Finite Group Databasebeta

← C47×#13755 #26053453

Subgroups — C47×#13755 #26053453

Order 12032.

1774 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊C4×C47C212yesmaximal
2C2×C47×#376C212yesmaximal
3C2×C43×C47C211yesmaximal, characteristic
4C47×C4⋊#95C214yesmaximal
5C47×#1434C214yesmaximal
6C4×#95×C47C212yesmaximal
7C22×C4⋊C4×C47C2214yes—
8C22×C42×C47C2212yes—
9C22×C42×C47C2212yes—
10C22×C42×C47C2211yescharacteristic
11C22×C42×C47C2214yes—
12C22×C42×C47C2211yescharacteristic
13C22×C42×C47C2211yescharacteristic
14C4×C4⋊C4×C47C4116yescomplemented
15C47×#376—28no—
16C2×#95×C47C2218yes—
17C2×#95×C47C2218yes—
18C2×#95×C47C2214yes—
19C43×C47C418yescomplemented
20C23×C4×C47C2318yes—
21C23×C4×C47C2311yescharacteristic
22C23×C4×C47C2312yes—
23C23×C4×C47C2312yes—
24C23×C4×C47C2311yescharacteristic
25C23×C4×C47C2311yescharacteristic
26C2×C4⋊C4×C47—216no—
27C2×C4⋊C4×C47C2×C4116yes—
28C2×C42×C47—28no—
29C2×C42×C47—28no—
30C2×C42×C47—28no—
31C2×C42×C47—24no—
32C2×C42×C47—28no—
33C2×C42×C47C2×C418yes—
34C2×C42×C47C2×C418yes—
35C2×C42×C47C2×C414yes—
36C2×C42×C47C2×C4116yes—
37C2×C42×C47C2×C414yes—
38C2×C42×C47C2×C418yes—
39C2×C42×C47C2×C414yes—
40C2×C42×C47D414yes—
41C2×C42×C47Q814yes—
42#95×C47—216no—
43#95×C47—216no—
44#95×C47C2×C4116yes—
45C24×C47C2411yescharacteristic
46C22×C4×C47—24no—
47C22×C4×C47—24no—
48C22×C4×C47—24no—
49C22×C4×C47—24no—
50C22×C4×C47—24no—
51C22×C4×C47—28no—
52C22×C4×C47—28no—
53C22×C4×C47—216no—
54C22×C4×C47—216no—
55C22×C4×C47C22×C4116yes—
56C22×C4×C47C22×C412yes—
57C22×C4×C47C22×C414yes—
58C22×C4×C47C22×C414yes—
59C22×C4×C47C22×C414yes—
60C22×C4×C47C22×C412yes—
61C22×C4×C47C22×C412yes—
62C22×C4×C47C22×C414yes—
63C22×C4×C47C22×C414yes—
64C22×C4×C47C2×D414yes—
65C22×C4×C47C2×D412yes—
66C22×C4×C47C2×Q814yes—
67C22×C4×C47C2×Q812yes—
68C22×C4×C47#4118yes—
69C22×C4×C47#4118yes—
70C22×C4×C47#4114yes—
71C4⋊C4×C47—232no—
72C42×C47—216no—
73C42×C47—216no—
74C42×C47—28no—
75C42×C47—232no—
76C42×C47C42116yes—
77C42×C47C4⋊C4116yescomplemented
78C23×C47C23×C411yescharacteristic
79C23×C47C23×C412yes—
80C23×C47C22×D411yescharacteristic
81C23×C47C22×Q811yescharacteristic
82C23×C47C2×#4114yes—
83C23×C47C2×#4112yes—
84C23×C47C2×#4114yes—
85C2×C4×C47—24no—
86C2×C4×C47—24no—
87C2×C4×C47—24no—
88C2×C4×C47—24no—
89C2×C4×C47—24no—
90C2×C4×C47—28no—
91C2×C4×C47—216no—
92C2×C4×C47—28no—
93C2×C4×C47—28no—
94C2×C4×C47—216no—
95C2×C4×C47—216no—
96C2×C4×C47—216no—
97C2×C4×C47—416no—
98C2×C4×C47—416no—
99C2×C4×C47C2×C4214yes—
100C2×C4×C47C2×C4214yes—
101C2×C4×C47C2×C4214yes—
102C2×C4×C47C2×C4⋊C418yes—
103C2×C4×C47C2×C4⋊C414yes—
104C2×C4×C47#11718yes—
105C2×C4×C47#11718yes—
106C2×C4×C47#117116yes—
107C2×C4×C47#11714yes—
108C2×C4×C47C4×D418yes—
109C2×C4×C47C4×Q818yes—
110#13755C4711yesmaximal, characteristic, direct product factor, Sylow
111C22×C47C22×C4211yescharacteristic
112C22×C47C22×C4⋊C411yescharacteristic
113C22×C47C2×#11712yes—
114C22×C47C2×#11714yes—
115C22×C47C2×#11711yescharacteristic
116C22×C47C2×#11712yes—
117C22×C47C2×C4×D412yes—
118C22×C47C2×C4×Q812yes—
119C22×C47C4×#4114yes—
120C22×C47#52918yes—
121C22×C47#53214yes—
122C22×C47#53314yes—
123C4×C47—24no—
124C4×C47—28no—
125C4×C47—28no—
126C4×C47—416no—
127C4×C47#37618yescomplemented
128C4×C47C4×C4⋊C418yes—
129C2×C4×C4⋊C4C2×C4712yes—
130C2×#376C2×C4712yes—
131C2×C43C2×C4711yescharacteristic
132C4⋊#95C2×C4714yes—
133#1434C2×C4714yes—
134C4×#95C2×C4712yes—
135C2×C47C2×#37611yescharacteristic
136C2×C47C2×C4×C4⋊C411yescharacteristic
137C2×C47C4×#11711yescharacteristic
138C2×C47#228312yes—
139C2×C47#228512yes—
