Finite Group Databasebeta

← C127×#7256 #44570929

Subgroups — C127×#7256 #44570929

Order 32512.

318 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 98 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×#349×C127C211yesmaximal, characteristic
2C127×#1811C211yesmaximal, characteristic
3C2×H(3,ℤ/4)×C127C211yesmaximal, characteristic
4C22×M16×C127C2211yescharacteristic
5C2×#117×C127—21no—
6C2×M32×C127—21no—
7C2×#108×C127C411yescharacteristic, complemented
8C2×#108×C127C411yescharacteristic, complemented
9#407×C127—21no—
10#349×C127—21no—
11#349×C127C412yescomplemented
12#341×C127—21no—
13H(3,ℤ/4)×C127—21no—
14H(3,ℤ/4)×C127C412yes—
15C23×C4×C127D411yescharacteristic
16C2×M16×C127—21no—
17C2×M16×C127—21no—
18C2×M16×C127—41no—
19C2×M16×C127—41no—
20C2×M16×C127C2×C411yescharacteristic
21C2×M16×C127C2×C411yescharacteristic
22C2×M16×C127D411yescharacteristic
23C2×M16×C127Q811yescharacteristic
24C22×C8×C127C2×C411yescharacteristic
25C22×C8×C127D411yescharacteristic
26#117×C127—22no—
27#117×C127—22no—
28#117×C127—41no—
29#117×C127—41no—
30C2×C4⋊C4×C127—41no—
31C2×C22⋊C4×C127—41no—
32C2×C42×C127—22no—
33M32×C127—22no—
34M32×C127—41no—
35C2×C16×C127—41no—
36#108×C127—41no—
37#108×C127—41no—
38C22⋊C8×C127—41no—
39C22⋊C8×C127—41no—
40C24×C127SD1611yescharacteristic
41C22×C4×C127—21no—
42C22×C4×C127—21no—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22×C4×C127—42no—
46C22×C4×C127C22⋊C411yescharacteristic
47C22×C4×C127C22⋊C411yescharacteristic
48C22×C4×C127C4⋊C411yescharacteristic
49C22×C4×C127D811yescharacteristic
50C22×C4×C127SD1611yescharacteristic
51C22×C4×C127Q1611yescharacteristic
52M16×C127—41no—
53M16×C127—41no—
54M16×C127—41no—
55M16×C127—41no—
56M16×C127—41no—
57M16×C127—41no—
58M16×C127—41no—
59M16×C127—41no—
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127—41no—
65C2×C8×C127—41no—
66C2×C8×C127C4211yescharacteristic
67C2×C8×C127C22⋊C411yescharacteristic
68C2×C8×C127C4⋊C411yescharacteristic
69C2×C8×C127C4⋊C411yescharacteristic
70C4⋊C4×C127—42no—
71C4⋊C4×C127—42no—
72C22⋊C4×C127—42no—
73C22⋊C4×C127—42no—
74C42×C127—24no—
75C42×C127—42no—
76C16×C127—42no—
77C23×C127—21no—
78C23×C127—21no—
79C23×C127—81no—
80C23×C127#10211yescharacteristic
81C23×C127#10311yescharacteristic
82C23×C127#10611yescharacteristic
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—44no—
94C2×C4×C127—84no—
95C2×C4×C127#9511yescharacteristic
96C2×C4×C127#9911yescharacteristic
97C2×C4×C127#9911yescharacteristic
98C2×C4×C127#10211yescharacteristic
99C2×C4×C127#10311yescharacteristic
100C2×C4×C127#10411yescharacteristic
101C2×C4×C127#10411yescharacteristic
102C2×C4×C127#10711yescharacteristic
103C8×C127—21no—
104C8×C127—21no—
105C8×C127—21no—
106C8×C127—21no—
107C8×C127—81no—
108C8×C127—81no—
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—81no—
116C22×C127#32811yescharacteristic
117C22×C127#32811yescharacteristic
118C22×C127#34011yescharacteristic
119C4×C127—21no—
120C4×C127—21no—
121C4×C127—84no—
122C4×C127#32711yescharacteristic
123C4×C127#32711yescharacteristic
124C4×C127#33911yescharacteristic
125C4×C127#34211yescharacteristic
126#7256C12711yesmaximal, characteristic, direct product factor, Sylow
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—21no—
130C2×C127—41no—
131C2×C127—41no—
132C2×C127#129411yescharacteristic
