Finite Group Databasebeta

← C127×#12876 #44576549

Subgroups — C127×#12876 #44576549

Order 32512.

370 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 138 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×#430×C127C211yesmaximal, characteristic
2C2×#349×C127C211yesmaximal, characteristic
3C127×#1811C211yesmaximal, characteristic
4C127×#1388C212yesmaximal, complemented
5C127×#1385C212yesmaximal, complemented
6C22×M16×C127—21no—
7C22×M16×C127C2211yescharacteristic
8C2×M32×C127C2212yes—
9#430×C127—22no—
10#430×C127C414yes—
11C2×#108×C127—21no—
12C2×#108×C127C2211yescharacteristic
13C2×#108×C127C2211yescharacteristic
14C2×#108×C127C2212yes—
15#407×C127—21no—
16#349×C127—22no—
17#341×C127—21no—
18C23×C4×C127D411yescharacteristic
19C2×M16×C127—22no—
20C2×M16×C127—22no—
21C2×M16×C127—24no—
22C2×M16×C127—22no—
23C2×M16×C127—21no—
24C2×M16×C127—21no—
25C2×M16×C127—42no—
26C2×M16×C127—41no—
27C2×M16×C127C2×C412yes—
28C2×M16×C127D412yes—
29C22×C8×C127—41no—
30C22×C8×C127D411yescharacteristic
31C22×C8×C127C2311yescharacteristic
32M32×C127—22no—
33M32×C127—22no—
34C2×C16×C127—22no—
35#108×C127—24no—
36#108×C127—22no—
37#108×C127—22no—
38#108×C127—22no—
39#108×C127C2×C414yes—
40C22⋊C8×C127—42no—
41C24×C127Q1611yescharacteristic
42C22×C4×C127—21no—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127C22⋊C412yes—
47C22×C4×C127SD1612yes—
48C22×C4×C127Q1611yescharacteristic
49C22×C4×C127C2×D411yescharacteristic
50M16×C127—24no—
51M16×C127—24no—
52M16×C127—24no—
53M16×C127—44no—
54M16×C127—42no—
55M16×C127—41no—
56M16×C127—41no—
57M16×C127—41no—
58M16×C127—41no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127—44no—
62C2×C8×C127—42no—
63C2×C8×C127—41no—
64C2×C8×C127—41no—
65C2×C8×C127C22⋊C412yes—
66C2×C8×C127C22×C411yescharacteristic
67C2×C8×C127C2×D411yescharacteristic
68C2×C8×C127#4112yes—
69C2×C8×C127#4112yes—
70C16×C127—42no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127—81no—
74C23×C127#10312yes—
75C23×C127C2×Q1611yescharacteristic
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127#10312yes—
83C2×C4×C127C2×C22⋊C411yescharacteristic
84C2×C4×C127#12311yescharacteristic
85C2×C4×C127#12311yescharacteristic
86C2×C4×C127C2×SD1611yescharacteristic
87C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
88C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
89C8×C127—22no—
90C8×C127—44no—
91C8×C127—81no—
92C8×C127—81no—
93C8×C127#11712yes—
94C8×C127#12312yes—
95C22×C127—22no—
96C22×C127—41no—
97C22×C127—42no—
98C22×C127—42no—
99C22×C127—81no—
100C22×C127C2×#10311yescharacteristic
101C22×C127#48411yescharacteristic
102C22×C127#48411yescharacteristic
103C4×C127—22no—
104C4×C127#38111yescharacteristic
105C4×C127#41811yescharacteristic
106C4×C127#48111yescharacteristic
107C4×C127#48111yescharacteristic
108#12876C12711yesmaximal, characteristic, direct product factor, Sylow
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—42no—
113C2×C127#182411yescharacteristic
114C2×#430C2×C12711yescharacteristic
115C2×#349C2×C12711yescharacteristic
116#1811C2×C12711yescharacteristic
117#1388C2×C12712yescomplemented
118#1385C2×C12712yescomplemented
119C127#1287611yescharacteristic, direct product factor, Sylow
120C22×M16—21no—
121C22×M16C22×C12711yescharacteristic
122C2×M32C22×C12712yes—
123#430—22no—
124#430C4×C12714yes—
125C2×#108—21no—
126C2×#108C22×C12711yescharacteristic
127C2×#108C22×C12711yescharacteristic
128C2×#108C22×C12712yes—
129#407—21no—
130#349—22no—
131#341—21no—
132C23×C4D4×C12711yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—24no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—42no—
140C2×M16—41no—
141C2×M16C2×C4×C12712yes—
142C2×M16D4×C12712yes—
143C22×C8—41no—
144C22×C8D4×C12711yescharacteristic
145C22×C8C23×C12711yescharacteristic
146M32—22no—
147M32—22no—
148C2×C16—22no—
149#108—24no—
150#108—22no—
151#108—22no—
152#108—22no—
153#108C2×C4×C12714yes—
154C22⋊C8—42no—
155C24Q16×C12711yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4C22⋊C4×C12712yes—
161C22×C4SD16×C12712yes—
162C22×C4Q16×C12711yescharacteristic
163C22×C4C2×D4×C12711yescharacteristic
164M16—24no—
165M16—24no—
166M16—24no—
167M16—44no—
168M16—42no—
169M16—41no—
170M16—41no—
171M16—41no—
172M16—41no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—44no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8C22⋊C4×C12712yes—
180C2×C8C22×C4×C12711yescharacteristic
181C2×C8C2×D4×C12711yescharacteristic
182C2×C8#41×C12712yes—
183C2×C8#41×C12712yes—
184C16—42no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23#103×C12712yes—
189C23C2×Q16×C12711yescharacteristic
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4#103×C12712yes—
197C2×C4C2×C22⋊C4×C12711yescharacteristic
198C2×C4#123×C12711yescharacteristic
199C2×C4#123×C12711yescharacteristic
200C2×C4C2×SD16×C12711yescharacteristic
201C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
202C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
203C8—22no—
204C8—44no—
205C8—81no—
206C8—81no—
207C8#117×C12712yes—
208C8#123×C12712yes—
209C22—22no—
210C22—41no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22C2×#103×C12711yescharacteristic
215C22#484×C12711yescharacteristic
216C22#484×C12711yescharacteristic
217C4—22no—
218C4#381×C12711yescharacteristic
219C4#418×C12711yescharacteristic
220C4#481×C12711yescharacteristic
221C4#481×C12711yescharacteristic
222C2—21no—
223C2—21no—
224C2—21no—
225C2—42no—
226C2C127×#182411yescharacteristic
subgroupG/H
centerC4×C127#481
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
socleC2×C127#1824
prime radicalC24×C127Q16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C22×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1388C2semidirect
5. C127×#1385C2semidirect
108. #12876C127direct (N × H)
117. #1388C2×C127semidirect
118. #1385C2×C127semidirect
119. C127#12876direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#430×C1272
C2×#349×C1272
C127×#18112
C127×#13882
C127×#13852
#12876127