Finite Group Databasebeta

← #26105532

Subgroups — #26105532

Order 12032.

422 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 87 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#20244916C211yesmaximal, characteristic
2#20244796C212yesmaximal, complemented
3#20241214C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2804C211yesmaximal, characteristic
6C2×#13093957—21no—
7C2×#13093948—22no—
8C2×#13093945—21no—
9#15688993C2211yescharacteristic
10#15688885—22no—
11#15688878C2212yes—
12#15688878C2211yescharacteristic
13C47×#400—21no—
14C47×#393—22no—
15C47×#391C2212yes—
16C47×#390C2211yescharacteristic
17C47×#386—21no—
18C47×#385—22no—
19C22×C47⋊C8—21no—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—22no—
23#13093957—42no—
24#13093948—44no—
25#13093945—42no—
26C22×Q8×C47D412yes—
27C22×D4×C47D411yescharacteristic
28C23×C4×C47—21no—
29C2×C4⋊C4×C47—21no—
30C2×C4⋊C4×C47D412yes—
31C2×C4⋊C4×C47D411yescharacteristic
32C2×C22⋊C4×C47—22no—
33C2×C22⋊C4×C47—22no—
34C2×C22⋊C4×C47—21no—
35C2×C42×C47C2311yescharacteristic
36#95×C47—22no—
37#95×C47—21no—
38#95×C47—22no—
39#95×C47—22no—
40C24×C47—21no—
41C2×Q8×C47—22no—
42C2×Q8×C47—22no—
43C2×Q8×C47—42no—
44C2×Q8×C47—42no—
45C2×D4×C47—21no—
46C2×D4×C47—41no—
47C2×D4×C47—41no—
48C2×D4×C47Q1612yes—
49C22×C4×C47—21no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—81no—
53C22×C4×C47—82no—
54C22×C4×C47C2×D411yescharacteristic
55C22×C4×C47C2×D412yes—
56C22×Dic47—41no—
57C4⋊C4×C47—42no—
58C4⋊C4×C47—41no—
59C4⋊C4×C47—81no—
60C22⋊C4×C47—42no—
61C22⋊C4×C47—42no—
62C22⋊C4×C47—82no—
63C22⋊C4×C47—81no—
64C42×C47—41no—
65C4⋊Dic47—41no—
66C4⋊Dic47—41no—
67C4⋊Dic47—42no—
68C4⋊Dic47—42no—
69C2×C47⋊C8—22no—
70C2×C47⋊C8—41no—
71C2×C47⋊C8—41no—
72C2×C47⋊C8—42no—
73C2×C47⋊C8—42no—
74C2×C47⋊C8—42no—
75C23×C47—22no—
76C23×C47—42no—
77C23×C47—41no—
78C23×C47—42no—
79C23×C47—41no—
80C23×C47#12011yescharacteristic
81Q8×C47—44no—
82Q8×C47—82no—
83D4×C47—42no—
84D4×C47—81no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—22no—
89C2×C4×C47—42no—
90C2×C4×C47—42no—
91C2×C4×C47—41no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—84no—
98C2×C4×C47C2×Q1611yescharacteristic
99C2×C4×C47#13512yes—
100C2×C4×C47AGL(1,ℤ/8)12yes—
101C2×C4×C47#13711yescharacteristic
102C2×Dic47—82no—
103C2×Dic47—81no—
104C47⋊C8—42no—
105C47⋊C8—82no—
106#7821—471nomaximal, Sylow
107C22×C47—42no—
108C22×C47—44no—
109C22×C47—82no—
110C22×C47—81no—
111C22×C47—81no—
112C22×C47#44912yes—
113C22×C47#45111yescharacteristic
114C22×C47#45311yescharacteristic
115C22×C47#45512yes—
116C22×C47#45811yescharacteristic
117C4×C47—22no—
118C4×C47—22no—
119C4×C47—21no—
120C4×C47—21no—
121C4×C47—82no—
122C4×C47—81no—
123C4×C47—82no—
124Dic47—161no—
125#2804D4711yescharacteristic
126#2093—471no—
127#2072—472no—
128#1272—472no—
129#1270—471no—
130C2×C47—82no—
131C2×C47#161212yes—
132C2×C47#161411yescharacteristic
133C2×C47#162011yescharacteristic
134C2×C47#164412yes—
135C2×C47#165911yescharacteristic
136C2×#107—941no—
137C2×#103—942no—
138C2×#102—941no—
139#400—21no—
140#393—22no—
141#391C2×D4712yes—
142#390C2×D4711yescharacteristic
143#386—21no—
144#385—22no—
145#384—471no—
146#340—942no—
147#336—472no—
148#336—471no—
149C47#782111yescharacteristic, complemented, Sylow
150C22×Q8C22⋊D4712yes—
151C22×D4C22⋊D4711yescharacteristic
152C23×C4—21no—
153C22×C8—941no—
154C22×C8—942no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—942no—
157C2×C4⋊C4—941no—
158C2×C4⋊C4C22⋊D4712yes—
159C2×C4⋊C4C22⋊D4711yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C22×D4711yescharacteristic
164#107—1882no—
165#103—1884no—
166#102—1882no—
167#95—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171C24—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#1062533712yes—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—81no—
184C22×C4—82no—
185C22×C4—1881no—
186C22×C4C2×C22⋊D4711yescharacteristic
187C22×C4C2×C22⋊D4712yes—
188C2×C8—942no—
189C2×C8—1882no—
190C2×C8—1882no—
191C2×C8—1882no—
192C2×C8—1881no—
193C2×C8—1881no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—1881no—
198C4⋊C4—1881no—
199C4⋊C4—1882no—
200C4⋊C4—1882no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1309399311yescharacteristic
212Q8—44no—
213Q8—82no—
214D4—42no—
215D4—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—3762no—
230C2×C4—3761no—
231C2×C4C2×#1062533711yescharacteristic
232C2×C4#1309404412yes—
233C2×C4#1309404912yes—
234C2×C4#1309405511yescharacteristic
235C8—3762no—
236C8—1882no—
237C22—44no—
238C22—42no—
239C22—81no—
240C22—81no—
241C22—82no—
242C22#1568926812yes—
243C22#1568927911yescharacteristic
244C22#1568929111yescharacteristic
245C22#1568930112yes—
246C22#1568931311yescharacteristic
247C4—22no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—81no—
253C4—82no—
254C4—7521no—
255C2—82no—
256C2#2024240212yes—
257C2#2024241311yescharacteristic
258C2#2024244711yescharacteristic
259C2#2024260212yes—
260C2#2024269411yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2804C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2804D47
hypocenterC47#7821
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #20244796C2semidirect
3. #20241214C2semidirect
149. C47#7821semidirect

Maximal subgroups

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

subgroupindex
#202449162
#202447962
#202412142
#202412092
C47×#28042
#782147