Finite Group Databasebeta

← C2×C127×#1735 #44584018

Subgroups — C2×C127×#1735 #44584018

Order 32512.

2398 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C8×C127C211yesmaximal, characteristic
2C22×#117×C127C212yesmaximal
3C127×#1735C218yesmaximal, direct product factor
4C2×#340×C127C214yesmaximal, complemented
5C24×C4×C127C2211yescharacteristic
6C23×C8×C127—28no—
7C23×C8×C127C418yescomplemented
8C23×C8×C127C2212yes—
9C23×C8×C127C2214yes—
10C2×#117×C127—28no—
11C2×#117×C127—24no—
12C2×#117×C127C4116yescomplemented
13C2×#117×C127C2218yes—
14C22×C4⋊C4×C127C2214yes—
15C22×C22⋊C4×C127—22no—
16C22×C42×C127—22no—
17#340×C127C22116yescomplemented
18C25×C127D411yescharacteristic
19C23×C4×C127—24no—
20C23×C4×C127C2×C414yes—
21C23×C4×C127C2×C418yes—
22C23×C4×C127D412yes—
23C23×C4×C127D418yes—
24C23×C4×C127D411yescharacteristic
25C23×C4×C127Q814yes—
26C23×C4×C127C2311yescharacteristic
27C23×C4×C127C2312yes—
28C22×C8×C127—232no—
29C22×C8×C127—28no—
30C22×C8×C127—216no—
31C22×C8×C127C2×C418yes—
32C22×C8×C127C2×C4116yescomplemented
33C22×C8×C127C2314yes—
34#117×C127—232no—
35#117×C127—216no—
36#117×C127C2×C4132yescomplemented
37C2×C4⋊C4×C127—24no—
38C2×C4⋊C4×C127—28no—
39C2×C4⋊C4×C127C2×C4116yes—
40C2×C4⋊C4×C127C2318yes—
41C2×C22⋊C4×C127—216no—
42C2×C22⋊C4×C127—28no—
43C2×C42×C127—216no—
44C2×C42×C127—28no—
45C24×C127—28no—
46C24×C127C22⋊C418yes—
47C24×C127C4⋊C414yes—
48C24×C127C2×D412yes—
49C24×C127C2×D411yescharacteristic
50C22×C4×C127—216no—
51C22×C4×C127—28no—
52C22×C4×C127—416no—
53C22×C4×C127C42116yes—
54C22×C4×C127C22⋊C418yes—
55C22×C4×C127C22⋊C4116yes—
56C22×C4×C127C22⋊C4116yes—
57C22×C4×C127C4⋊C418yes—
58C22×C4×C127C4⋊C414yes—
59C22×C4×C127C4⋊C4132yes—
60C22×C4×C127C22×C412yes—
61C22×C4×C127C22×C414yes—
62C22×C4×C127C22×C414yes—
63C22×C4×C127C22×C418yes—
64C22×C4×C127C2×D412yes—
65C22×C4×C127C2×D411yescharacteristic
66C22×C4×C127C2×D414yes—
67C22×C4×C127C2×D418yes—
68C22×C4×C127C2×Q814yes—
69C22×C4×C127C2×Q812yes—
70C22×C4×C127C2411yescharacteristic
71C2×C8×C127—216no—
72C2×C8×C127—232no—
73C2×C8×C127—28no—
74C2×C8×C127C22×C418yes—
75C4⋊C4×C127—216no—
76C4⋊C4×C127—28no—
77C4⋊C4×C127C22×C4116yes—
78C22⋊C4×C127—232no—
79C42×C127—232no—
80C23×C127—28no—
81C23×C127—232no—
82C23×C127—216no—
83C23×C127#95116yes—
84C23×C127C2×C22⋊C418yes—
85C23×C127C2×C22⋊C414yes—
86C23×C127C2×C4⋊C414yes—
87C23×C127C2×C4⋊C412yes—
88C23×C127#13518yes—
89C23×C127C22×D411yescharacteristic
90C2×C4×C127—216no—
91C2×C4×C127—416no—
92C2×C4×C127—432no—
93C2×C4×C127#95132yes—
94C2×C4×C127#95116yes—
95C2×C4×C127C2×C4218yes—
96C2×C4×C127C2×C4214yes—
