Finite Group Databasebeta

← #26105520

Subgroups — #26105520

Order 12032.

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

normal subgroups only

This group has 434 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#20244864C212yesmaximal, complemented
3#20241211C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2724C211yesmaximal, characteristic
6C2×#13093957—21no—
7C2×#13093955—22no—
8C2×#13093945—21no—
9#15688993C2211yescharacteristic
10#15688885—22no—
11#15688878C2212yes—
12#15688878C2211yescharacteristic
13C2×#123×C47—21no—
14C47×#398—21no—
15C47×#397—22no—
16C47×#390C2211yescharacteristic
17C47×#389C2212yes—
18C47×#385—22no—
19C22×C47⋊C8—21no—
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—22no—
23#13093957—42no—
24#13093955—44no—
25#13093945—42no—
26C22×D4×C47D411yescharacteristic
27C23×C4×C47—21no—
28#123×C47—42no—
29#123×C47—42no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47D412yes—
33C2×C4⋊C4×C47D412yes—
34C2×C4⋊C4×C47D411yescharacteristic
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47—21no—
37C2×C22⋊C4×C47—22no—
38C2×C42×C47C2311yescharacteristic
39#95×C47—22no—
40#95×C47—21no—
41#95×C47—22no—
42C24×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47—41no—
46C2×D4×C47SD1612yes—
47C22×C4×C47—21no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—42no—
51C22×C4×C47—81no—
52C22×C4×C47—81no—
53C22×C4×C47C2×D411yescharacteristic
54C22×C4×C47C2×D412yes—
55C22×Dic47—41no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47—22no—
58C4⋊C4×C47—42no—
59C4⋊C4×C47—41no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—81no—
63C22⋊C4×C47—42no—
64C22⋊C4×C47—44no—
65C22⋊C4×C47—42no—
66C22⋊C4×C47—42no—
67C42×C47—41no—
68C4⋊Dic47—41no—
69C4⋊Dic47—41no—
70C4⋊Dic47—42no—
71C4⋊Dic47—42no—
72C2×C47⋊C8—22no—
73C2×C47⋊C8—24no—
74C2×C47⋊C8—42no—
75C2×C47⋊C8—41no—
76C2×C47⋊C8—42no—
77C2×C47⋊C8—41no—
78C23×C47—22no—
79C23×C47—42no—
80C23×C47—41no—
81C23×C47—42no—
82C23×C47—41no—
83C23×C47#12011yescharacteristic
84D4×C47—42no—
85D4×C47—81no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—21no—
90C2×C4×C47—42no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—42no—
94C2×C4×C47—41no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—84no—
99C2×C4×C47C2×SD1611yescharacteristic
100C2×C4×C47#13512yes—
101C2×C4×C47AGL(1,ℤ/8)12yes—
102C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
103C2×Dic47—82no—
104C2×Dic47—81no—
105C47⋊C8—44no—
106C47⋊C8—42no—
107#7819—471nomaximal, Sylow
108C22×C47—42no—
109C22×C47—44no—
110C22×C47—82no—
111C22×C47—81no—
112C22×C47—81no—
113C22×C47#44912yes—
114C22×C47#45011yescharacteristic
115C22×C47#45311yescharacteristic
116C22×C47#45612yes—
117C22×C47#45811yescharacteristic
118C4×C47—22no—
119C4×C47—22no—
120C4×C47—21no—
121C4×C47—21no—
122C4×C47—82no—
123C4×C47—81no—
124C4×C47—82no—
125Dic47—161no—
126#2724D4711yescharacteristic
127#2093—471no—
128#2084—472no—
129#1271—472no—
130#1270—471no—
131C2×C47—82no—
132C2×C47#160912yes—
133C2×C47#161311yescharacteristic
134C2×C47#162011yescharacteristic
135C2×C47#165412yes—
136C2×C47#166011yescharacteristic
137C2×#123—21no—
138C2×#107—941no—
139C2×#106—942no—
140C2×#102—941no—
141#398—21no—
142#397—22no—
143#390C2×D4711yescharacteristic
144#389C2×D4712yes—
145#385—22no—
146#384—471no—
147#340—942no—
148#336—472no—
149#336—471no—
150C47#781911yescharacteristic, complemented, Sylow
151C22×D4C22⋊D4711yescharacteristic
152C23×C4—21no—
153C22×C8—942no—
154C22×C8—941no—
155#123—42no—
156#123—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—942no—
160C2×C4⋊C4—941no—
161C2×C4⋊C4C22⋊D4712yes—
162C2×C4⋊C4C22⋊D4712yes—
163C2×C4⋊C4C22⋊D4711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C42C22×D4711yescharacteristic
168#107—1882no—
169#106—1884no—
170#102—1882no—
171#95—22no—
172#95—21no—
173#95—22no—
174C24—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4#1062533512yes—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—1881no—
186C22×C4C2×C22⋊D4711yescharacteristic
187C22×C4C2×C22⋊D4712yes—
188C2×C8—942no—
189C2×C8—944no—
190C2×C8—1882no—
191C2×C8—1881no—
192C2×C8—1882no—
193C2×C8—1881no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—1881no—
202C4⋊C4—1881no—
203C4⋊C4—1882no—
204C4⋊C4—1882no—
205C22⋊C4—42no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—41no—
210C23—22no—
211C23—42no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23#1309399311yescharacteristic
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—3762no—
232C2×C4—3761no—
233C2×C4C2×#1062533511yescharacteristic
234C2×C4#1309404412yes—
235C2×C4#1309404912yes—
236C2×C4#1309405011yescharacteristic
237C8—1884no—
238C8—1882no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#1568926812yes—
245C22#1568927411yescharacteristic
246C22#1568929111yescharacteristic
247C22#1568930612yes—
248C22#1568931311yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—82no—
255C4—82no—
256C4—7521no—
257C2—82no—
258C2#2024238512yes—
259C2#2024240811yescharacteristic
260C2#2024244711yescharacteristic
261C2#2024267212yes—
262C2#2024269911yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2724C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2724D47
hypocenterC47#7819
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #20244864C2semidirect
3. #20241211C2semidirect
150. C47#7819semidirect

Maximal subgroups

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

subgroupindex
#202449162
#202448642
#202412112
#202412092
C47×#27242
#781947