133C2×#349C2×C12711yescharacteristic
134#1811C2×C12711yescharacteristic
135C2×H(3,ℤ/4)C2×C12711yescharacteristic
136C127#725611yescharacteristic, direct product factor, Sylow
137C22×M16C22×C12711yescharacteristic
138C2×#117—21no—
139C2×M32—21no—
140C2×#108C4×C12711yescharacteristic, complemented
141C2×#108C4×C12711yescharacteristic, complemented
142#407—21no—
143#349—21no—
144#349C4×C12712yescomplemented
145#341—21no—
146H(3,ℤ/4)—21no—
147H(3,ℤ/4)C4×C12712yes—
148C23×C4D4×C12711yescharacteristic
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—41no—
152C2×M16—41no—
153C2×M16C2×C4×C12711yescharacteristic
154C2×M16C2×C4×C12711yescharacteristic
155C2×M16D4×C12711yescharacteristic
156C2×M16Q8×C12711yescharacteristic
157C22×C8C2×C4×C12711yescharacteristic
158C22×C8D4×C12711yescharacteristic
159#117—22no—
160#117—22no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C42—22no—
166M32—22no—
167M32—41no—
168C2×C16—41no—
169#108—41no—
170#108—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C24SD16×C12711yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4C22⋊C4×C12711yescharacteristic
180C22×C4C22⋊C4×C12711yescharacteristic
181C22×C4C4⋊C4×C12711yescharacteristic
182C22×C4D8×C12711yescharacteristic
183C22×C4SD16×C12711yescharacteristic
184C22×C4Q16×C12711yescharacteristic
185M16—41no—
186M16—41no—
187M16—41no—
188M16—41no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—41no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8C42×C12711yescharacteristic
200C2×C8C22⋊C4×C12711yescharacteristic
201C2×C8C4⋊C4×C12711yescharacteristic
202C2×C8C4⋊C4×C12711yescharacteristic
203C4⋊C4—42no—
204C4⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C42—24no—
208C42—42no—
209C16—42no—
210C23—21no—
211C23—21no—
212C23—81no—
213C23#102×C12711yescharacteristic
214C23#103×C12711yescharacteristic
215C23#106×C12711yescharacteristic
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—44no—
227C2×C4—84no—
228C2×C4#95×C12711yescharacteristic
229C2×C4#99×C12711yescharacteristic
230C2×C4#99×C12711yescharacteristic
231C2×C4#102×C12711yescharacteristic
232C2×C4#103×C12711yescharacteristic
233C2×C4#104×C12711yescharacteristic
234C2×C4#104×C12711yescharacteristic
235C2×C4#107×C12711yescharacteristic
236C8—21no—
237C8—21no—
238C8—21no—
239C8—21no—
240C8—81no—
241C8—81no—
242C22—21no—
243C22—21no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—42no—
248C22—81no—
249C22#328×C12711yescharacteristic
250C22#328×C12711yescharacteristic
251C22#340×C12711yescharacteristic
252C4—21no—
253C4—21no—
254C4—84no—
255C4#327×C12711yescharacteristic
256C4#327×C12711yescharacteristic
257C4#339×C12711yescharacteristic
258C4#342×C12711yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—41no—
263C2—41no—
264C2C127×#129411yescharacteristic
subgroupG/H
centerC4×C127#327
commutatorC2×C8C42×C127
Frattini, cosocleC22×M16C22×C127
socleC2×C127#1294
prime radicalC24×C127SD16
LCSG ▷ C2×C8 ▷ C2×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
7. C2×#108×C127C4semidirect
8. C2×#108×C127C4semidirect
11. #349×C127C4semidirect
126. #7256C127direct (N × H)
136. C127#7256direct (N × H)
140. C2×#108C4×C127semidirect
141. C2×#108C4×C127semidirect
144. #349C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#349×C1272
C127×#18112
C2×H(3,ℤ/4)×C1272
#7256127