Finite Group Databasebeta

← C127×#14341 #44578014

Subgroups — C127×#14341 #44578014

Order 32512.

3246 conjugacy classes of subgroups; only the 542 normal subgroups are shown.

normal subgroups only

This group has 3246 conjugacy classes of subgroups — too many to draw. Only its 542 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
1C23×C22⋊C4×C127C211yesmaximal, characteristic
2C2×#379×C127C212yesmaximal
3C127×#1440C218yesmaximal, complemented
4C127×C22⋊#95C214yesmaximal, complemented
5C26×C127C2211yescharacteristic
6C24×C4×C127C2212yes—
7C22×C22⋊C4×C127—28no—
8C22×C22⋊C4×C127C418yescomplemented
9C22×C22⋊C4×C127C2214yes—
10C22×C22⋊C4×C127C2214yes—
11C22×C22⋊C4×C127C2218yes—
12#379×C127—28no—
13C2×#95×C127C22116yescomplemented
14C25×C127—28no—
15C25×C127—48no—
16C25×C127C2×C414yes—
17C25×C127D418yes—
18C25×C127C2312yes—
19C25×C127C2311yescharacteristic
20C23×C4×C127—28no—
21C23×C4×C127—48no—
22C23×C4×C127C2×C418yes—
23C23×C4×C127C2318yes—
24C23×C4×C127C2314yes—
25C2×C22⋊C4×C127—232no—
26C2×C22⋊C4×C127—216no—
27C2×C22⋊C4×C127—216no—
28C2×C22⋊C4×C127—28no—
29C2×C22⋊C4×C127—216no—
30C2×C22⋊C4×C127—416no—
31C2×C22⋊C4×C127C2×C4116yes—
32#95×C127—416no—
33C24×C127—28no—
34C24×C127—216no—
35C24×C127—232no—
36C24×C127—28no—
37C24×C127—216no—
38C24×C127—416no—
39C24×C127—432no—
40C24×C127—464no—
41C24×C127C22⋊C4116yes—
42C24×C127C22×C414yes—
43C24×C127C22×C412yes—
44C24×C127C2×D418yes—
45C24×C127C2×D414yes—
46C24×C127#4118yes—
47C24×C127C2411yescharacteristic
48C22×C4×C127—216no—
49C22×C4×C127—28no—
50C22×C4×C127—216no—
51C22×C4×C127—28no—
52C22×C4×C127—216no—
53C22×C4×C127—232no—
54C22×C4×C127—416no—
55C22×C4×C127—432no—
56C22×C4×C127C22×C418yes—
57C22⋊C4×C127—232no—
58C22⋊C4×C127—432no—
59C22⋊C4×C127—416no—
60C23×C127—28no—
61C23×C127—216no—
62C23×C127—232no—
63C23×C127—28no—
64C23×C127—216no—
65C23×C127—216no—
66C23×C127—216no—
67C23×C127—216no—
68C23×C127—232no—
69C23×C127—432no—
70C23×C127—464no—
71C23×C127—4128no—
72C23×C127—432no—
73C23×C127C2×C22⋊C418yes—
74C23×C127C2×C22⋊C414yes—
75C23×C127#11718yes—
76C23×C127#123116yes—
77C23×C127C23×C411yescharacteristic
78C23×C127C22×D412yes—
79C23×C127C2×#4114yes—
80C23×C127H(5,𝔽2)18yes—
81C2×C4×C127—28no—
82C2×C4×C127—216no—
83C2×C4×C127—432no—
84C2×C4×C127—416no—
85C2×C4×C127—48no—
86C2×C4×C127—416no—
87C2×C4×C127—432no—
88C22×C127—28no—
89C22×C127—216no—
90C22×C127—232no—
91C22×C127—216no—
92C22×C127—28no—
93C22×C127—464no—
94C22×C127—416no—
95C22×C127—432no—
96C22×C127#38118yes—
97C22×C127C22×C22⋊C411yescharacteristic
98C22×C127C2×#11712yes—
99C22×C127#51814yes—
100C22×C127C2×#12314yes—
101C22×C127#53418yes—
102C22×C127#538116yes—
103C4×C127—48no—
104C4×C127—88no—
105#14341C12711yesmaximal, characteristic, direct product factor, Sylow
106C2×C127—28no—
107C2×C127—48no—
108C2×C127C2×#38111yescharacteristic
109C2×C127#230812yes—
