Finite Group Databasebeta

← C127×#13776 #44577449

Subgroups — C127×#13776 #44577449

Order 32512.

2318 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C127C211yesmaximal, characteristic
2C2×#381×C127C212yesmaximal
3C2×C4×C22⋊C4×C127C212yesmaximal
4C127×#1436C218yesmaximal, complemented
5C127×C22⋊#95C212yesmaximal
6C24×C4×C127C2211yescharacteristic
7C24×C4×C127C2211yescharacteristic
8C24×C4×C127C2211yescharacteristic
9C22×C4⋊C4×C127—28no—
10C22×C4⋊C4×C127C418yescomplemented
11C22×C4⋊C4×C127C2214yes—
12C22×C22⋊C4×C127C2214yes—
13C22×C42×C127C2214yes—
14#381×C127C4116yescomplemented
15C4×C22⋊C4×C127—28no—
16C2×#95×C127C2214yes—
17C2×#95×C127C2218yes—
18C2×#95×C127C2218yes—
19C25×C127C2311yescharacteristic
20C23×C4×C127—24no—
21C23×C4×C127—24no—
22C23×C4×C127—24no—
23C23×C4×C127—44no—
24C23×C4×C127—44no—
25C23×C4×C127C2×C414yes—
26C23×C4×C127C2×C418yes—
27C23×C4×C127C2×C414yes—
28C23×C4×C127C2×C414yes—
29C23×C4×C127D418yes—
30C23×C4×C127C2318yes—
31C23×C4×C127C2312yes—
32C23×C4×C127C2312yes—
33C23×C4×C127C2312yes—
34C2×C4⋊C4×C127—232no—
35C2×C4⋊C4×C127—28no—
36C2×C4⋊C4×C127—28no—
37C2×C4⋊C4×C127C2×C4116yes—
38C2×C22⋊C4×C127—28no—
39C2×C22⋊C4×C127—28no—
40C2×C22⋊C4×C127C2×C4116yes—
41C2×C42×C127—28no—
42C2×C42×C127—48no—
43#95×C127—28no—
44#95×C127—48no—
45#95×C127C2×C4132yescomplemented
46C24×C127—24no—
47C24×C127—24no—
48C24×C127C22×C412yes—
49C24×C127C22×C414yes—
50C24×C127C2×D414yes—
51C24×C127#4114yes—
52C24×C127C2411yescharacteristic
53C22×C4×C127—28no—
54C22×C4×C127—28no—
55C22×C4×C127—232no—
56C22×C4×C127—24no—
57C22×C4×C127—24no—
58C22×C4×C127—28no—
59C22×C4×C127—28no—
60C22×C4×C127—28no—
61C22×C4×C127—24no—
62C22×C4×C127—28no—
63C22×C4×C127—216no—
64C22×C4×C127—28no—
65C22×C4×C127—216no—
66C22×C4×C127—48no—
67C22×C4×C127—48no—
68C22×C4×C127—416no—
69C22×C4×C127—416no—
70C22×C4×C127C42116yescomplemented
71C22×C4×C127C22⋊C4116yes—
72C22×C4×C127C22×C4116yes—
73C22×C4×C127C22×C414yes—
74C22×C4×C127C22×C418yes—
75C22×C4×C127C22×C414yes—
76C22×C4×C127C22×C414yes—
77C22×C4×C127C2×D414yes—
78C22×C4×C127C2×D414yes—
79C4⋊C4×C127—232no—
80C22⋊C4×C127—232no—
81C42×C127—416no—
82C23×C127—24no—
83C23×C127—24no—
84C23×C127—216no—
85C23×C127—28no—
86C23×C127—28no—
87C23×C127—216no—
88C23×C127C2×C4214yes—
89C23×C127C2×C22⋊C414yes—
90C23×C127#11714yes—
91C23×C127C4×D4116yes—
92C23×C127C23×C411yescharacteristic
93C23×C127C23×C412yes—
94C23×C127C22×D412yes—
95C23×C127C2×#4112yes—
96C23×C127H(5,𝔽2)14yes—
97C23×C1272−1+414yes—
98C2×C4×C127—24no—
99C2×C4×C127—24no—
100C2×C4×C127—216no—
101C2×C4×C127—28no—
102C2×C4×C127—28no—
103C2×C4×C127—28no—
104C2×C4×C127—232no—
105C2×C4×C127—24no—
106C2×C4×C127—216no—
107C2×C4×C127—28no—
108C2×C4×C127—232no—
109C2×C4×C127—416no—
110C2×C4×C127—416no—
111C2×C4×C127—48no—
112C2×C4×C127—48no—
113C2×C4×C127—48no—
114C2×C4×C127—416no—
115C2×C4×C127C2×C4218yes—
116C2×C4×C127C2×C22⋊C418yes—
117C22×C127—24no—
118C22×C127—24no—
119C22×C127—28no—
120C22×C127—232no—
121C22×C127—28no—
122C22×C127C4×C22⋊C418yes—
123C22×C127C22×C4211yescharacteristic
124C22×C127C22×C22⋊C411yescharacteristic
125C22×C127C2×#11711yescharacteristic
126C22×C127C2×C4×D414yes—
127C22×C127#51812yes—
128C22×C127#51912yes—
129C22×C127#52018yes—
130C22×C127#53414yes—
131C22×C127#53614yes—
132C22×C127#53918yes—
133C4×C127—28no—
134C4×C127—28no—
135C4×C127—44no—
136C4×C127—416no—
137#13776C12711yesmaximal, characteristic, direct product factor, Sylow
138C2×C127—28no—
139C2×C127C2×C4×C22⋊C411yescharacteristic
140C2×C127#227412yes—
141C2×C127#230811yescharacteristic
142C2×C127#231011yescharacteristic
143C2×C127#231312yes—
144C2×C127#232218yes—
145C23×C4⋊C4C2×C12711yescharacteristic
