Finite Group Databasebeta

← #26111332

Subgroups — #26111332

Order 12032.

616 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#15689053C211yesmaximal, characteristic
2#20243472C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241266C212yesmaximal
5C47×#2313C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×C22⋊(C47⋊C8)C2212yes—
9#15689053—22no—
10#15689053C414yescomplemented
11C2×C47⋊#95—21no—
12C2×C47⋊#95C2211yescharacteristic
13#15688885—24no—
14#15688849—22no—
15C22×C4⋊C4×C47C2211yescharacteristic
16C47×#387—24no—
17C47×#381—21no—
18C47×#380C2212yes—
19C4×C22⋊C4×C47—21no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—24no—
28C2×Dic47⋊C4—44no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C47⋊#95—24no—
32C47⋊#95—21no—
33C47⋊#95C2×C412yes—
34C23×C4×C47D411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47C2×C412yes—
40C2×C4⋊C4×C47D412yes—
41C2×C4⋊C4×C47D414yes—
42C2×C22⋊C4×C47—22no—
43C2×C42×C47—22no—
44#95×C47—22no—
45#95×C47—24no—
46#95×C47—22no—
47C24×C47C2×D411yescharacteristic
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—21no—
54C22×C4×C47—24no—
55C22×C4×C47—42no—
56C22×C4×C47—42no—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D412yes—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47#4112yes—
63C22×Dic47—24no—
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—42no—
67C22×Dic47—42no—
68C22×Dic47—41no—
69C22×Dic47—41no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72C4⋊C4×C47—24no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—44no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—42no—
77C42×C47—82no—
78C4⋊Dic47—24no—
79C4⋊Dic47—42no—
80Dic47⋊C4—48no—
81C2×C47⋊C8—44no—
82C2×C47⋊C8—42no—
83C23×C47—22no—
84C23×C47—22no—
85C23×C47C2×C22⋊C411yescharacteristic
86C23×C47#12011yescharacteristic
87C23×C47#12311yescharacteristic
88C23×C47#13512yes—
89C23×C47AGL(1,ℤ/8)12yes—
90C2×C4×C47—24no—
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
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—44no—
103C2×C4×C47—44no—
104C2×C4×C47—44no—
105C2×C4×C47—81no—
106C2×C4×C47C4×D412yes—
107C2×C4×C47#12112yes—
108C2×Dic47—24no—
109C2×Dic47—48no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—82no—
116C2×Dic47—82no—
117C47⋊C8—82no—
118#9026—471nomaximal, Sylow
119C22×C47—24no—
120C22×C47—24no—
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—21no—
125C22×C47#38611yescharacteristic
126C22×C47#41611yescharacteristic
127C22×C47#41811yescharacteristic
128C22×C47#44912yes—
129C22×C47#45612yes—
130C22×C47#45812yes—
131C22×C47#48212yes—
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—42no—
135C4×C47—42no—
136C4×C47—84no—
137Dic47—44no—
138Dic47—82no—
139C2×#398—471no—
140#2313D4711yescharacteristic
141#1823—472no—
142C2×#324—471no—
143#1294—472no—
144C2×C47—21no—
145C2×C47—21no—
146C2×C47—22no—
147C2×C47#160914yes—
148C2×C47#187511yescharacteristic
149C2×C47#188611yescharacteristic
150C2×C47#189011yescharacteristic
151C22×C4⋊C4—471no—
152C22×C4⋊C4—941no—
153C22×C4⋊C4C2×D4711yescharacteristic
154C2×C22⋊C8—472no—
155#398—474no—
156#398—942no—
157#387—24no—
158#381—21no—
159#380C2×D4712yes—
160C4×C22⋊C4—21no—
161C2×#95—471no—
162C2×#95—941no—
163#340—944no—
164#324—942no—
165C47#902611yescharacteristic, complemented, Sylow
166C23×C4—941no—
167C23×C4—941no—
168C23×C4C22⋊D4711yescharacteristic
169C23×C4C22⋊D4711yescharacteristic
170C23×C4C22×D4711yescharacteristic
171C22×C8—942no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—472no—
175C2×C4⋊C4—942no—
176C2×C4⋊C4—942no—
177C2×C4⋊C4—944no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4—1884no—
180C2×C4⋊C4C4×D4712yes—
181C2×C4⋊C4D18812yes—
182C2×C4⋊C4C22⋊D4714yes—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C22⋊C8—942no—
186C22⋊C8—942no—
187#95—22no—
188#95—24no—
189#95—22no—
190#95—472no—
191#95—944no—
192#95—941no—
193C24C2×C22⋊D4711yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—944no—
204C22×C4—942no—
205C22×C4—942no—
206C22×C4—1881no—
207C22×C4—1881no—
208C22×C4—1882no—
209C22×C4—1882no—
210C22×C4—1882no—
211C22×C4—1882no—
212C22×C4(C2×D47)⋊C412yes—
213C22×C4(C2×D47)⋊C412yes—
214C22×C4C2×C4×D4711yescharacteristic
215C22×C4C2×D18811yescharacteristic
216C22×C4#1062535012yes—
217C22×C4C2×C22⋊D4712yes—
218C2×C8—1884no—
219C2×C8—1882no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—44no—
223C4⋊C4—944no—
224C4⋊C4—1888no—
225C4⋊C4—1882no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—82no—
229C23—22no—
230C23—22no—
231C23C2×(C2×D47)⋊C411yescharacteristic
232C23#1309399311yescharacteristic
233C23#1309400711yescharacteristic
234C23#1309404412yes—
235C23#1309404912yes—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—81no—
252C2×C4—944no—
253C2×C4—3762no—
254C2×C4—3762no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1888no—
260C2×C4—1882no—
261C2×C4C4×C22⋊D4712yes—
262C2×C4#1309399812yes—
263C8—3762no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—24no—
270C22#1568900211yescharacteristic
271C22#1568911711yescharacteristic
272C22#1568912711yescharacteristic
273C22#1568926812yes—
274C22#1568930612yes—
275C22#1568931312yes—
276C22#1568944712yes—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—84no—
282C4—3762no—
283C4—1884no—
284C2—21no—
285C2—21no—
286C2—22no—
287C2#2024238514yes—
288C2#2024371011yescharacteristic
289C2#2024378111yescharacteristic
290C2#2024380111yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2313C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2313D47
hypocenterC47#9026
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
10. #15689053C4semidirect
165. C47#9026semidirect

Maximal subgroups

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

subgroupindex
C2×#156890532
#202434722
C2×#156888492
#202412662
C47×#23132
#902647