Finite Group Databasebeta

← C127×#14657 #44578330

Subgroups — C127×#14657 #44578330

Order 32512.

3274 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 3274 conjugacy classes of subgroups — too many to draw. Only its 462 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
1C22×#120×C127C211yesmaximal, characteristic
2C2×#379×C127C212yesmaximal, complemented
3C127×#1440C216yesmaximal, complemented
4C127×#1438C216yesmaximal, complemented
5C26×C127C2211yescharacteristic
6C23×D4×C127C2213yes—
7C2×#120×C127—28no—
8C2×#120×C127C418yescomplemented
9C22×C22⋊C4×C127C2213yes—
10C22×C22⋊C4×C127C22112yes—
11C22×C22⋊C4×C127C2214yescomplemented
12#379×C127—28no—
13C2×#95×C127C22112yescomplemented
14C25×C127—412no—
15C25×C127C2×C414yes—
16C25×C127D418yes—
17C25×C127C2311yescharacteristic
18C25×C127C2313yes—
19C22×D4×C127—212no—
20C22×D4×C127—412no—
21C22×D4×C127C2×C4112yes—
22C23×C4×C127C2318yes—
23C23×C4×C127C2313yes—
24#120×C127—232no—
25C2×C22⋊C4×C127—212no—
26C2×C22⋊C4×C127—224no—
27C2×C22⋊C4×C127—424no—
28C2×C22⋊C4×C127—412no—
29C2×C22⋊C4×C127C2×C4112yes—
30#95×C127—412no—
31C24×C127—224no—
32C24×C127—212no—
33C24×C127—22no—
34C24×C127—212no—
35C24×C127—424no—
36C24×C127—448no—
37C24×C127—448no—
38C24×C127—416no—
39C24×C127C22⋊C4116yescomplemented
40C24×C127C22×C412yes—
41C24×C127C22×C416yes—
42C24×C127C2×D4112yes—
43C24×C127C2411yescharacteristic
44C2×D4×C127—224no—
45C2×D4×C127—424no—
46C2×D4×C127—448no—
47C22×C4×C127—26no—
48C22×C4×C127—212no—
49C22×C4×C127—28no—
50C22×C4×C127—224no—
51C22×C4×C127—224no—
52C22×C4×C127C22×C416yes—
53C22⋊C4×C127—224no—
54C22⋊C4×C127—448no—
55C22⋊C4×C127—416no—
56C23×C127—212no—
57C23×C127—224no—
58C23×C127—224no—
59C23×C127—224no—
60C23×C127—224no—
61C23×C127—22no—
62C23×C127—212no—
63C23×C127—448no—
64C23×C127—464no—
65C23×C127—4128no—
66C23×C127—436no—
67C23×C127—412no—
68C23×C127—412no—
69C23×C127—412no—
70C23×C127—44no—
71C23×C127C2×C22⋊C4112yes—
72C23×C127C23×C411yescharacteristic
73C23×C127C22×D412yes—
74C23×C127H(5,𝔽2)112yes—
75D4×C127—448no—
76C2×C4×C127—26no—
77C2×C4×C127—212no—
78C2×C4×C127—412no—
79C2×C4×C127—424no—
80C2×C4×C127—424no—
81C2×C4×C127—48no—
82C22×C127—212no—
83C22×C127—224no—
84C22×C127—464no—
85C22×C127—424no—
86C22×C127—448no—
87C22×C127—46no—
88C22×C127—48no—
89C22×C127—88no—
90C22×C127C22×C22⋊C411yescharacteristic
91C22×C127#51816yes—
92C22×C127#534112yes—
93C22×C127#560112yes—
94C22×C127H(3,𝔽4)14yes—
95C4×C127—46no—
96C4×C127—88no—
97#14657C12711yesmaximal, characteristic, direct product factor, Sylow
98C2×C127—412no—
99C2×C127—82no—
100C2×C127#230813yes—
101C2×C127#237514yes—
102C2×C127#281218yes—
103C22×#120C2×C12711yescharacteristic
104C2×#379C2×C12712yescomplemented
