Finite Group Databasebeta

← C127×#7131 #44570804

Subgroups — C127×#7131 #44570804

Order 32512.

476 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1446C211yesmaximal, characteristic
2C127×#1280C212yesmaximal
3C23×Q8×C127C411yescharacteristic
4C2×C22⋊C8×C127—22no—
5C2×#95×C127—22no—
6C2×#95×C127C411yescharacteristic
7C2×#95×C127C2211yescharacteristic
8C22×Q8×C127—21no—
9C22×Q8×C127—44no—
10C22×Q8×C127—41no—
11C22×Q8×C127—41no—
12C22×Q8×C127C812yescomplemented
13C23×C4×C127—21no—
14C23×C4×C127—21no—
15C23×C4×C127C2×C411yescharacteristic
16C23×C4×C127D412yes—
17C22×C8×C127—42no—
18C22⋊C8×C127—24no—
19C22⋊C8×C127—42no—
20#95×C127—41no—
21#95×C127—44no—
22#95×C127—41no—
23C24×C127C22⋊C411yescharacteristic
24C2×Q8×C127—44no—
25C2×Q8×C127—44no—
26C2×Q8×C127—42no—
27C2×Q8×C127—42no—
28C2×Q8×C127—48no—
29C2×Q8×C127—44no—
30C2×Q8×C127—44no—
31C22×C4×C127—24no—
32C22×C4×C127—22no—
33C22×C4×C127—22no—
34C22×C4×C127—21no—
35C22×C4×C127—21no—
36C22×C4×C127—22no—
37C22×C4×C127—22no—
38C22×C4×C127—42no—
39C22×C4×C127—41no—
40C22×C4×C127—41no—
41C22×C4×C127—42no—
42C22×C4×C127—42no—
43C22×C4×C127—42no—
44C22×C4×C127—42no—
45C22×C4×C127—41no—
46C22×C4×C127—41no—
47C22×C4×C127—42no—
48C22×C4×C127C2×C811yescharacteristic
49C22×C4×C127M1611yescharacteristic
50C22×C4×C127SD1612yes—
51C22×C4×C127Q1612yes—
52C2×C8×C127—44no—
53C2×C8×C127—82no—
54C2×C8×C127—82no—
55C23×C127—22no—
56C23×C127—21no—
57C23×C127—21no—
58C23×C127C22⋊C811yescharacteristic
59C23×C127#9911yescharacteristic
60C23×C127#10011yescharacteristic
61C23×C127#10312yes—
62C23×C127#10412yes—
63Q8×C127—44no—
64Q8×C127—44no—
65Q8×C127—48no—
66C2×C4×C127—24no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—42no—
72C2×C4×C127—44no—
73C2×C4×C127—42no—
74C2×C4×C127—42no—
75C2×C4×C127—42no—
76C2×C4×C127—44no—
77C2×C4×C127—42no—
78C2×C4×C127—42no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—42no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127—44no—
85C2×C4×C127—82no—
86C2×C4×C127—81no—
87C2×C4×C127—82no—
88C2×C4×C127—81no—
89C8×C127—84no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—22no—
95C22×C127—22no—
96C22×C127—22no—
97C22×C127—22no—
98C22×C127#32311yescharacteristic
99C22×C127Q8⋊C812yes—
100C22×C127#32812yes—
101C22×C127#32911yescharacteristic
102C22×C127#33311yescharacteristic
103C22×C127AΣL(1,𝔽16)12yes—
104C22×C127#35411yescharacteristic
105C22×C127#35611yescharacteristic
106C4×C127—44no—
107C4×C127—41no—
108C4×C127—41no—
109C4×C127—42no—
110C4×C127—42no—
111C4×C127—82no—
112#7131C12711yesmaximal, characteristic, direct product factor, Sylow
113C2×C127—21no—
114C2×C127—22no—
115C2×C127—21no—
116C2×C127#127211yescharacteristic
117C2×C127#131611yescharacteristic
118C2×C127#131911yescharacteristic
119C2×C127#135112yes—
120C2×C127#135412yes—
121#1446C2×C12711yescharacteristic
122#1280C2×C12712yes—
123C127#713111yescharacteristic, direct product factor, Sylow
124C23×Q8C4×C12711yescharacteristic
125C2×C22⋊C8—22no—
126C2×#95—22no—
127C2×#95C4×C12711yescharacteristic
128C2×#95C22×C12711yescharacteristic
129C22×Q8—21no—
130C22×Q8—44no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8C8×C12712yescomplemented
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×C4×C12711yescharacteristic
137C23×C4D4×C12712yes—
138C22×C8—42no—
139C22⋊C8—24no—
140C22⋊C8—42no—
141#95—41no—
142#95—41no—
143#95—44no—
144C24C22⋊C4×C12711yescharacteristic
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—48no—
150C2×Q8—44no—
151C2×Q8—44no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4C2×C8×C12711yescharacteristic
170C22×C4M16×C12711yescharacteristic
171C22×C4SD16×C12712yes—
172C22×C4Q16×C12712yes—
173C2×C8—44no—
174C2×C8—82no—
175C2×C8—82no—
176C23—22no—
177C23—21no—
178C23—21no—
179C23C22⋊C8×C12711yescharacteristic
180C23#99×C12711yescharacteristic
181C23#100×C12711yescharacteristic
182C23#103×C12712yes—
183C23#104×C12712yes—
184Q8—44no—
185Q8—44no—
186Q8—48no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—82no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—81no—
210C8—84no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22#323×C12711yescharacteristic
220C22Q8⋊C8×C12712yes—
221C22#328×C12712yes—
222C22#329×C12711yescharacteristic
223C22#333×C12711yescharacteristic
224C22AΣL(1,𝔽16)×C12712yes—
225C22#354×C12711yescharacteristic
226C22#356×C12711yescharacteristic
227C4—44no—
228C4—41no—
229C4—41no—
230C4—42no—
231C4—42no—
232C4—82no—
233C2—21no—
234C2—22no—
235C2—21no—
236C2C127×#127211yescharacteristic
237C2C127×#131611yescharacteristic
238C2C127×#131911yescharacteristic
239C2C127×#135112yes—
240C2C127×#135412yes—
subgroupG/H
center, socleC23×C127#99
commutatorC22×C4C2×C8×C127
Frattini, cosocleC2×#95C22×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C127 ◁ C2×#95×C127 ◁ G

Complements

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

NH ≅ G/Ntype
12. C22×Q8×C127C8semidirect
112. #7131C127direct (N × H)
123. C127#7131direct (N × H)
133. C22×Q8C8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#14462
C127×#12802
#7131127