Finite Group Databasebeta

← C127×#11488 #44575161

Subgroups — C127×#11488 #44575161

Order 32512.

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

normal subgroups only

This group has 418 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×C8×D4×C127C211yesmaximal, characteristic
2C127×#2163C212yesmaximal, complemented
3C127×#1314C214yesmaximal, complemented
4C23×C8×C127C412yes—
5C2×C4×D4×C127C411yescharacteristic
6C8×D4×C127—22no—
7C8×D4×C127—22no—
8C2×C4⋊C8×C127C411yescharacteristic
9C2×C22⋊C8×C127C2212yes—
10C2×C4×C8×C127C2211yescharacteristic
11C4⋊C16×C127—22no—
12C22⋊C16×C127C2214yes—
13#346×C127—22no—
14C22×D4×C127C811yescharacteristic
15C23×C4×C127C2×C412yes—
16C22×C8×C127—22no—
17C22×C8×C127—22no—
18C22×C8×C127—44no—
19C22×C8×C127C2×C411yescharacteristic
20C22×C8×C127C2×C412yes—
21C22×C8×C127C2311yescharacteristic
22C4×D4×C127—22no—
23C4×D4×C127—22no—
24C2×C4⋊C4×C127C811yescharacteristic
25C2×C22⋊C4×C127C812yes—
26C2×C42×C127C2×C411yescharacteristic
27C2×C16×C127—24no—
28C4⋊C8×C127—22no—
29C22⋊C8×C127—24no—
30C4×C8×C127D412yes—
31C4×C8×C127D412yes—
32C24×C127C2×C812yes—
33C2×D4×C127—24no—
34C2×D4×C127—22no—
35C22×C4×C127—22no—
36C22×C4×C127—22no—
37C22×C4×C127—42no—
38C22×C4×C127—42no—
39C22×C4×C127C2×C811yescharacteristic
40C22×C4×C127C2×C812yes—
41C22×C4×C127C2×C811yescharacteristic
42C22×C4×C127C22×C411yescharacteristic
43C2×C8×C127—22no—
44C2×C8×C127—22no—
45C2×C8×C127—41no—
46C2×C8×C127—44no—
47C2×C8×C127—44no—
48C2×C8×C127—42no—
49C2×C8×C127C22⋊C411yescharacteristic
50C2×C8×C127C22⋊C411yescharacteristic
51C2×C8×C127C2×D412yes—
52C4⋊C4×C127—22no—
53C22⋊C4×C127—24no—
54C42×C127C22⋊C412yes—
55C42×C127M1612yes—
56C16×C127—44no—
57C23×C127—22no—
58C23×C127—22no—
59C23×C127—42no—
60C23×C127—42no—
61C23×C127M3214yes—
62C23×C127C22×C811yescharacteristic
63D4×C127—28no—
64C2×C4×C127—22no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—22no—
68C2×C4×C127—41no—
69C2×C4×C127—44no—
70C2×C4×C127—42no—
71C2×C4×C127—42no—
72C2×C4×C127—42no—
73C2×C4×C127C22⋊C812yes—
74C2×C4×C127C22⋊C812yes—
75C2×C4×C127C2×C22⋊C411yescharacteristic
76C2×C4×C127C2×M1611yescharacteristic
77C8×C127—22no—
78C8×C127—21no—
79C8×C127—42no—
80C8×C127#10012yes—
81C22×C127—21no—
82C22×C127—21no—
83C22×C127—24no—
84C22×C127—44no—
85C22×C127—44no—
86C22×C127—42no—
87C22×C127—42no—
88C22×C127C2×C22⋊C811yescharacteristic
89C22×C127C2×M3212yes—
90C4×C127—22no—
91C4×C127—22no—
92C4×C127—21no—
93C4×C127—42no—
94C4×C127#41011yescharacteristic
95C4×C127C2×#10011yescharacteristic
96#11488C12711yesmaximal, characteristic, direct product factor, Sylow
97C2×C127—22no—
98C2×C127—44no—
99C2×C127#146011yescharacteristic
100C2×C127#211211yescharacteristic
101C2×C127#211511yescharacteristic
102C2×C8×D4C2×C12711yescharacteristic
103#2163C2×C12712yescomplemented
104#1314C2×C12714yescomplemented
105C127#1148811yescharacteristic, direct product factor, Sylow
106C23×C8C4×C12712yes—
107C2×C4×D4C4×C12711yescharacteristic
108C8×D4—22no—
109C8×D4—22no—
110C2×C4⋊C8C4×C12711yescharacteristic
111C2×C22⋊C8C22×C12712yes—
112C2×C4×C8C22×C12711yescharacteristic
113C4⋊C16—22no—
114C22⋊C16C22×C12714yes—
115#346—22no—
116C22×D4C8×C12711yescharacteristic
117C23×C4C2×C4×C12712yes—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—44no—
121C22×C8C2×C4×C12711yescharacteristic
122C22×C8C2×C4×C12712yes—
123C22×C8C23×C12711yescharacteristic
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4C8×C12711yescharacteristic
127C2×C22⋊C4C8×C12712yes—
128C2×C42C2×C4×C12711yescharacteristic
129C2×C16—24no—
130C4⋊C8—22no—
131C22⋊C8—24no—
132C4×C8D4×C12712yes—
133C4×C8D4×C12712yes—
134C24C2×C8×C12712yes—
135C2×D4—24no—
136C2×D4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4C2×C8×C12711yescharacteristic
142C22×C4C2×C8×C12712yes—
143C22×C4C2×C8×C12711yescharacteristic
144C22×C4C22×C4×C12711yescharacteristic
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—41no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—42no—
151C2×C8C22⋊C4×C12711yescharacteristic
152C2×C8C22⋊C4×C12711yescharacteristic
153C2×C8C2×D4×C12712yes—
154C4⋊C4—22no—
155C22⋊C4—24no—
156C42C22⋊C4×C12712yes—
157C42M16×C12712yes—
158C16—44no—
159C23—22no—
160C23—22no—
161C23—42no—
162C23—42no—
163C23M32×C12714yes—
164C23C22×C8×C12711yescharacteristic
165D4—28no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4C22⋊C8×C12712yes—
176C2×C4C22⋊C8×C12712yes—
177C2×C4C2×C22⋊C4×C12711yescharacteristic
178C2×C4C2×M16×C12711yescharacteristic
179C8—22no—
180C8—21no—
181C8—42no—
182C8#100×C12712yes—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—44no—
187C22—44no—
188C22—42no—
189C22—42no—
190C22C2×C22⋊C8×C12711yescharacteristic
191C22C2×M32×C12712yes—
192C4—22no—
193C4—22no—
194C4—21no—
195C4—42no—
196C4#410×C12711yescharacteristic
197C4C2×#100×C12711yescharacteristic
198C2—22no—
199C2—44no—
200C2C127×#146011yescharacteristic
201C2C127×#211211yescharacteristic
202C2C127×#211511yescharacteristic
subgroupG/H
centerC2×C4×C127C2×C22⋊C4
commutatorC23C22×C8×C127
Frattini, cosocleC22×C8C23×C127
socleC22×C127C2×C22⋊C8
prime radicalC22×D4×C127C8
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2163C2semidirect
3. C127×#1314C2semidirect
96. #11488C127direct (N × H)
103. #2163C2×C127semidirect
104. #1314C2×C127semidirect
105. C127#11488direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C8×D4×C1272
C127×#21632
C127×#13142
#11488127