Finite Group Databasebeta

← C127×#12170 #44575843

Subgroups — C127×#12170 #44575843

Order 32512.

356 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C127×#3259C211yesmaximal, characteristic, complemented
2C127×#2190C212yesmaximal, complemented
3C127×#2053C211yesmaximal, characteristic, complemented
4C127×#1339C212yesmaximal, complemented
5C127×#1328C211yesmaximal, characteristic, complemented
6#575×C127C2211yescharacteristic
7#540×C127—21no—
8#509×C127—22no—
9#478×C127—22no—
10#469×C127C2212yes—
11#466×C127—21no—
12#459×C127—22no—
13#424×C127C2211yescharacteristic
14#416×C127—21no—
15#410×C127—21no—
16#360×C127—22no—
17#349×C127C2212yes—
18H(3,ℤ/4)×C127C2211yescharacteristic
19C2×#41×C127—21no—
20#137×C127—41no—
21AGL(1,ℤ/8)×C127—41no—
22#135×C127—22no—
23C2×D8×C127—21no—
24C2×D8×C127D412yes—
25C2×M16×C127C2311yescharacteristic
26C22×C8×C127D411yescharacteristic
27#127×C127—21no—
28#125×C127—21no—
29#123×C127—42no—
30#121×C127—22no—
31#121×C127—42no—
32C4×D4×C127—22no—
33#117×C127—21no—
34#117×C127D411yescharacteristic
35SD32×C127—42no—
36D16×C127—42no—
37M32×C127—22no—
38#107×C127—21no—
39#106×C127D412yes—
40C4⋊C8×C127—22no—
41#103×C127—41no—
42#102×C127—22no—
43#102×C127—22no—
44#102×C127—41no—
45#99×C127—42no—
46#41×C127—41no—
47#41×C127—41no—
48#41×C127—41no—
49#41×C127—41no—
50C2×Q8×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—22no—
53C2×D4×C127—42no—
54C2×D4×C127—41no—
55C2×D4×C127—42no—
56C22×C4×C127—41no—
57C22×C4×C127—42no—
58C22×C4×C127C2×D411yescharacteristic
59Q16×C127—22no—
60SD16×C127—42no—
61D8×C127—22no—
62D8×C127—42no—
63D8×C127—81no—
64M16×C127—41no—
65C2×C8×C127—21no—
66C2×C8×C127—41no—
67C2×C8×C127C2×D412yes—
68C4⋊C4×C127—21no—
69C4⋊C4×C127—22no—
70C4⋊C4×C127—42no—
71C4⋊C4×C127—41no—
72C22⋊C4×C127—42no—
73C22⋊C4×C127—42no—
74C22⋊C4×C127—41no—
75C22⋊C4×C127D812yes—
76C42×C127—21no—
77C42×C127—41no—
78C16×C127—42no—
79C23×C127—41no—
80C23×C127—42no—
81C23×C127C2×D811yescharacteristic
82Q8×C127—41no—
83Q8×C127—41no—
84D4×C127—41no—
85D4×C127—42no—
86D4×C127—41no—
87D4×C127—42no—
88D4×C127—81no—
89D4×C127—81no—
90D4×C127—162no—
91D4×C127—162no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—41no—
96C2×C4×C127—42no—
97C2×C4×C127—81no—
98C2×C4×C127—81no—
99C2×C4×C127—81no—
100C2×C4×C127—162no—
101C2×C4×C127#12011yescharacteristic
102C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
103C8×C127—22no—
104C8×C127—41no—
105C22×C127—21no—
106C22×C127—41no—
107C22×C127—82no—
108C22×C127—82no—
109C22×C127—81no—
110C22×C127—82no—
111C22×C127—161no—
112C22×C127#44711yescharacteristic
113C4×C127—21no—
114C4×C127—42no—
115C4×C127—82no—
116C4×C127—81no—
117C4×C127—161no—
118C4×C127#45411yescharacteristic
119C4×C127B(4,𝔽2)11yescharacteristic
120#12170C12711yesmaximal, characteristic, direct product factor, Sylow
121C2×C127—21no—
122C2×C127—41no—
123C2×C127—162no—
124C2×C127—161no—
125C2×C127#159811yescharacteristic
126#3259C2×C12711yescharacteristic, complemented
127#2190C2×C12712yescomplemented
128#2053C2×C12711yescharacteristic, complemented
129#1339C2×C12712yescomplemented
130#1328C2×C12711yescharacteristic, complemented
131C127#1217011yescharacteristic, direct product factor, Sylow
132#575C22×C12711yescharacteristic
133#540—21no—
134#509—22no—
135#478—22no—
136#469C22×C12712yes—
137#466—21no—
138#459—22no—
139#424C22×C12711yescharacteristic
140#416—21no—
141#410—21no—
142#360—22no—
143#349C22×C12712yes—
144H(3,ℤ/4)C22×C12711yescharacteristic
145C2×#41—21no—
146#137—41no—
147AGL(1,ℤ/8)—41no—
148#135—22no—
149C2×D8—21no—
150C2×D8D4×C12712yes—
151C2×M16C23×C12711yescharacteristic
152C22×C8D4×C12711yescharacteristic
153#127—21no—
154#125—21no—
155#123—42no—
156#121—22no—
157#121—42no—
158C4×D4—22no—
159#117—21no—
160#117D4×C12711yescharacteristic
161SD32—42no—
162D16—42no—
163M32—22no—
164#107—21no—
165#106D4×C12712yes—
166C4⋊C8—22no—
167#103—41no—
168#102—22no—
169#102—22no—
170#102—41no—
171#99—42no—
172#41—41no—
173#41—41no—
174#41—41no—
175#41—41no—
176C2×Q8—21no—
177C2×D4—21no—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4C2×D4×C12711yescharacteristic
185Q16—22no—
186SD16—42no—
187D8—22no—
188D8—42no—
189D8—81no—
190M16—41no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8C2×D4×C12712yes—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4D8×C12712yes—
202C42—21no—
203C42—41no—
204C16—42no—
205C23—41no—
206C23—42no—
207C23C2×D8×C12711yescharacteristic
208Q8—41no—
209Q8—41no—
210D4—41no—
211D4—42no—
212D4—41no—
213D4—42no—
214D4—81no—
215D4—81no—
216D4—162no—
217D4—162no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—162no—
227C2×C4#120×C12711yescharacteristic
228C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
229C8—22no—
230C8—41no—
231C22—21no—
232C22—41no—
233C22—82no—
234C22—82no—
235C22—81no—
236C22—82no—
237C22—161no—
238C22#447×C12711yescharacteristic
239C4—21no—
240C4—42no—
241C4—82no—
242C4—81no—
243C4—161no—
244C4#454×C12711yescharacteristic
245C4B(4,𝔽2)×C12711yescharacteristic
246C2—21no—
247C2—41no—
248C2—162no—
249C2—161no—
250C2C127×#159811yescharacteristic
subgroupG/H
center, socleC2×C127#1598
commutator, Frattini, cosocleC2×M16C23×C127
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3259C2semidirect
2. C127×#2190C2semidirect
3. C127×#2053C2semidirect
4. C127×#1339C2semidirect
5. C127×#1328C2semidirect
120. #12170C127direct (N × H)
126. #3259C2×C127semidirect
127. #2190C2×C127semidirect
128. #2053C2×C127semidirect
129. #1339C2×C127semidirect
130. #1328C2×C127semidirect
131. C127#12170direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32592
C127×#21902
C127×#20532
C127×#13392
C127×#13282
#12170127