140C2×C47#229314yes—
141C2×C47#229514yes—
142C22×C4⋊C4C22×C4714yes—
143C22×C42C22×C4711yescharacteristic
144C22×C42C22×C4714yes—
145C22×C42C22×C4712yes—
146C22×C42C22×C4712yes—
147C22×C42C22×C4711yescharacteristic
148C22×C42C22×C4711yescharacteristic
149C4×C4⋊C4C4×C47116yescomplemented
150#376—28no—
151C2×#95C22×C4718yes—
152C2×#95C22×C4718yes—
153C2×#95C22×C4714yes—
154C43C4×C4718yescomplemented
155C47#1375511yescharacteristic, direct product factor, Sylow
156C23×C4C23×C4718yes—
157C23×C4C23×C4711yescharacteristic
158C23×C4C23×C4712yes—
159C23×C4C23×C4712yes—
160C23×C4C23×C4711yescharacteristic
161C23×C4C23×C4711yescharacteristic
162C2×C4⋊C4—216no—
163C2×C4⋊C4C2×C4×C47116yes—
164C2×C42—28no—
165C2×C42—28no—
166C2×C42—28no—
167C2×C42—24no—
168C2×C42—28no—
169C2×C42C2×C4×C4718yes—
170C2×C42C2×C4×C4718yes—
171C2×C42C2×C4×C4714yes—
172C2×C42C2×C4×C47116yes—
173C2×C42C2×C4×C4714yes—
174C2×C42C2×C4×C4718yes—
175C2×C42C2×C4×C4714yes—
176C2×C42D4×C4714yes—
177C2×C42Q8×C4714yes—
178#95—216no—
179#95—216no—
180#95C2×C4×C47116yes—
181C24C24×C4711yescharacteristic
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—216no—
189C22×C4—28no—
190C22×C4—216no—
191C22×C4C22×C4×C47116yes—
192C22×C4C22×C4×C4714yes—
193C22×C4C22×C4×C4712yes—
194C22×C4C22×C4×C4714yes—
195C22×C4C22×C4×C4714yes—
196C22×C4C22×C4×C4712yes—
197C22×C4C22×C4×C4712yes—
198C22×C4C22×C4×C4714yes—
199C22×C4C22×C4×C4714yes—
200C22×C4C2×D4×C4714yes—
201C22×C4C2×D4×C4712yes—
202C22×C4C2×Q8×C4714yes—
203C22×C4C2×Q8×C4712yes—
204C22×C4#41×C4718yes—
205C22×C4#41×C4718yes—
206C22×C4#41×C4714yes—
207C4⋊C4—232no—
208C42—216no—
209C42—216no—
210C42—28no—
211C42—232no—
212C42C42×C47116yes—
213C42C4⋊C4×C47116yescomplemented
214C23C23×C4×C4711yescharacteristic
215C23C23×C4×C4712yes—
216C23C22×D4×C4711yescharacteristic
217C23C22×Q8×C4711yescharacteristic
218C23C2×#41×C4714yes—
219C23C2×#41×C4712yes—
220C23C2×#41×C4714yes—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—28no—
227C2×C4—216no—
228C2×C4—28no—
229C2×C4—28no—
230C2×C4—216no—
231C2×C4—216no—
232C2×C4—216no—
233C2×C4—416no—
234C2×C4—416no—
235C2×C4C2×C42×C4714yes—
236C2×C4C2×C42×C4714yes—
237C2×C4C2×C42×C4714yes—
238C2×C4C2×C4⋊C4×C4718yes—
239C2×C4C2×C4⋊C4×C4714yes—
240C2×C4#117×C4718yes—
241C2×C4#117×C4718yes—
242C2×C4#117×C47116yes—
243C2×C4#117×C4714yes—
244C2×C4C4×D4×C4718yes—
245C2×C4C4×Q8×C4718yes—
246C22C22×C42×C4711yescharacteristic
247C22C22×C4⋊C4×C4711yescharacteristic
248C22C2×#117×C4712yes—
249C22C2×#117×C4714yes—
250C22C2×#117×C4711yescharacteristic
251C22C2×#117×C4712yes—
252C22C2×C4×D4×C4712yes—
253C22C2×C4×Q8×C4712yes—
254C22C4×#41×C4714yes—
255C22C47×#52918yes—
256C22C47×#53214yes—
257C22C47×#53314yes—
258C4—24no—
259C4—28no—
260C4—28no—
261C4—416no—
262C4C47×#37618yescomplemented
263C4C4×C4⋊C4×C4718yes—
264C2C2×C47×#37611yescharacteristic
265C2C2×C4×C4⋊C4×C4711yescharacteristic
266C2C4×#117×C4711yescharacteristic
267C2C47×#228312yes—
268C2C47×#228512yes—
269C2C47×#229314yes—
270C2C47×#229514yes—
subgroupG/H
centerC23×C4×C47C23
commutatorC22C22×C42×C47
Frattini, cosocleC24C24×C47
socle, prime radicalC24×C47C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
14. C4×C4⋊C4×C47C4semidirect
19. C43×C47C4semidirect
77. C42×C47C4⋊C4semidirect
110. #13755C47direct (N × H)
127. C4×C47#376semidirect
149. C4×C4⋊C4C4×C47semidirect
154. C43C4×C47semidirect
155. C47#13755direct (N × H)
213. C42C4⋊C4×C47semidirect
262. C4C47×#376semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C4×C472
C2×C47×#3762
C2×C43×C472
C47×C4⋊#952
C47×#14342
C4×#95×C472
#1375547