Finite Group Databasebeta

← C127×#17273 #44580946

Subgroups — C127×#17273 #44580946

Order 32512.

1426 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 302 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×#1445C214yesmaximal, complemented
2C127×#1439C218yesmaximal, complemented
3C127×#1438C211yesmaximal, characteristic
4C127×#1438C212yesmaximal
5C23×D4×C127C2211yescharacteristic
6C22×C4⋊C4×C127C2214yes—
7C22×C22⋊C4×C127C2214yes—
8C22×C22⋊C4×C127C2214yes—
9C22×C22⋊C4×C127C2214yes—
10C2×#95×C127C2214yes—
11C2×#95×C127C2212yes—
12C2×#95×C127C2218yes—
13C2×#95×C127C2214yes—
14C25×C127C2312yes—
15C22×D4×C127—42no—
16C22×D4×C127—44no—
17C22×D4×C127Q814yes—
18C23×C4×C127C2318yes—
19C23×C4×C127C2311yescharacteristic
20C23×C4×C127C2314yes—
21C2×C4⋊C4×C127—28no—
22C2×C4⋊C4×C127—44no—
23C2×C4⋊C4×C127—44no—
24C2×C22⋊C4×C127—216no—
25C2×C22⋊C4×C127—44no—
26C2×C22⋊C4×C127—44no—
27C2×C22⋊C4×C127—48no—
28C2×C22⋊C4×C127—48no—
29C2×C22⋊C4×C127—44no—
30#95×C127—48no—
31#95×C127—42no—
32#95×C127—44no—
33#95×C127—44no—
34C24×C127—28no—
35C24×C127—28no—
36C24×C127—24no—
37C24×C127—24no—
38C24×C127—24no—
39C24×C127C2×Q814yes—
40C24×C127C2411yescharacteristic
41C2×D4×C127—48no—
42C2×D4×C127—44no—
43C2×D4×C127—416no—
44C22×C4×C127—28no—
45C22×C4×C127—28no—
46C22×C4×C127—24no—
47C22×C4×C127—24no—
48C22×C4×C127—22no—
49C22×C4×C127—24no—
50C22×C4×C127—28no—
51C22×C4×C127—28no—
52C22×C4×C127—28no—
53C22×C4×C127—28no—
54C22×C4×C127—28no—
55C22×C4×C127—28no—
56C22×C4×C127—28no—
57C22×C4×C127C2×Q812yes—
58C22×C4×C127#4118yes—
59C4⋊C4×C127—416no—
60C22⋊C4×C127—416no—
61C22⋊C4×C127—416no—
62C22⋊C4×C127—416no—
63C23×C127—24no—
64C23×C127—24no—
65C23×C127—24no—
66C23×C127—216no—
67C23×C127—48no—
68C23×C127—48no—
69C23×C127—48no—
70C23×C127—44no—
71C23×C127—44no—
72C23×C127—44no—
73C23×C127—44no—
74C23×C127—48no—
75C23×C127—44no—
76C23×C127—44no—
77C23×C127C22×Q811yescharacteristic
78C23×C127C2×#4114yes—
79C23×C127H(5,𝔽2)12yes—
80C23×C127H(5,𝔽2)14yes—
81C23×C127H(5,𝔽2)14yes—
82D4×C127—416no—
83C2×C4×C127—28no—
84C2×C4×C127—24no—
85C2×C4×C127—22no—
86C2×C4×C127—216no—
87C2×C4×C127—48no—
88C2×C4×C127—44no—
89C2×C4×C127—44no—
90C2×C4×C127—44no—
91C2×C4×C127—48no—
92C2×C4×C127—48no—
93C2×C4×C127—48no—
94C2×C4×C127—48no—
95C2×C4×C127—48no—
96C2×C4×C127—48no—
97C2×C4×C127—48no—
98C22×C127—44no—
99C22×C127—48no—
100C22×C127—48no—
101C22×C127—48no—
102C22×C127—88no—
103C22×C127—88no—
104C22×C127#53814yes—
105C22×C127#54018yes—
106C22×C127#54312yes—
107C22×C127#54311yescharacteristic
108C22×C127#55114yes—
109C22×C127#55414yes—
110C22×C127#55912yes—
111C22×C127#56014yes—
112C22×C127#56014yes—
113C22×C127H(3,𝔽4)12yes—
114C4×C127—42no—
115C4×C127—48no—
116C4×C127—88no—
