Finite Group Databasebeta

← #26114167

Subgroups — #26114167

Order 12032.

458 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 99 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#20248965C211yesmaximal, characteristic
2#20241836C212yesmaximal, complemented
3#20241722C211yesmaximal, characteristic
4#20241253C212yesmaximal, complemented
5C47×#2921C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689180C2212yes—
8#15689053C411yescharacteristic
9#15689048C411yescharacteristic
10#15689043C411yescharacteristic
11#15689035C411yescharacteristic
12#15689011—21no—
13#15689007—21no—
14#15689004—21no—
15#15688996—21no—
16#15688991C2211yescharacteristic
17#15688855—24no—
18#15688849—21no—
19#15688847—21no—
20C2×#117×C47C2211yescharacteristic
21C47×#433—24no—
22C47×#423—21no—
23C47×#407—21no—
24C4×M16×C47C2212yes—
25C22×Dic94C2×C411yescharacteristic
26C4×Dic94—22no—
27C4×Dic94—22no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C4×Dic47—21no—
35C4⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37#13093935—22no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—22no—
48#117×C47—22no—
49#117×C47—22no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—24no—
57C22⋊C8×C47—24no—
58#97×C47—22no—
59C4×C8×C47D414yes—
60C24×C47C22⋊C411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C2×Dic94—22no—
76C2×Dic94—42no—
77M16×C47—42no—
78M16×C47—42no—
79C2×C8×C47—24no—
80C4⋊C4×C47—22no—
81C4⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C42×C47—21no—
85C42×C47—21no—
86C42×C47C2×D412yes—
87C42×C47#4112yes—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C4⋊Dic47—22no—
91Dic47⋊C4—42no—
92Dic47⋊C4—42no—
93Dic47⋊C4—41no—
94Dic47⋊C4—41no—
95C4×Dic47—42no—
96C2×C47⋊C8—42no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47C2×C22⋊C411yescharacteristic
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12112yes—
119C2×C4×C47#12311yescharacteristic
120C2×Dic47—22no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132Dic94—24no—
133Dic94—42no—
134C8×C47—44no—
135C47⋊C8—82no—
136#9493—471nomaximal, Sylow
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—42no—
140C22×C47—41no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—81no—
144C22×C47#38611yescharacteristic
145C22×C47#42111yescharacteristic
146C22×C47#42111yescharacteristic
147C4×C47—22no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—22no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154Dic47—42no—
155Dic47—81no—
156Dic47—81no—
157Dic47—81no—
158#2921D4711yescharacteristic
159#2599—471no—
160#1491—472no—
161#1464—471no—
162#1288—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—81no—
166C2×C47#189111yescharacteristic
167C2×C47#189711yescharacteristic
168C2×C47#190911yescharacteristic
169C2×C4×Q8—471no—
170C2×#117C2×D4711yescharacteristic
171#433—24no—
172#432—472no—
173#423—21no—
174#407—21no—
175C4×M16C2×D4712yes—
176#398—471no—
177#397—471no—
178#395—471no—
179#393—471no—
180#388—941no—
181#387—941no—
182#387—941no—
183#385—941no—
184#383—471no—
185Q8⋊C8—944no—
186#324—941no—
187#323—941no—
188C47#949311yescharacteristic, complemented, Sylow
189C22×Q8—471no—
190C23×C4C22⋊D4711yescharacteristic
191C2×M16—22no—
192C4×Q8—942no—
193C4×Q8—942no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—471no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4—941no—
200C2×C4⋊C4C4×D4711yescharacteristic
201C2×C4⋊C4D18811yescharacteristic
202C2×C22⋊C4—941no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4C4×D4711yescharacteristic
205C2×C22⋊C4D18811yescharacteristic
206C2×C42—941no—
207C2×C42C22⋊D4711yescharacteristic
208C2×C42C22×D4711yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—942no—
211C22⋊C8—24no—
212C22⋊C8—942no—
213#97—22no—
214#97—942no—
215C4×C8C22⋊D4714yes—
216#95—471no—
217#95—471no—
218#95—941no—
219#95—941no—
220#95—941no—
221#95—941no—
222#95—941no—
223#95—941no—
224C24(C2×D47)⋊C411yescharacteristic
225C2×Q8—942no—
226C2×Q8—1882no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—941no—
232C22×C4—941no—
233C22×C4—941no—
234C22×C4—941no—
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4(C2×D47)⋊C411yescharacteristic
238C22×C4C2×C4×D4711yescharacteristic
239C22×C4C2×D18811yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—24no—
244C2×C8—1882no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—942no—
248C4⋊C4—1882no—
249C4⋊C4—1882no—
250C4⋊C4—1881no—
251C4⋊C4—1881no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—1882no—
255C22⋊C4—1882no—
256C42—21no—
257C42—21no—
258C42—1882no—
259C42#1062535012yes—
260C42C2×C22⋊D4712yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×(C2×D47)⋊C411yescharacteristic
267Q8—944no—
268Q8—1882no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—942no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4C4×C22⋊D4712yes—
295C2×C4#1309399311yescharacteristic
296C2×C4#1309399812yes—
297C2×C4#1309400711yescharacteristic
298C8—44no—
299C8—3762no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1568900211yescharacteristic
308C22#1568914111yescharacteristic
309C22#1568914111yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—3761no—
318C4—3761no—
319C4—3761no—
320C4—1882no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2024380611yescharacteristic
325C2#2024383611yescharacteristic
326C2#2024391011yescharacteristic
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#9493
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. #20241836C2semidirect
4. #20241253C2semidirect
188. C47#9493semidirect

Maximal subgroups

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

subgroupindex
#202489652
#202418362
#202417222
#202412532
C47×#29212
#949347