97C2×C4×C127C2×C22⋊C418yes—
98C2×C4×C127C2×C22⋊C414yes—
99C2×C4×C127C2×C22⋊C418yes—
100C2×C4×C127C2×C22⋊C414yes—
101C2×C4×C127C2×C4⋊C4116yes—
102C2×C4×C127C2×C4⋊C418yes—
103C2×C4×C127C2×C4⋊C412yes—
104C2×C4×C127C2×C4⋊C414yes—
105C2×C4×C127C23×C411yescharacteristic
106C2×C4×C127C23×C412yes—
107C2×C4×C127C22×D412yes—
108C2×C4×C127C22×Q811yescharacteristic
109C8×C127—28no—
110C22×C127—216no—
111C22×C127—28no—
112C22×C127—232no—
113C22×C127C2×#9518yes—
114C22×C127C2×#9514yes—
115C22×C127#416116yes—
116C22×C127#42718yes—
117C22×C127C22×C22⋊C412yes—
118C22×C127C22×C4⋊C411yescharacteristic
119C22×C127C2×#13514yes—
120C4×C127—416no—
121C4×C127C2×#9518yes—
122C4×C127C2×#9514yes—
123C4×C127C22×C4211yescharacteristic
124C4×C127C22×C22⋊C411yescharacteristic
125C4×C127C22×C4⋊C412yes—
126C2×#1735C12711yesmaximal, characteristic, direct product factor, Sylow
127C2×C127—28no—
128C2×C127#173518yesdirect product factor
129C2×C127C22×#9511yescharacteristic
130C2×C127C2×#41614yes—
131C2×C127C2×#42712yes—
132C24×C8C2×C12711yescharacteristic
133C22×#117C2×C12712yes—
134#1735C2×C12718yesdirect product factor
135C2×#340C2×C12714yescomplemented
136C127C2×#173511yescharacteristic, direct product factor, Sylow
137C24×C4C22×C12711yescharacteristic
138C23×C8—28no—
139C23×C8C4×C12718yescomplemented
140C23×C8C22×C12712yes—
141C23×C8C22×C12714yes—
142C2×#117—28no—
143C2×#117—24no—
144C2×#117C4×C127116yescomplemented
145C2×#117C22×C12718yes—
146C22×C4⋊C4C22×C12714yes—
147C22×C22⋊C4—22no—
148C22×C42—22no—
149#340C22×C127116yescomplemented
150C25D4×C12711yescharacteristic
151C23×C4—24no—
152C23×C4C2×C4×C12714yes—
153C23×C4C2×C4×C12718yes—
154C23×C4D4×C12712yes—
155C23×C4D4×C12711yescharacteristic
156C23×C4D4×C12718yes—
157C23×C4Q8×C12714yes—
158C23×C4C23×C12711yescharacteristic
159C23×C4C23×C12712yes—
160C22×C8—232no—
161C22×C8—28no—
162C22×C8—216no—
163C22×C8C2×C4×C12718yes—
164C22×C8C2×C4×C127116yescomplemented
165C22×C8C23×C12714yes—
166#117—232no—
167#117—216no—
168#117C2×C4×C127132yescomplemented
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4C2×C4×C127116yes—
172C2×C4⋊C4C23×C12718yes—
173C2×C22⋊C4—216no—
174C2×C22⋊C4—28no—
175C2×C42—216no—
176C2×C42—28no—
177C24—28no—
178C24C22⋊C4×C12718yes—
179C24C4⋊C4×C12714yes—
180C24C2×D4×C12712yes—
181C24C2×D4×C12711yescharacteristic
182C22×C4—216no—
183C22×C4—28no—
184C22×C4—416no—
185C22×C4C42×C127116yes—
186C22×C4C22⋊C4×C12718yes—
187C22×C4C22⋊C4×C127116yes—
188C22×C4C22⋊C4×C127116yes—
189C22×C4C4⋊C4×C12718yes—
190C22×C4C4⋊C4×C12714yes—
191C22×C4C4⋊C4×C127132yes—
192C22×C4C22×C4×C12712yes—