117#17273C12711yesmaximal, characteristic, direct product factor, Sylow
118C2×C127—84no—
119C2×C127#274814yes—
120C2×C127#275214yes—
121C2×C127#280412yes—
122C2×C127#282814yes—
123C2×C127#284211yescharacteristic
124#1445C2×C12714yescomplemented
125#1439C2×C12718yescomplemented
126#1438C2×C12711yescharacteristic
127#1438C2×C12712yes—
128C127#1727311yescharacteristic, direct product factor, Sylow
129C23×D4C22×C12711yescharacteristic
130C22×C4⋊C4C22×C12714yes—
131C22×C22⋊C4C22×C12714yes—
132C22×C22⋊C4C22×C12714yes—
133C22×C22⋊C4C22×C12714yes—
134C2×#95C22×C12714yes—
135C2×#95C22×C12712yes—
136C2×#95C22×C12718yes—
137C2×#95C22×C12714yes—
138C25C23×C12712yes—
139C22×D4—42no—
140C22×D4—44no—
141C22×D4Q8×C12714yes—
142C23×C4C23×C12718yes—
143C23×C4C23×C12711yescharacteristic
144C23×C4C23×C12714yes—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C22⋊C4—216no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—48no—
152C2×C22⋊C4—48no—
153C2×C22⋊C4—44no—
154#95—48no—
155#95—42no—
156#95—44no—
157#95—44no—
158C24—28no—
159C24—28no—
160C24—24no—
161C24—24no—
162C24—24no—
163C24C2×Q8×C12714yes—
164C24C24×C12711yescharacteristic
165C2×D4—48no—
166C2×D4—44no—
167C2×D4—416no—
168C22×C4—28no—
169C22×C4—28no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4C2×Q8×C12712yes—
182C22×C4#41×C12718yes—
183C4⋊C4—416no—
184C22⋊C4—416no—
185C22⋊C4—416no—
186C22⋊C4—416no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—216no—
191C23—48no—
192C23—48no—
193C23—48no—
194C23—44no—
195C23—44no—
196C23—44no—
197C23—44no—
198C23—48no—
199C23—44no—
200C23—44no—
201C23C22×Q8×C12711yescharacteristic
202C23C2×#41×C12714yes—
203C23H(5,𝔽2)×C12712yes—
204C23H(5,𝔽2)×C12714yes—
205C23H(5,𝔽2)×C12714yes—
206D4—416no—
207C2×C4—28no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—216no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—48no—
217C2×C4—48no—
218C2×C4—48no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—48no—
222C22—44no—
223C22—48no—
224C22—48no—
225C22—48no—
226C22—88no—
227C22—88no—
228C22#538×C12714yes—
229C22#540×C12718yes—
230C22#543×C12712yes—
231C22#543×C12711yescharacteristic
232C22#551×C12714yes—
233C22#554×C12714yes—
234C22#559×C12712yes—
235C22#560×C12714yes—
236C22#560×C12714yes—
237C22H(3,𝔽4)×C12712yes—
238C4—42no—
239C4—48no—
240C4—88no—
241C2—84no—
242C2C127×#274814yes—
243C2C127×#275214yes—
244C2C127×#280412yes—
245C2C127×#282814yes—
246C2C127×#284211yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#1445C2semidirect
2. C127×#1439C2semidirect
117. #17273C127direct (N × H)
124. #1445C2×C127semidirect
125. #1439C2×C127semidirect
128. C127#17273direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#14452
C127×#14392
C127×#14382
#17273127