Finite Group Databasebeta

← C2×C4×#107×C127 #44584121

Subgroups — C2×C4×#107×C127 #44584121

Order 32512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#107×C127C212yesmaximal
2C22×C4×C8×C127C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C127C212yesmaximal
4C4×#107×C127C218yesmaximal, direct product factor
5C2×#340×C127C212yesmaximal
6C23×C8×C127C2211yescharacteristic
7C23×C8×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2214yes—
9C22×C42×C127—22no—
10C22×C42×C127C2211yescharacteristic
11C2×#107×C127C4116yesdirect product factor
12C2×#107×C127C2218yes—
13C2×C4×C8×C127C418yescomplemented
14C2×C4×C8×C127C2214yes—
15C4×C4⋊C4×C127—24no—
16C4×C4⋊C4×C127C2218yes—
17C2×#95×C127—22no—
18#340×C127C2218yes—
19C23×C4×C127—24no—
20C23×C4×C127D411yescharacteristic
21C23×C4×C127D411yescharacteristic
22C23×C4×C127C2311yescharacteristic
23C22×C8×C127—24no—
24C22×C8×C127C2×C414yes—
25C22×C8×C127C2×C414yes—
26C22×C8×C127C2×C418yes—
27C22×C8×C127C2312yes—
28C22×C8×C127C2312yes—
29C2×C4⋊C4×C127—24no—
30C2×C4⋊C4×C127—28no—
31C2×C4⋊C4×C127C2×C4116yes—
32C2×C4⋊C4×C127C2318yes—
33C2×C42×C127—28no—
34C2×C42×C127—48no—
35C2×C42×C127C2×C414yes—
36C2×C42×C127D412yes—
37C2×C42×C127Q814yes—
38C2×C42×C127C2312yes—
39#107×C127C2×C4132yesdirect product factor
40C4×C8×C127C2×C4116yescomplemented
41#95×C127—28no—
42C24×C127C2×D411yescharacteristic
43C22×C4×C127—24no—
44C22×C4×C127—216no—
45C22×C4×C127—28no—
46C22×C4×C127—48no—
47C22×C4×C127—48no—
48C22×C4×C127C4⋊C414yes—
49C22×C4×C127C4⋊C414yes—
50C22×C4×C127D814yes—
51C22×C4×C127Q1614yes—
52C22×C4×C127C22×C412yes—
53C22×C4×C127C22×C414yes—
54C22×C4×C127C2×D412yes—
55C22×C4×C127C2×D411yescharacteristic
56C22×C4×C127C2×D412yes—
57C22×C4×C127C2×Q812yes—
58C22×C4×C127#4114yes—
59C22×C4×C127C2411yescharacteristic
60C2×C8×C127—28no—
61C2×C8×C127—24no—
62C2×C8×C127C42116yescomplemented
63C2×C8×C127C22×C414yes—
64C2×C8×C127C22×C414yes—
65C2×C8×C127C22×C418yes—
66C4⋊C4×C127—216no—
67C4⋊C4×C127—28no—
68C4⋊C4×C127C22×C4116yes—
69C42×C127—416no—
70C42×C127C4⋊C418yes—
71C42×C127C22×C414yes—
72C42×C127C2×Q814yes—
73C23×C127C2×C4⋊C412yes—
74C23×C127C4×D414yes—
75C23×C127C2×D812yes—
76C23×C127C2×Q1612yes—
77C23×C127#13514yes—
78C23×C127C22×D411yescharacteristic
79C2×C4×C127—24no—
80C2×C4×C127—28no—
81C2×C4×C127—216no—
82C2×C4×C127—48no—
83C2×C4×C127—48no—
84C2×C4×C127—432no—
85C2×C4×C127#107116yesdirect product factor
86C2×C4×C127C2×C4214yes—
87C2×C4×C127C2×C4⋊C414yes—
88C2×C4×C127C2×C4⋊C412yes—
89C2×C4×C127C2×C4⋊C414yes—
90C2×C4×C127#11714yes—
91C2×C4×C127C4×D414yes—
92C2×C4×C127C4×Q818yes—
93C2×C4×C127C2×D814yes—
94C2×C4×C127C2×Q1614yes—
95C2×C4×C127C23×C411yescharacteristic
