Finite Group Databasebeta

← C127×#7202 #44570875

Subgroups — C127×#7202 #44570875

Order 32512.

304 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2112C211yesmaximal, characteristic
2C2×C22⋊C16×C127C211yesmaximal, characteristic
3C8×C22⋊C4×C127C211yesmaximal, characteristic
4C23×C8×C127C2211yescharacteristic
5C2×M32×C127C412yescomplemented
6C22×C16×C127—21no—
7C2×C22⋊C8×C127C411yescharacteristic
8C2×C4×C8×C127—21no—
9C4×C22⋊C4×C127C411yescharacteristic
10C22⋊C16×C127—21no—
11C22⋊C16×C127—21no—
12C22⋊C16×C127—21no—
13C22⋊C16×C127C412yescomplemented
14#336×C127—21no—
15C23×C4×C127C2×C411yescharacteristic
16C22×C8×C127—21no—
17C22×C8×C127—42no—
18C22×C8×C127C2×C411yescharacteristic
19C22×C8×C127C2×C411yescharacteristic
20C22×C8×C127D411yescharacteristic
21C22×C8×C127D411yescharacteristic
22C22×C8×C127D411yescharacteristic
23C22×C8×C127Q811yescharacteristic
24C2×C22⋊C4×C127—21no—
25C2×C42×C127—21no—
26M32×C127—42no—
27C2×C16×C127—22no—
28C2×C16×C127—22no—
29C2×C16×C127—42no—
30C22⋊C8×C127—21no—
31C22⋊C8×C127C812yes—
32C4×C8×C127—42no—
33#95×C127C812yes—
34C24×C127M1611yescharacteristic
35C22×C4×C127—22no—
36C22×C4×C127—41no—
37C22×C4×C127—41no—
38C22×C4×C127C4211yescharacteristic
39C22×C4×C127C22⋊C411yescharacteristic
40C22×C4×C127C4⋊C411yescharacteristic
41C22×C4×C127C2×C811yescharacteristic
42C22×C4×C127C2×C811yescharacteristic
43C22×C4×C127M1611yescharacteristic
44C2×C8×C127—22no—
45C2×C8×C127—22no—
46C2×C8×C127—22no—
47C2×C8×C127—22no—
48C2×C8×C127—41no—
49C2×C8×C127—41no—
50C2×C8×C127—41no—
51C2×C8×C127—41no—
52C2×C8×C127—41no—
53C2×C8×C127C22⋊C411yescharacteristic
54C2×C8×C127C22⋊C411yescharacteristic
55C2×C8×C127C4⋊C411yescharacteristic
56C2×C8×C127C4⋊C411yescharacteristic
57C22⋊C4×C127—24no—
58C42×C127—41no—
59C42×C127—41no—
60C16×C127—42no—
61C16×C127—42no—
62C23×C127—22no—
63C23×C127—22no—
64C23×C127—41no—
65C23×C127#9711yescharacteristic
66C23×C127C22⋊C811yescharacteristic
67C23×C127C4⋊C811yescharacteristic
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—24no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—42no—
79C2×C4×C127—42no—
80C2×C4×C127#9511yescharacteristic
81C2×C4×C127C4×C811yescharacteristic
82C2×C4×C127C22⋊C811yescharacteristic
83C2×C4×C127C4⋊C811yescharacteristic
84C8×C127—21no—
85C8×C127—21no—
86C8×C127—21no—
87C8×C127—42no—
88C8×C127#9912yes—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—44no—
97C22×C127#33611yescharacteristic
98C22×C127#35012yes—
99C22×C127#36412yes—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—21no—
103C4×C127—44no—
104C4×C127#34211yescharacteristic
105C4×C127#34411yescharacteristic
106#7202C12711yesmaximal, characteristic, direct product factor, Sylow
107C2×C127—22no—
108C2×C127—22no—
109C2×C127—41no—
110C2×C127#128211yescharacteristic
111C2×C127#137611yescharacteristic
112C2×C127#137911yescharacteristic
113#2112C2×C12711yescharacteristic
114C2×C22⋊C16C2×C12711yescharacteristic
115C8×C22⋊C4C2×C12711yescharacteristic
116C127#720211yescharacteristic, direct product factor, Sylow
117C23×C8C22×C12711yescharacteristic
118C2×M32C4×C12712yescomplemented
119C22×C16—21no—
120C2×C22⋊C8C4×C12711yescharacteristic
121C2×C4×C8—21no—
122C4×C22⋊C4C4×C12711yescharacteristic
123C22⋊C16—21no—
124C22⋊C16—21no—
125C22⋊C16—21no—
126C22⋊C16C4×C12712yescomplemented
127#336—21no—
128C23×C4C2×C4×C12711yescharacteristic
129C22×C8—21no—
130C22×C8—42no—
131C22×C8C2×C4×C12711yescharacteristic
132C22×C8C2×C4×C12711yescharacteristic
133C22×C8D4×C12711yescharacteristic
134C22×C8D4×C12711yescharacteristic
135C22×C8D4×C12711yescharacteristic
136C22×C8Q8×C12711yescharacteristic
137C2×C22⋊C4—21no—
138C2×C42—21no—
139M32—42no—
140C2×C16—22no—
141C2×C16—22no—
142C2×C16—42no—
143C22⋊C8—21no—
144C22⋊C8C8×C12712yes—
145C4×C8—42no—
146#95C8×C12712yes—
147C24M16×C12711yescharacteristic
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4C42×C12711yescharacteristic
152C22×C4C22⋊C4×C12711yescharacteristic
153C22×C4C4⋊C4×C12711yescharacteristic
154C22×C4C2×C8×C12711yescharacteristic
155C22×C4C2×C8×C12711yescharacteristic
156C22×C4M16×C12711yescharacteristic
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8C22⋊C4×C12711yescharacteristic
167C2×C8C22⋊C4×C12711yescharacteristic
168C2×C8C4⋊C4×C12711yescharacteristic
169C2×C8C4⋊C4×C12711yescharacteristic
170C22⋊C4—24no—
171C42—41no—
172C42—41no—
173C16—42no—
174C16—42no—
175C23—22no—
176C23—22no—
177C23—41no—
178C23#97×C12711yescharacteristic
179C23C22⋊C8×C12711yescharacteristic
180C23C4⋊C8×C12711yescharacteristic
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4#95×C12711yescharacteristic
194C2×C4C4×C8×C12711yescharacteristic
195C2×C4C22⋊C8×C12711yescharacteristic
196C2×C4C4⋊C8×C12711yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8#99×C12712yes—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22#336×C12711yescharacteristic
211C22#350×C12712yes—
212C22#364×C12712yes—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—44no—
217C4#342×C12711yescharacteristic
218C4#344×C12711yescharacteristic
219C2—22no—
220C2—22no—
221C2—41no—
222C2C127×#128211yescharacteristic
223C2C127×#137611yescharacteristic
224C2C127×#137911yescharacteristic
subgroupG/H
centerC2×C8×C127C22⋊C4
commutatorC2×C4C4×C8×C127
Frattini, cosocleC23×C8C22×C127
socleC22×C127#336
prime radicalC24×C127M16
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×M32×C127C4semidirect
13. C22⋊C16×C127C4semidirect
106. #7202C127direct (N × H)
116. C127#7202direct (N × H)
118. C2×M32C4×C127semidirect
126. C22⋊C16C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#21122
C2×C22⋊C16×C1272
C8×C22⋊C4×C1272
#7202127