193C22×C4C22×C4×C12714yes—
194C22×C4C22×C4×C12714yes—
195C22×C4C22×C4×C12718yes—
196C22×C4C2×D4×C12712yes—
197C22×C4C2×D4×C12711yescharacteristic
198C22×C4C2×D4×C12714yes—
199C22×C4C2×D4×C12718yes—
200C22×C4C2×Q8×C12714yes—
201C22×C4C2×Q8×C12712yes—
202C22×C4C24×C12711yescharacteristic
203C2×C8—216no—
204C2×C8—232no—
205C2×C8—28no—
206C2×C8C22×C4×C12718yes—
207C4⋊C4—216no—
208C4⋊C4—28no—
209C4⋊C4C22×C4×C127116yes—
210C22⋊C4—232no—
211C42—232no—
212C23—28no—
213C23—232no—
214C23—216no—
215C23#95×C127116yes—
216C23C2×C22⋊C4×C12718yes—
217C23C2×C22⋊C4×C12714yes—
218C23C2×C4⋊C4×C12714yes—
219C23C2×C4⋊C4×C12712yes—
220C23#135×C12718yes—
221C23C22×D4×C12711yescharacteristic
222C2×C4—216no—
223C2×C4—416no—
224C2×C4—432no—
225C2×C4#95×C127132yes—
226C2×C4#95×C127116yes—
227C2×C4C2×C42×C12718yes—
228C2×C4C2×C42×C12714yes—
229C2×C4C2×C22⋊C4×C12718yes—
230C2×C4C2×C22⋊C4×C12714yes—
231C2×C4C2×C22⋊C4×C12718yes—
232C2×C4C2×C22⋊C4×C12714yes—
233C2×C4C2×C4⋊C4×C127116yes—
234C2×C4C2×C4⋊C4×C12718yes—
235C2×C4C2×C4⋊C4×C12712yes—
236C2×C4C2×C4⋊C4×C12714yes—
237C2×C4C23×C4×C12711yescharacteristic
238C2×C4C23×C4×C12712yes—
239C2×C4C22×D4×C12712yes—
240C2×C4C22×Q8×C12711yescharacteristic
241C8—28no—
242C22—216no—
243C22—28no—
244C22—232no—
245C22C2×#95×C12718yes—
246C22C2×#95×C12714yes—
247C22#416×C127116yes—
248C22#427×C12718yes—
249C22C22×C22⋊C4×C12712yes—
250C22C22×C4⋊C4×C12711yescharacteristic
251C22C2×#135×C12714yes—
252C4—416no—
253C4C2×#95×C12718yes—
254C4C2×#95×C12714yes—
255C4C22×C42×C12711yescharacteristic
256C4C22×C22⋊C4×C12711yescharacteristic
257C4C22×C4⋊C4×C12712yes—
258C2—28no—
259C2C127×#173518yesdirect product factor
260C2C22×#95×C12711yescharacteristic
261C2C2×#416×C12714yes—
262C2C2×#427×C12712yes—
subgroupG/H
centerC23×C4×C127D4
commutatorC4C22×C42×C127
Frattini, cosocleC22×C4C24×C127
socleC24×C127C2×D4
prime radicalC25×C127D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1735C2direct (N × H)
4. C2×#340×C127C2semidirect
7. C23×C8×C127C4semidirect
12. C2×#117×C127C4semidirect
17. #340×C127C22semidirect
32. C22×C8×C127C2×C4semidirect
36. #117×C127C2×C4semidirect
126. C2×#1735C127direct (N × H)
128. C2×C127#1735direct (N × H)
134. #1735C2×C127direct (N × H)
135. C2×#340C2×C127semidirect
136. C127C2×#1735direct (N × H)
139. C23×C8C4×C127semidirect
144. C2×#117C4×C127semidirect
149. #340C22×C127semidirect
164. C22×C8C2×C4×C127semidirect
168. #117C2×C4×C127semidirect
259. C2C127×#1735direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C8×C1272
C22×#117×C1272
C127×#17352
C2×#340×C1272
C2×#1735127