96C2×C4×C127C23×C412yes—
97C2×C4×C127C22×Q811yescharacteristic
98C2×C4×C127C2×#4112yes—
99C8×C127—24no—
100C8×C127C2×C4218yescomplemented
101C22×C127C4×C4⋊C414yes—
102C22×C127C2×#10714yes—
103C22×C127#42714yes—
104C22×C127C4×D818yes—
105C22×C127C4×Q1618yes—
106C22×C127C22×C4⋊C411yescharacteristic
107C22×C127C2×C4×D412yes—
108C22×C127C22×D811yescharacteristic
109C22×C127C22×Q1611yescharacteristic
110C22×C127C2×#13512yes—
111C4×C127—24no—
112C4×C127—416no—
113C4×C127C4×C4⋊C414yes—
114C4×C127C2×#10718yesdirect product factor
115C4×C127C22×C4211yescharacteristic
116C4×C127C2×#11711yescharacteristic
117C4×C127C2×C4×Q812yes—
118C2×C4×#107C12711yesmaximal, characteristic, direct product factor, Sylow
119C2×C127C4×#10718yesdirect product factor
120C2×C127C2×C4×C4⋊C411yescharacteristic
121C2×C127C22×#10711yescharacteristic
122C2×C127C2×#42711yescharacteristic
123C2×C127C2×C4×D812yes—
124C2×C127C2×C4×Q1612yes—
125C22×#107C2×C12712yes—
126C22×C4×C8C2×C12711yescharacteristic
127C2×C4×C4⋊C4C2×C12712yes—
128C4×#107C2×C12718yesdirect product factor
129C2×#340C2×C12712yes—
130C127C2×C4×#10711yescharacteristic, direct product factor, Sylow
131C23×C8C22×C12711yescharacteristic
132C23×C8C22×C12711yescharacteristic
133C22×C4⋊C4C22×C12714yes—
134C22×C42—22no—
135C22×C42C22×C12711yescharacteristic
136C2×#107C4×C127116yesdirect product factor
137C2×#107C22×C12718yes—
138C2×C4×C8C4×C12718yescomplemented
139C2×C4×C8C22×C12714yes—
140C4×C4⋊C4—24no—
141C4×C4⋊C4C22×C12718yes—
142C2×#95—22no—
143#340C22×C12718yes—
144C23×C4—24no—
145C23×C4D4×C12711yescharacteristic
146C23×C4D4×C12711yescharacteristic
147C23×C4C23×C12711yescharacteristic
148C22×C8—24no—
149C22×C8C2×C4×C12714yes—
150C22×C8C2×C4×C12714yes—
151C22×C8C2×C4×C12718yes—
152C22×C8C23×C12712yes—
153C22×C8C23×C12712yes—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4C2×C4×C127116yes—
157C2×C4⋊C4C23×C12718yes—
158C2×C42—28no—
159C2×C42—48no—
160C2×C42C2×C4×C12714yes—
161C2×C42D4×C12712yes—
162C2×C42Q8×C12714yes—
163C2×C42C23×C12712yes—
164#107C2×C4×C127132yesdirect product factor
165C4×C8C2×C4×C127116yescomplemented
166#95—28no—
167C24C2×D4×C12711yescharacteristic
168C22×C4—24no—
169C22×C4—216no—
170C22×C4—28no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4C4⋊C4×C12714yes—
174C22×C4C4⋊C4×C12714yes—
175C22×C4D8×C12714yes—
176C22×C4Q16×C12714yes—
177C22×C4C22×C4×C12712yes—
178C22×C4C22×C4×C12714yes—
179C22×C4C2×D4×C12712yes—
180C22×C4C2×D4×C12711yescharacteristic
181C22×C4C2×D4×C12712yes—
182C22×C4C2×Q8×C12712yes—
183C22×C4#41×C12714yes—
184C22×C4C24×C12711yescharacteristic
185C2×C8—28no—
186C2×C8—24no—
187C2×C8C42×C127116yescomplemented
