Finite Group Databasebeta

← C127×#8526 #44572199

Subgroups — C127×#8526 #44572199

Order 32512.

784 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#540×C127C211yesmaximal, characteristic
2C127×#1555C212yesmaximal, complemented
3C127×#1534C212yesmaximal, complemented
4C127×#1464C212yesmaximal, complemented
5#540×C127—24no—
6#540×C127—24no—
7C2×#127×C127C2211yescharacteristic
8C2×#125×C127C2211yescharacteristic
9C2×#123×C127—22no—
10C2×#121×C127—22no—
11C2×#121×C127C412yes—
12C2×C4×D4×C127C412yes—
13C2×#117×C127C2211yescharacteristic
14#432×C127C2214yes—
15#331×C127—24no—
16#330×C127—24no—
17#324×C127—22no—
18#323×C127—22no—
19C22×D4×C127—22no—
20C22×D4×C127—22no—
21C22×D4×C127C2×C412yes—
22C23×C4×C127—22no—
23C23×C4×C127C2×C411yescharacteristic
24#127×C127—22no—
25#127×C127D414yes—
26#125×C127—22no—
27#125×C127D414yes—
28#123×C127—216no—
29#121×C127—216no—
30#121×C127—24no—
31#121×C127—24no—
32C4×D4×C127—24no—
33C4×D4×C127—24no—
34#117×C127—22no—
35#117×C127—22no—
36C2×C4⋊C4×C127—22no—
37C2×C4⋊C4×C127C2×C412yes—
38C2×C4⋊C4×C127D412yes—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127D412yes—
42C2×C42×C127C2×C411yescharacteristic
43C2×C42×C127C2311yescharacteristic
44C4⋊C8×C127—24no—
45C22⋊C8×C127—24no—
46#97×C127—24no—
47C24×C127—22no—
48C24×C127C22⋊C411yescharacteristic
49C2×D4×C127—28no—
50C2×D4×C127—22no—
51C2×D4×C127—28no—
52C2×D4×C127—44no—
53C2×D4×C127—84no—
54C2×D4×C127—84no—
55C2×D4×C127C22⋊C414yes—
56C22×C4×C127—22no—
57C22×C4×C127—28no—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—42no—
61C22×C4×C127—84no—
62C22×C4×C127C22⋊C411yescharacteristic
63C22×C4×C127C22⋊C412yes—
64C22×C4×C127C22×C411yescharacteristic
65C22×C4×C127C2×D412yes—
66C2×C8×C127—44no—
67C4⋊C4×C127—22no—
68C4⋊C4×C127—28no—
69C4⋊C4×C127—24no—
70C4⋊C4×C127C22⋊C414yes—
71C22⋊C4×C127—28no—
72C22⋊C4×C127—28no—
73C22⋊C4×C127—24no—
74C42×C127—21no—
75C42×C127—21no—
76C42×C127C2×D412yes—
77C42×C127C2×D412yes—
78C23×C127—21no—
79C23×C127—24no—
80C23×C127—22no—
81C23×C127—44no—
82C23×C127—44no—
83C23×C127—42no—
84C23×C127—44no—
85C23×C127C2×C22⋊C411yescharacteristic
86D4×C127—44no—
87D4×C127—44no—
88D4×C127—88no—
89D4×C127—88no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—24no—
94C2×C4×C127—22no—
95C2×C4×C127—42no—
96C2×C4×C127—42no—
97C2×C4×C127—42no—
98C2×C4×C127—44no—
99C2×C4×C127—42no—
100C2×C4×C127—88no—
101C2×C4×C127C2×C22⋊C411yescharacteristic
102C2×C4×C127C2×C22⋊C411yescharacteristic
103C2×C4×C127#12014yes—
104C8×C127—84no—
105C22×C127—22no—
106C22×C127—44no—
107C22×C127—42no—
108C22×C127—44no—
109C22×C127—41no—
110C22×C127—44no—
111C22×C127—82no—
112C22×C127—88no—
113C22×C127#37911yescharacteristic
114C22×C127#41312yes—
115C4×C127—24no—
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—44no—
119C4×C127—42no—
120C4×C127—41no—
121#8526C12711yesmaximal, characteristic, direct product factor, Sylow