110C2×C127#231214yes—
111C2×C127#262218yes—
112C23×C22⋊C4C2×C12711yescharacteristic
113C2×#379C2×C12712yes—
114#1440C2×C12718yescomplemented
115C22⋊#95C2×C12714yescomplemented
116C127#1434111yescharacteristic, direct product factor, Sylow
117C26C22×C12711yescharacteristic
118C24×C4C22×C12712yes—
119C22×C22⋊C4—28no—
120C22×C22⋊C4C4×C12718yescomplemented
121C22×C22⋊C4C22×C12714yes—
122C22×C22⋊C4C22×C12714yes—
123C22×C22⋊C4C22×C12718yes—
124#379—28no—
125C2×#95C22×C127116yescomplemented
126C25—28no—
127C25—48no—
128C25C2×C4×C12714yes—
129C25D4×C12718yes—
130C25C23×C12712yes—
131C25C23×C12711yescharacteristic
132C23×C4—28no—
133C23×C4—48no—
134C23×C4C2×C4×C12718yes—
135C23×C4C23×C12718yes—
136C23×C4C23×C12714yes—
137C2×C22⋊C4—232no—
138C2×C22⋊C4—216no—
139C2×C22⋊C4—216no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—216no—
142C2×C22⋊C4—416no—
143C2×C22⋊C4C2×C4×C127116yes—
144#95—416no—
145C24—28no—
146C24—216no—
147C24—232no—
148C24—28no—
149C24—216no—
150C24—416no—
151C24—432no—
152C24—464no—
153C24C22⋊C4×C127116yes—
154C24C22×C4×C12714yes—
155C24C22×C4×C12712yes—
156C24C2×D4×C12718yes—
157C24C2×D4×C12714yes—
158C24#41×C12718yes—
159C24C24×C12711yescharacteristic
160C22×C4—216no—
161C22×C4—28no—
162C22×C4—216no—
163C22×C4—28no—
164C22×C4—216no—
165C22×C4—232no—
166C22×C4—416no—
167C22×C4—432no—
168C22×C4C22×C4×C12718yes—
169C22⋊C4—232no—
170C22⋊C4—432no—
171C22⋊C4—416no—
172C23—28no—
173C23—216no—
174C23—232no—
175C23—28no—
176C23—216no—
177C23—216no—
178C23—216no—
179C23—216no—
180C23—232no—
181C23—432no—
182C23—464no—
183C23—4128no—
184C23—432no—
185C23C2×C22⋊C4×C12718yes—
186C23C2×C22⋊C4×C12714yes—
187C23#117×C12718yes—
188C23#123×C127116yes—
189C23C23×C4×C12711yescharacteristic
190C23C22×D4×C12712yes—
191C23C2×#41×C12714yes—
192C23H(5,𝔽2)×C12718yes—
193C2×C4—28no—
194C2×C4—216no—
195C2×C4—432no—
196C2×C4—416no—
197C2×C4—48no—
198C2×C4—416no—
199C2×C4—432no—
200C22—216no—
201C22—232no—
202C22—28no—
203C22—28no—
204C22—216no—
205C22—464no—
206C22—416no—
207C22—432no—
208C22#381×C12718yes—
209C22C22×C22⋊C4×C12711yescharacteristic
210C22C2×#117×C12712yes—
211C22#518×C12714yes—
212C22C2×#123×C12714yes—
213C22#534×C12718yes—
214C22#538×C127116yes—
215C4—48no—
216C4—88no—
217C2—28no—
218C2—48no—
219C2C2×#381×C12711yescharacteristic
220C2C127×#230812yes—
221C2C127×#231214yes—
222C2C127×#262218yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC26×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1440C2semidirect
4. C127×C22⋊#95C2semidirect
8. C22×C22⋊C4×C127C4semidirect
13. C2×#95×C127C22semidirect
105. #14341C127direct (N × H)
114. #1440C2×C127semidirect
115. C22⋊#95C2×C127semidirect
116. C127#14341direct (N × H)
120. C22×C22⋊C4C4×C127semidirect
125. C2×#95C22×C127semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C1272
C2×#379×C1272
C127×#14402
C127×C22⋊#952
#14341127