188C2×C8C22×C4×C12714yes—
189C2×C8C22×C4×C12714yes—
190C2×C8C22×C4×C12718yes—
191C4⋊C4—216no—
192C4⋊C4—28no—
193C4⋊C4C22×C4×C127116yes—
194C42—416no—
195C42C4⋊C4×C12718yes—
196C42C22×C4×C12714yes—
197C42C2×Q8×C12714yes—
198C23C2×C4⋊C4×C12712yes—
199C23C4×D4×C12714yes—
200C23C2×D8×C12712yes—
201C23C2×Q16×C12712yes—
202C23#135×C12714yes—
203C23C22×D4×C12711yescharacteristic
204C2×C4—24no—
205C2×C4—28no—
206C2×C4—216no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—432no—
210C2×C4#107×C127116yesdirect product factor
211C2×C4C2×C42×C12714yes—
212C2×C4C2×C4⋊C4×C12714yes—
213C2×C4C2×C4⋊C4×C12712yes—
214C2×C4C2×C4⋊C4×C12714yes—
215C2×C4#117×C12714yes—
216C2×C4C4×D4×C12714yes—
217C2×C4C4×Q8×C12718yes—
218C2×C4C2×D8×C12714yes—
219C2×C4C2×Q16×C12714yes—
220C2×C4C23×C4×C12711yescharacteristic
221C2×C4C23×C4×C12712yes—
222C2×C4C22×Q8×C12711yescharacteristic
223C2×C4C2×#41×C12712yes—
224C8—24no—
225C8C2×C42×C12718yescomplemented
226C22C4×C4⋊C4×C12714yes—
227C22C2×#107×C12714yes—
228C22#427×C12714yes—
229C22C4×D8×C12718yes—
230C22C4×Q16×C12718yes—
231C22C22×C4⋊C4×C12711yescharacteristic
232C22C2×C4×D4×C12712yes—
233C22C22×D8×C12711yescharacteristic
234C22C22×Q16×C12711yescharacteristic
235C22C2×#135×C12712yes—
236C4—24no—
237C4—416no—
238C4C4×C4⋊C4×C12714yes—
239C4C2×#107×C12718yesdirect product factor
240C4C22×C42×C12711yescharacteristic
241C4C2×#117×C12711yescharacteristic
242C4C2×C4×Q8×C12712yes—
243C2C4×#107×C12718yesdirect product factor
244C2C2×C4×C4⋊C4×C12711yescharacteristic
245C2C22×#107×C12711yescharacteristic
246C2C2×#427×C12711yescharacteristic
247C2C2×C4×D8×C12712yes—
248C2C2×C4×Q16×C12712yes—
subgroupG/H
centerC23×C4×C127D4
commutatorC4C22×C42×C127
Frattini, cosocleC22×C4C24×C127
socle, prime radicalC24×C127C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C4×#107×C127C2direct (N × H)
11. C2×#107×C127C4direct (N × H)
13. C2×C4×C8×C127C4semidirect
39. #107×C127C2×C4direct (N × H)
40. C4×C8×C127C2×C4semidirect
62. C2×C8×C127C42semidirect
85. C2×C4×C127#107direct (N × H)
100. C8×C127C2×C42semidirect
114. C4×C127C2×#107direct (N × H)
118. C2×C4×#107C127direct (N × H)
119. C2×C127C4×#107direct (N × H)
128. C4×#107C2×C127direct (N × H)
130. C127C2×C4×#107direct (N × H)
136. C2×#107C4×C127direct (N × H)
138. C2×C4×C8C4×C127semidirect
164. #107C2×C4×C127direct (N × H)
165. C4×C8C2×C4×C127semidirect
187. C2×C8C42×C127semidirect
210. C2×C4#107×C127direct (N × H)
225. C8C2×C42×C127semidirect
239. C4C2×#107×C127direct (N × H)
243. C2C4×#107×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#107×C1272
C22×C4×C8×C1272
C2×C4×C4⋊C4×C1272
C4×#107×C1272
C2×#340×C1272
C2×C4×#107127