Finite Group Databasebeta

← #26129097

Subgroups — #26129097

Order 12032.

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

normal subgroups only

This group has 460 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#20244544C211yesmaximal, characteristic
2#20242361C212yesmaximal, complemented
3#20241418C212yesmaximal, complemented
4#20241320C211yesmaximal, characteristic
5C47×#2683C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689279—22no—
8#15689128—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C2×C22⋊Q8×C47—22no—
14C47×#396—22no—
15C47×#395C2212yes—
16C47×#390C2211yescharacteristic
17C47×#389—21no—
18C47×#386—21no—
19C2×#10625337—42no—
20C2×#10625329—21no—
21#13094023—41no—
22#13094015—41no—
23#13094003—22no—
24#13093983—21no—
25#13093948—42no—
26#13093945—41no—
27#13093945—41no—
28C22⋊(C47⋊C8)—22no—
29C22×Q8×C47D412yes—
30C22×D4×C47D411yescharacteristic
31C23×C4×C47—21no—
32C22⋊Q8×C47—48no—
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47C2311yescharacteristic
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—21no—
41C2×C42×C47D411yescharacteristic
42#95×C47—22no—
43#95×C47D412yes—
44C24×C47—21no—
45C2×Q8×C47—44no—
46C2×Q8×C47—42no—
47C2×D4×C47—21no—
48C2×D4×C47—41no—
49C2×D4×C47—41no—
50C2×D4×C47Q1612yescomplemented
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—44no—
55C22×C4×C47—81no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C2×Dic94—22no—
59C4⋊C4×C47—24no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—44no—
62C4⋊C4×C47—44no—
63C4⋊C4×C47—81no—
64C22⋊C4×C47—44no—
65C22⋊C4×C47—44no—
66C22⋊C4×C47—81no—
67C42×C47—41no—
68C22⋊Dic47—41no—
69#10625337—84no—
70C4⋊Dic47—21no—
71C4⋊Dic47—21no—
72C4⋊Dic47—22no—
73Dic47⋊C4—42no—
74C4×Dic47—41no—
75#10625329—41no—
76#10625329—41no—
77C2×C47⋊C8—42no—
78C2×C47⋊C8—41no—
79C23×C47—22no—
80C23×C47—42no—
81C23×C47—42no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47#12011yescharacteristic
85Q8×C47—44no—
86Q8×C47—82no—
87D4×C47—42no—
88D4×C47—81no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—22no—
92C2×C4×C47—24no—
93C2×C4×C47—22no—
94C2×C4×C47—42no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—84no—
103C2×C4×C47C2×Q1611yescharacteristic
104C2×C4×C47#13711yescharacteristic
105C2×Dic47—41no—
106C2×Dic47—41no—
107Dic94—82no—
108C47⋊C8—81no—
109C47⋊C8—82no—
110#12220—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—21no—
113C22×C47—44no—
114C22×C47—42no—
115C22×C47—82no—
116C22×C47—81no—
117C22×C47—81no—
118C22×C47#45111yescharacteristic
119C22×C47#45511yescharacteristic
120C22×C47B(4,𝔽2)11yescharacteristic
121C4×C47—21no—
122C4×C47—21no—
123C4×C47—42no—
124C4×C47—42no—
125C4×C47—44no—
126C4×C47—82no—
127Dic47—81no—
128Dic47—81no—
129#2683D4711yescharacteristic
130#2027—471no—
131#1605—472no—
132#1354—472no—
133#1318—471no—
134C2×C47—21no—
135C2×C47—21no—
136C2×C47—82no—
137C2×C47#160211yescharacteristic
138C2×C47#219611yescharacteristic
139C2×C47#220311yescharacteristic
140#544—471no—
141C2×C22⋊Q8—22no—
142#451—942no—
143#418—941no—
144#396—22no—
145#395C2×D4712yes—
146#390C2×D4711yescharacteristic
147#389—21no—
148#386—21no—
149#343—471no—
150#339—941no—
151#328—942no—
152#324—472no—
153C47#1222011yescharacteristic, complemented, Sylow
154C22×Q8C22⋊D4712yes—
155C22×D4C22⋊D4711yescharacteristic
156C23×C4—21no—
157C2×Q16—1882no—
158C2×M16—941no—
159C4⋊Q8—1881no—
160#125—1881no—
161C22⋊Q8—48no—
162C22⋊Q8—942no—
163#117—941no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×D4711yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C42C22⋊D4711yescharacteristic
173#103—1882no—
174#102—1881no—
175#102—1881no—
176C22⋊C8—942no—
177#95—22no—
178#95C22⋊D4712yes—
179C24—21no—
180C2×Q8—44no—
181C2×Q8—42no—
182C2×Q8—942no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#1062533712yescomplemented
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—81no—
192C22×C4C2×C22⋊D4711yescharacteristic
193C22×C4C2×C22⋊D4712yes—
194Q16—3764no—
195M16—1881no—
196M16—1881no—
197C2×C8—1882no—
198C2×C8—1881no—
199C4⋊C4—24no—
200C4⋊C4—41no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—941no—
205C4⋊C4—941no—
206C4⋊C4—942no—
207C4⋊C4—1882no—
208C22⋊C4—44no—
209C22⋊C4—44no—
210C22⋊C4—81no—
211C22⋊C4—1881no—
212C42—41no—
213C42—1881no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1309399311yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—3762no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—84no—
239C2×C4—1881no—
240C2×C4—1881no—
241C2×C4C2×#1062533711yescharacteristic
242C2×C4#1309405511yescharacteristic
243C8—3761no—
244C8—3762no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1568927911yescharacteristic
253C22#1568930111yescharacteristic
254C22#1568931011yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—44no—
260C4—82no—
261C4—3761no—
262C4—3761no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2024234511yescharacteristic
267C2#2024537211yescharacteristic
268C2#2024540911yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2683C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
hypercenter#2683D47
hypocenterC47#12220
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2683

Complements

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

NH ≅ G/Ntype
2. #20242361C2semidirect
3. #20241418C2semidirect
50. C2×D4×C47Q16semidirect
153. C47#12220semidirect
186. C2×D4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202445442
#202423612
#202414182
#202413202
C47×#26832
#1222047