122C2×C127—22no—
123C2×C127—81no—
124C2×C127—84no—
125C2×C127#178511yescharacteristic
126C2×C127#179512yes—
127C2×#540C2×C12711yescharacteristic
128#1555C2×C12712yescomplemented
129#1534C2×C12712yescomplemented
130#1464C2×C12712yescomplemented
131C127#852611yescharacteristic, direct product factor, Sylow
132#540—24no—
133#540—24no—
134C2×#127C22×C12711yescharacteristic
135C2×#125C22×C12711yescharacteristic
136C2×#123—22no—
137C2×#121—22no—
138C2×#121C4×C12712yes—
139C2×C4×D4C4×C12712yes—
140C2×#117C22×C12711yescharacteristic
141#432C22×C12714yes—
142#331—24no—
143#330—24no—
144#324—22no—
145#323—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×C4×C12712yes—
149C23×C4—22no—
150C23×C4C2×C4×C12711yescharacteristic
151#127—22no—
152#127D4×C12714yes—
153#125—22no—
154#125D4×C12714yes—
155#123—216no—
156#121—216no—
157#121—24no—
158#121—24no—
159C4×D4—24no—
160C4×D4—24no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×C4×C12712yes—
165C2×C4⋊C4D4×C12712yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4D4×C12712yes—
169C2×C42C2×C4×C12711yescharacteristic
170C2×C42C23×C12711yescharacteristic
171C4⋊C8—24no—
172C22⋊C8—24no—
173#97—24no—
174C24—22no—
175C24C22⋊C4×C12711yescharacteristic
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—44no—
180C2×D4—84no—
181C2×D4—84no—
182C2×D4C22⋊C4×C12714yes—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—84no—
189C22×C4C22⋊C4×C12711yescharacteristic
190C22×C4C22⋊C4×C12712yes—
191C22×C4C22×C4×C12711yescharacteristic
192C22×C4C2×D4×C12712yes—
193C2×C8—44no—
194C4⋊C4—22no—
195C4⋊C4—28no—
196C4⋊C4—24no—
197C4⋊C4C22⋊C4×C12714yes—
198C22⋊C4—28no—
199C22⋊C4—28no—
200C22⋊C4—24no—
201C42—21no—
202C42—21no—
203C42C2×D4×C12712yes—
204C42C2×D4×C12712yes—
205C23—21no—
206C23—24no—
207C23—22no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—44no—
212C23C2×C22⋊C4×C12711yescharacteristic
213D4—44no—
214D4—44no—
215D4—88no—
216D4—88no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—88no—
228C2×C4C2×C22⋊C4×C12711yescharacteristic
229C2×C4C2×C22⋊C4×C12711yescharacteristic
230C2×C4#120×C12714yes—
231C8—84no—
232C22—22no—
233C22—41no—
234C22—44no—
235C22—44no—
236C22—44no—
237C22—42no—
238C22—82no—
239C22—88no—
240C22#379×C12711yescharacteristic
241C22#413×C12712yes—
242C4—21no—
243C4—21no—
244C4—24no—
245C4—41no—
246C4—44no—
247C4—42no—
248C2—22no—
249C2—84no—
250C2—81no—
251C2C127×#178511yescharacteristic
252C2C127×#179512yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#540×C127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1555C2semidirect
3. C127×#1534C2semidirect
4. C127×#1464C2semidirect
121. #8526C127direct (N × H)
128. #1555C2×C127semidirect
129. #1534C2×C127semidirect
130. #1464C2×C127semidirect
131. C127#8526direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#540×C1272
C127×#15552
C127×#15342
C127×#14642
#8526127