105#1440C2×C12716yescomplemented
106#1438C2×C12716yescomplemented
107C127#1465711yescharacteristic, direct product factor, Sylow
108C26C22×C12711yescharacteristic
109C23×D4C22×C12713yes—
110C2×#120—28no—
111C2×#120C4×C12718yescomplemented
112C22×C22⋊C4C22×C12713yes—
113C22×C22⋊C4C22×C127112yes—
114C22×C22⋊C4C22×C12714yescomplemented
115#379—28no—
116C2×#95C22×C127112yescomplemented
117C25—412no—
118C25C2×C4×C12714yes—
119C25D4×C12718yes—
120C25C23×C12711yescharacteristic
121C25C23×C12713yes—
122C22×D4—212no—
123C22×D4—412no—
124C22×D4C2×C4×C127112yes—
125C23×C4C23×C12718yes—
126C23×C4C23×C12713yes—
127#120—232no—
128C2×C22⋊C4—212no—
129C2×C22⋊C4—224no—
130C2×C22⋊C4—424no—
131C2×C22⋊C4—412no—
132C2×C22⋊C4C2×C4×C127112yes—
133#95—412no—
134C24—224no—
135C24—212no—
136C24—22no—
137C24—212no—
138C24—424no—
139C24—448no—
140C24—448no—
141C24—416no—
142C24C22⋊C4×C127116yescomplemented
143C24C22×C4×C12712yes—
144C24C22×C4×C12716yes—
145C24C2×D4×C127112yes—
146C24C24×C12711yescharacteristic
147C2×D4—224no—
148C2×D4—424no—
149C2×D4—448no—
150C22×C4—26no—
151C22×C4—212no—
152C22×C4—28no—
153C22×C4—224no—
154C22×C4—224no—
155C22×C4C22×C4×C12716yes—
156C22⋊C4—224no—
157C22⋊C4—448no—
158C22⋊C4—416no—
159C23—212no—
160C23—224no—
161C23—224no—
162C23—224no—
163C23—224no—
164C23—22no—
165C23—212no—
166C23—448no—
167C23—464no—
168C23—4128no—
169C23—436no—
170C23—412no—
171C23—412no—
172C23—412no—
173C23—44no—
174C23C2×C22⋊C4×C127112yes—
175C23C23×C4×C12711yescharacteristic
176C23C22×D4×C12712yes—
177C23H(5,𝔽2)×C127112yes—
178D4—448no—
179C2×C4—26no—
180C2×C4—212no—
181C2×C4—412no—
182C2×C4—424no—
183C2×C4—424no—
184C2×C4—48no—
185C22—212no—
186C22—224no—
187C22—464no—
188C22—424no—
189C22—448no—
190C22—48no—
191C22—46no—
192C22—88no—
193C22C22×C22⋊C4×C12711yescharacteristic
194C22#518×C12716yes—
195C22#534×C127112yes—
196C22#560×C127112yes—
197C22H(3,𝔽4)×C12714yes—
198C4—46no—
199C4—88no—
200C2—412no—
201C2—82no—
202C2C127×#230813yes—
203C2C127×#237514yes—
204C2C127×#281218yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC22×#120×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#379×C127C2semidirect
3. C127×#1440C2semidirect
4. C127×#1438C2semidirect
8. C2×#120×C127C4semidirect
11. C22×C22⋊C4×C127C22semidirect
13. C2×#95×C127C22semidirect
39. C24×C127C22⋊C4semidirect
97. #14657C127direct (N × H)
104. C2×#379C2×C127semidirect
105. #1440C2×C127semidirect
106. #1438C2×C127semidirect
107. C127#14657direct (N × H)
111. C2×#120C4×C127semidirect
114. C22×C22⋊C4C22×C127semidirect
116. C2×#95C22×C127semidirect
142. C24C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#120×C1272
C2×#379×C1272
C127×#14402
C127×#14382
#14657127