146C2×#381C2×C12712yes—
147C2×C4×C22⋊C4C2×C12712yes—
148#1436C2×C12718yescomplemented
149C22⋊#95C2×C12712yes—
150C127#1377611yescharacteristic, direct product factor, Sylow
151C24×C4C22×C12711yescharacteristic
152C24×C4C22×C12711yescharacteristic
153C24×C4C22×C12711yescharacteristic
154C22×C4⋊C4—28no—
155C22×C4⋊C4C4×C12718yescomplemented
156C22×C4⋊C4C22×C12714yes—
157C22×C22⋊C4C22×C12714yes—
158C22×C42C22×C12714yes—
159#381C4×C127116yescomplemented
160C4×C22⋊C4—28no—
161C2×#95C22×C12714yes—
162C2×#95C22×C12718yes—
163C2×#95C22×C12718yes—
164C25C23×C12711yescharacteristic
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×C4×C12714yes—
171C23×C4C2×C4×C12718yes—
172C23×C4C2×C4×C12714yes—
173C23×C4C2×C4×C12714yes—
174C23×C4D4×C12718yes—
175C23×C4C23×C12718yes—
176C23×C4C23×C12712yes—
177C23×C4C23×C12712yes—
178C23×C4C23×C12712yes—
179C2×C4⋊C4—232no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4C2×C4×C127116yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4C2×C4×C127116yes—
186C2×C42—28no—
187C2×C42—48no—
188#95—28no—
189#95—48no—
190#95C2×C4×C127132yescomplemented
191C24—24no—
192C24—24no—
193C24C22×C4×C12712yes—
194C24C22×C4×C12714yes—
195C24C2×D4×C12714yes—
196C24#41×C12714yes—
197C24C24×C12711yescharacteristic
198C22×C4—28no—
199C22×C4—28no—
200C22×C4—232no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—216no—
209C22×C4—28no—
210C22×C4—216no—
211C22×C4—48no—
212C22×C4—48no—
213C22×C4—416no—
214C22×C4—416no—
215C22×C4C42×C127116yescomplemented
216C22×C4C22⋊C4×C127116yes—
217C22×C4C22×C4×C127116yes—
218C22×C4C22×C4×C12714yes—
219C22×C4C22×C4×C12718yes—
220C22×C4C22×C4×C12714yes—
221C22×C4C22×C4×C12714yes—
222C22×C4C2×D4×C12714yes—
223C22×C4C2×D4×C12714yes—
224C4⋊C4—232no—
225C22⋊C4—232no—
226C42—416no—
227C23—24no—
228C23—24no—
229C23—216no—
230C23—28no—
231C23—28no—
232C23—216no—
233C23C2×C42×C12714yes—
234C23C2×C22⋊C4×C12714yes—
235C23#117×C12714yes—
236C23C4×D4×C127116yes—
237C23C23×C4×C12711yescharacteristic
238C23C23×C4×C12712yes—
239C23C22×D4×C12712yes—
240C23C2×#41×C12712yes—
241C23H(5,𝔽2)×C12714yes—
242C232−1+4×C12714yes—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—216no—
246C2×C4—28no—
247C2×C4—28no—
248C2×C4—28no—
249C2×C4—232no—
250C2×C4—24no—
251C2×C4—216no—
252C2×C4—28no—
253C2×C4—232no—
254C2×C4—416no—
255C2×C4—416no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—416no—
260C2×C4C2×C42×C12718yes—
261C2×C4C2×C22⋊C4×C12718yes—
262C22—28no—
263C22—232no—
264C22—28no—
265C22—24no—
266C22—24no—
267C22C4×C22⋊C4×C12718yes—
268C22C22×C42×C12711yescharacteristic
269C22C22×C22⋊C4×C12711yescharacteristic
270C22C2×#117×C12711yescharacteristic
271C22C2×C4×D4×C12714yes—
272C22#518×C12712yes—
273C22#519×C12712yes—
274C22#520×C12718yes—
275C22#534×C12714yes—
276C22#536×C12714yes—
277C22#539×C12718yes—
278C4—28no—
279C4—28no—
280C4—416no—
281C4—44no—
282C2—28no—
283C2C2×C4×C22⋊C4×C12711yescharacteristic
284C2C127×#227412yes—
285C2C127×#230811yescharacteristic
286C2C127×#231011yescharacteristic
287C2C127×#231312yes—
288C2C127×#232218yes—
subgroupG/H
center, socleC24×C127C24
commutatorC22C22×C42×C127
Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1436C2semidirect
10. C22×C4⋊C4×C127C4semidirect
14. #381×C127C4semidirect
45. #95×C127C2×C4semidirect
70. C22×C4×C127C42semidirect
137. #13776C127direct (N × H)
148. #1436C2×C127semidirect
150. C127#13776direct (N × H)
155. C22×C4⋊C4C4×C127semidirect
159. #381C4×C127semidirect
190. #95C2×C4×C127semidirect
215. C22×C4C42×C127semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C1272
C2×#381×C1272
C2×C4×C22⋊C4×C1272
C127×#14362
C127×C22⋊#952
#13776127