Finite Group Databasebeta

← #26129333

Subgroups — #26129333

Order 12032.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#20244733C211yesmaximal, characteristic
2#20242361C212yesmaximal
3#20241418C212yesmaximal
4#20241334C211yesmaximal, characteristic
5C47×#2551C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689279—22no—
8#15689163—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C2×C4⋊Q8×C47—22no—
14C47×#395—21no—
15C47×#395C2212yes—
16C47×#389—22no—
17C47×#384C2211yescharacteristic
18C4×C4⋊C4×C47—21no—
19C2×#10625337—42no—
20C2×#10625329—21no—
21#13094023—41no—
22#13094015—41no—
23#13094003—22no—
24#13093983—21no—
25#13093957—41no—
26#13093955—41no—
27#13093948—42no—
28C22⋊(C47⋊C8)—22no—
29C22×Q8×C47—21no—
30C22×Q8×C47D412yes—
31C4⋊Q8×C47—48no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C4⋊C4×C47C2311yescharacteristic
39C2×C42×C47—22no—
40C2×C42×C47D411yescharacteristic
41#95×C47—21no—
42#95×C47—22no—
43#95×C47D412yes—
44C2×Q8×C47—44no—
45C2×Q8×C47—44no—
46C2×Q8×C47—42no—
47C2×Q8×C47—81no—
48C22×C4×C47—21no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—21no—
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47C2×D412yes—
54C2×Dic94—22no—
55C4⋊C4×C47—24no—
56C4⋊C4×C47—24no—
57C4⋊C4×C47—41no—
58C4⋊C4×C47—44no—
59C4⋊C4×C47—44no—
60C4⋊C4×C47—44no—
61C4⋊C4×C47—41no—
62C42×C47—21no—
63C42×C47—44no—
64C42×C47Q1612yescomplemented
65C22⋊Dic47—41no—
66#10625337—84no—
67C4⋊Dic47—22no—
68C4⋊Dic47—21no—
69C4⋊Dic47—21no—
70Dic47⋊C4—42no—
71C4×Dic47—41no—
72#10625329—22no—
73#10625329—41no—
74C2×C47⋊C8—41no—
75C2×C47⋊C8—42no—
76C23×C47#12011yescharacteristic
77Q8×C47—44no—
78Q8×C47—82no—
79Q8×C47—84no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—24no—
86C2×C4×C47—24no—
87C2×C4×C47—22no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—41no—
95C2×C4×C47C2×Q1611yescharacteristic
96C2×C4×C47#13711yescharacteristic
97C2×Dic47—41no—
98C2×Dic47—41no—
99Dic94—82no—
100C47⋊C8—42no—
101C47⋊C8—82no—
102#12256—471nomaximal, Sylow
103C22×C47—21no—
104C22×C47—21no—
105C22×C47#45111yescharacteristic
106C22×C47#45611yescharacteristic
107C22×C47B(4,𝔽2)11yescharacteristic
108C4×C47—24no—
109C4×C47—21no—
110C4×C47—21no—
111C4×C47—44no—
112C4×C47—42no—
113C4×C47—44no—
114C4×C47—81no—
115Dic47—81no—
116Dic47—81no—
117#2551D4711yescharacteristic
118#2060—471no—
119#1605—472no—
120#1354—472no—
121#1325—471no—
122C2×C47—21no—
123C2×C47—21no—
124C2×C47#160511yescharacteristic
125C2×C47#219611yescharacteristic
126C2×C47#220511yescharacteristic
127#544—471no—
128C2×C4⋊Q8—22no—
129#451—942no—
130#428—941no—
131#395—21no—
132#395C2×D4712yes—
133#389—22no—
134#384C2×D4711yescharacteristic
135C4×C4⋊C4—21no—
136#343—471no—
137#339—941no—
138#328—942no—
139#324—472no—
140C47#1225611yescharacteristic, complemented, Sylow
141C22×Q8—21no—
142C22×Q8C22⋊D4712yes—
143C2×Q16—1882no—
144C2×M16—941no—
145C4⋊Q8—48no—
146C4⋊Q8—1881no—
147#125—1881no—
148C22⋊Q8—942no—
149#117—941no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D4711yescharacteristic
156C2×C4⋊C4C22×D4711yescharacteristic
157C2×C42—22no—
158C2×C42C22⋊D4711yescharacteristic
159#107—1881no—
160#106—1881no—
161#103—1882no—
162C22⋊C8—942no—
163#95—21no—
164#95—22no—
165#95C22⋊D4712yes—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—942no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×C22⋊D4711yescharacteristic
176C22×C4C2×C22⋊D4712yes—
177Q16—3764no—
178M16—942no—
179M16—1881no—
180C2×C8—1882no—
181C2×C8—1881no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—942no—
190C4⋊C4—941no—
191C4⋊C4—941no—
192C4⋊C4—1882no—
193C22⋊C4—1881no—
194C42—21no—
195C42—44no—
196C42—1881no—
197C42#1062533712yescomplemented
198C23#1309399311yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—3762no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—1881no—
219C2×C4—1881no—
220C2×C4C2×#1062533711yescharacteristic
221C2×C4#1309405511yescharacteristic
222C8—3762no—
223C8—1882no—
224C22—21no—
225C22—21no—
226C22#1568927911yescharacteristic
227C22#1568930611yescharacteristic
228C22#1568931011yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—3761no—
237C4—3761no—
238C2—21no—
239C2—21no—
240C2#2024236211yescharacteristic
241C2#2024537211yescharacteristic
242C2#2024541811yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2551C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2551D47
hypocenterC47#12256
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
64. C42×C47Q16semidirect
140. C47#12256semidirect
197. C42#10625337semidirect

Maximal subgroups

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

subgroupindex
#202447332
#202423612
#202414182
#202413342
C47×#25512
#1225647