C2×C8C2×D4×C12711yescharacteristic
206C16—42no—
207C23—24no—
208C23—22no—
209C23—416no—
210C23—44no—
211C23—48no—
212C23—832no—
213C23—88no—
214C23—88no—
215C23—88no—
216C23#102×C12712yes—
217C23C2×SD16×C12711yescharacteristic
218D4—28no—
219D4—416no—
220D4—44no—
221D4—48no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4#102×C12712yes—
227C2×C4C2×C22⋊C4×C12711yescharacteristic
228C2×C4#120×C12712yes—
229C2×C4C2×D8×C12711yescharacteristic
230C2×C4AGL(1,ℤ/8)×C12712yes—
231C8—22no—
232C8—82no—
233C8C2×C22⋊C4×C12712yes—
234C8#120×C12712yes—
235C22—28no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—48no—
240C22—832no—
241C22—816no—
242C22—88no—
243C22#361×C12714yescomplemented
244C22C2×#102×C12711yescharacteristic
245C22#450×C12712yes—
246C4—22no—
247C4#379×C12711yescharacteristic
248C4#418×C12711yescharacteristic
249C4#447×C12712yes—
250C2—24no—
251C2—22no—
252C2—88no—
253C2C127×#178611yescharacteristic
254C2C2×#361×C12712yes—
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC23×D8×C127C2
LCSG ▷ C2×C8 ▷ 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
2. C2×#361×C127C2semidirect
3. C127×#2112C2semidirect
4. C127×#1810C2semidirect
12. C2×M32×C127C22semidirect
13. C2×#108×C127C22semidirect
115. C22×C127#361semidirect
122. #12736C127direct (N × H)
129. C2×#361C2×C127semidirect
130. #2112C2×C127semidirect
131. #1810C2×C127semidirect
132. C127#12736direct (N × H)
140. C2×M32C22×C127semidirect
141. C2×#108C22×C127semidirect
243. C22#361×C127semidirect

Maximal subgroups

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

subgroupindex
C23×D8×C1272
C2×#361×C1272
C127×#21122
C127×#18102
#12736127