Finite Group Databasebeta

← #26112304

Subgroups — #26112304

Order 12032.

478 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 111 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#20252026C211yesmaximal, characteristic
2#20249154C211yesmaximal, characteristic
3#20242059C212yesmaximal, complemented
4#20241247C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7C2×#13094003—21no—
8#15689184—24no—
9#15689147C2212yes—
10#15689087—21no—
11C4×#10625329—21no—
12#15689053—21no—
13#15689053C412yes—
14#15689048—21no—
15#15689048C412yes—
16#15689029—21no—
17#15688989C2211yescharacteristic
18#15688868—24no—
19C2×#117×C47C2211yescharacteristic
20C47×#431C2212yes—
21C47×#324—21no—
22C47×#323—21no—
23C22×Dic94—21no—
24C2×#10625329—22no—
25#13094024—22no—
26#13094024—22no—
27#13094003—44no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C4⋊(C47⋊C8)—22no—
34C4⋊(C47⋊C8)D414yes—
35C22⋊(C47⋊C8)—24no—
36#13093935—22no—
37C4×C47⋊C8—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C47⋊#95—21no—
41C47⋊#95C2×C412yes—
42C23×C4×C47D411yescharacteristic
43#117×C47—22no—
44#117×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47C2311yescharacteristic
51C4⋊C8×C47—22no—
52C22⋊C8×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—
71C2×C8×C47—42no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—22no—
74C22⋊C4×C47—22no—
75C22⋊C4×C47—22no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47C2×D412yes—
79C42×C47#4112yes—
80C22⋊Dic47—44no—
81C4⋊Dic47—24no—
82C4⋊Dic47—42no—
83C4⋊Dic47—42no—
84Dic47⋊C4—44no—
85#10625329—42no—
86#10625329—42no—
87C2×C47⋊C8—24no—
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—82no—
120C47⋊C8—44no—
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#2921—471no—
142#2621—471no—
143#1546—472no—
144#1458D4711yescharacteristic
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—944no—
156#431C2×D4712yes—
157#423—472no—
158#407—941no—
159C4×M16—941no—
160#398—472no—
161#398—941no—
162#397—472no—
163#397—941no—
164#391—941no—
165#382—471no—
166#332—944no—
167#324—21no—
168#323—21no—
169C47#919611yescharacteristic, complemented, Sylow
170C22×Q8—941no—
171C23×C4D18811yescharacteristic
172C2×M16—942no—
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⋊C4C22⋊D4711yescharacteristic
184C2×C22⋊C4—942no—
185C2×C22⋊C4C4×D4711yescharacteristic
186C2×C22⋊C4C22⋊D4711yescharacteristic
187C2×C42D18811yescharacteristic
188C2×C42C22×D4711yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—474no—
191C4⋊C8—942no—
192C22⋊C8—22no—
193C22⋊C8—944no—
194#97—942no—
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—1882no—
219M16—1882no—
220C2×C8—42no—
221C2×C8—944no—
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—
233C42D4×D4712yes—
234C42#1062535212yes—
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×C4(C22⋊D47)⋊C412yes—
261C2×C4#1309399411yescharacteristic
262C2×C4#1309399612yes—
263C2×C4#1309400811yescharacteristic
264C2×C4#1309404612yes—
265C2×C4#1309405212yes—
266C8—82no—
267C8—1884no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1568900011yescharacteristic
276C22#1568912011yescharacteristic
277C22#1568912911yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—3762no—
284C4—3762no—
285C4#1568927012yes—
286C4#1568944812yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2024370811yescharacteristic
291C2#2024390811yescharacteristic
292C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#9196
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20242059C2semidirect
4. #20241247C2semidirect
169. C47#9196semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202491542
#202420592
#202412472
C47×#14582
#919647