Finite Group Databasebeta

← #26112306

Subgroups — #26112306

Order 12032.

478 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 115 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#20249154C211yesmaximal, characteristic
2#20242061C212yesmaximal, complemented
3#20241702C211yesmaximal, characteristic
4#20241248C212yesmaximal, complemented
5C47×#2921C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7C2×#13094003—21no—
8#15689176C2212yes—
9#15689053—21no—
10#15689053C412yes—
11#15689048—21no—
12#15689048C412yes—
13#15689029—21no—
14#15688989C2211yescharacteristic
15#15688870—24no—
16#15688849—21no—
17#15688847—21no—
18C2×#117×C47C2211yescharacteristic
19C47×#433—24no—
20C47×#423C2212yes—
21C47×#407—21no—
22C4×M16×C47—21no—
23C22×Dic94—21no—
24#13094024—22no—
25#13094024—22no—
26#13094003—44no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47C2×C412yes—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C4⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C4×C47⋊C8—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—21no—
38C47⋊#95C2×C412yes—
39C23×C4×C47D411yescharacteristic
40C2×M16×C47—22no—
41#117×C47—22no—
42#117×C47—22no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47C2311yescharacteristic
49C4⋊C8×C47—22no—
50C4⋊C8×C47D414yes—
51C22⋊C8×C47—24no—
52#97×C47—22no—
53C4×C8×C47—22no—
54C24×C47C22⋊C411yescharacteristic
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×Dic47—22no—
66C22×Dic47—22no—
67C2×Dic94—41no—
68C2×Dic94—41no—
69C2×Dic94—42no—
70C2×Dic94—42no—
71M16×C47—42no—
72M16×C47—42no—
73C2×C8×C47—24no—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—22no—
76C22⋊C4×C47—22no—
77C22⋊C4×C47—22no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47C2×D412yes—
81C42×C47#4112yes—
82C22⋊Dic47—44no—
83C4⋊Dic47—24no—
84C4⋊Dic47—42no—
85C4⋊Dic47—42no—
86Dic47⋊C4—44no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47C4×D412yes—
106C2×C4×C47#12011yescharacteristic
107C2×C4×C47#12112yes—
108C2×C4×C47#12311yescharacteristic
109C2×C4×C47#13512yes—
110C2×C4×C47AGL(1,ℤ/8)12yes—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117Dic94—82no—
118Dic94—82no—
119C8×C47—44no—
120C47⋊C8—82no—
121#9196—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—81no—
129C22×C47#38611yescharacteristic
130C22×C47#41611yescharacteristic
131C22×C47#41811yescharacteristic
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47#44912yes—
138C4×C47#48212yes—
139Dic47—82no—
140Dic47—82no—
141#2921D4711yescharacteristic
142#2621—471no—
143#1546—472no—
144#1458—471no—
145#1286—472no—
146C2×C47—21no—
147C2×C47—21no—
148C2×C47—81no—
149C2×C47#187511yescharacteristic
150C2×C47#190911yescharacteristic
151C2×C47#191511yescharacteristic
152C2×C4⋊Q8—471no—
153C2×C22⋊Q8—941no—
154C2×#117C2×D4711yescharacteristic
155#433—24no—
156#431—472no—
157#423C2×D4712yes—
158#407—21no—
159C4×M16—21no—
160#398—472no—
161#398—941no—
162#397—472no—
163#397—941no—
164#391—941no—
165#382—471no—
166#332—944no—
167#324—941no—
168#323—941no—
169C47#919611yescharacteristic, complemented, Sylow
170C22×Q8—941no—
171C23×C4C22⋊D4711yescharacteristic
172C2×M16—22no—
173C4⋊Q8—942no—
174C4⋊Q8—942no—
175C22⋊Q8—1884no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—472no—
179C2×C4⋊C4—942no—
180C2×C4⋊C4—941no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4C4×D4711yescharacteristic
183C2×C4⋊C4D18811yescharacteristic
184C2×C22⋊C4—942no—
185C2×C22⋊C4C4×D4711yescharacteristic
186C2×C22⋊C4D18811yescharacteristic
187C2×C42C22⋊D4711yescharacteristic
188C2×C42C22×D4711yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—942no—
191C4⋊C8C22⋊D4714yes—
192C22⋊C8—24no—
193C22⋊C8—942no—
194#97—22no—
195C4×C8—22no—
196C4×C8—942no—
197#95—472no—
198#95—942no—
199#95—942no—
200#95—941no—
201C24(C2×D47)⋊C411yescharacteristic
202C2×Q8—1881no—
203C2×Q8—1881no—
204C2×Q8—1882no—
205C2×Q8—1882no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—942no—
211C22×C4—942no—
212C22×C4(C2×D47)⋊C411yescharacteristic
213C22×C4(C2×D47)⋊C411yescharacteristic
214C22×C4(C2×D47)⋊C411yescharacteristic
215C22×C4C2×C4×D4711yescharacteristic
216C22×C4C2×D18811yescharacteristic
217C22×C4C2×C22⋊D4711yescharacteristic
218M16—42no—
219M16—42no—
220C2×C8—24no—
221C2×C8—1882no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—944no—
225C4⋊C4—1884no—
226C4⋊C4—1882no—
227C4⋊C4—1882no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—1884no—
231C42—21no—
232C42—21no—
233C42#1062535012yes—
234C42C2×C22⋊D4712yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×(C2×D47)⋊C411yescharacteristic
241Q8—3762no—
242Q8—3762no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1882no—
260C2×C4C4×C22⋊D4712yes—
261C2×C4#1309399311yescharacteristic
262C2×C4#1309399812yes—
263C2×C4#1309400711yescharacteristic
264C2×C4#1309404412yes—
265C2×C4#1309404912yes—
266C8—44no—
267C8—3762no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1568900211yescharacteristic
276C22#1568911711yescharacteristic
277C22#1568912711yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—3762no—
284C4—3762no—
285C4#1568926812yes—
286C4#1568944712yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2024371011yescharacteristic
291C2#2024391011yescharacteristic
292C2#2024394011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2921C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2921D47
hypocenterC47#9196
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20242061C2semidirect
4. #20241248C2semidirect
169. C47#9196semidirect

Maximal subgroups

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

subgroupindex
#202491542
#202420612
#202417022
#202412482
C47×#29212
#919647