Finite Group Databasebeta

← #26103600

Subgroups — #26103600

Order 12032.

374 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 71 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#20241718C211yesmaximal, characteristic
2#20241656C211yesmaximal, characteristic
3C47×#1463C211yesmaximal, characteristic
4#15689087—21no—
5C2×C47⋊#95—21no—
6C2×C47⋊#95—21no—
7C2×C47⋊#95C411yescharacteristic
8C2×C47⋊#95C411yescharacteristic
9#15688849—21no—
10#15688849C412yescomplemented
11C22×C4⋊C4×C47C2211yescharacteristic
12C47×#407—21no—
13C47×#324—21no—
14C47×#324C412yescomplemented
15C23×Dic47—21no—
16C23×Dic47—21no—
17C2×#10625329—41no—
18C22⋊(C47⋊C8)—22no—
19C22⋊(C47⋊C8)—41no—
20C47⋊#95—42no—
21C47⋊#95—42no—
22C47⋊#95—41no—
23C47⋊#95—41no—
24C23×C4×C47C2×C411yescharacteristic
25C23×C4×C47D411yescharacteristic
26C23×C4×C47D411yescharacteristic
27C2×M16×C47—41no—
28C2×C4⋊C4×C47—41no—
29C2×C4⋊C4×C47—41no—
30C2×C4⋊C4×C47C2×C411yescharacteristic
31C2×C4⋊C4×C47C2×C411yescharacteristic
32C2×C4⋊C4×C47D411yescharacteristic
33C2×C4⋊C4×C47Q811yescharacteristic
34C22⋊C8×C47—22no—
35C22⋊C8×C47—41no—
36C24×C47C22⋊C411yescharacteristic
37C22×C4×C47—21no—
38C22×C4×C47—21no—
39C22×C4×C47—21no—
40C22×C4×C47—21no—
41C22×C4×C47—21no—
42C22×C4×C47—21no—
43C22×C4×C47—41no—
44C22×C4×C47—41no—
45C22×C4×C47—41no—
46C22×C4×C47—41no—
47C22×C4×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47C4211yescharacteristic
50C22×C4×C47C22⋊C411yescharacteristic
51C22×C4×C47C22⋊C411yescharacteristic
52C22×C4×C47C4⋊C411yescharacteristic
53C22×C4×C47C4⋊C411yescharacteristic
54C22×C4×C47C4⋊C411yescharacteristic
55C22×Dic47—22no—
56C22×Dic47—22no—
57C22×Dic47—41no—
58C22×Dic47—41no—
59C22×Dic47—41no—
60C22×Dic47—41no—
61C22×Dic47—41no—
62C22×Dic47—41no—
63C22×Dic47—41no—
64C22×Dic47—41no—
65C22×Dic47—41no—
66C22×Dic47—41no—
67C22×Dic47—41no—
68C22×Dic47—41no—
69M16×C47—81no—
70M16×C47—81no—
71C2×C8×C47—42no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76#10625329—81no—
77#10625329—81no—
78C2×C47⋊C8—42no—
79C23×C47—21no—
80C23×C47—21no—
81C23×C47—21no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47#9511yescharacteristic
86C23×C47#9911yescharacteristic
87C23×C47#9911yescharacteristic
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—81no—
109C2×C4×C47—81no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—81no—
117C2×Dic47—81no—
118C2×Dic47—81no—
119C2×Dic47—81no—
120C2×Dic47—81no—
121C2×Dic47—81no—
122C2×Dic47—81no—
123C2×Dic47—81no—
124C8×C47—82no—
125C47⋊C8—82no—
126#7326—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47#34211yescharacteristic
142C22×C47#34411yescharacteristic
143C22×C47#34411yescharacteristic
144C22×C47#35311yescharacteristic
145C22×C47#35411yescharacteristic
146C22×C47#35511yescharacteristic
147C22×C47#35611yescharacteristic
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154Dic47—82no—
155Dic47—82no—
156#1463—471no—
157#1463D4711yescharacteristic
158#1445—471no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#130611yescharacteristic
166C2×C47#139311yescharacteristic
167C2×C47#139411yescharacteristic
168C22×C4⋊C4C2×D4711yescharacteristic
169#407—21no—
170#407—941no—
171C2×#95—471no—
172C2×#95—471no—
173C2×#95—941no—
174C2×#95—941no—
175#324—21no—
176#324—472no—
177#324—941no—
178#324Dic4712yescomplemented
179C47#732611yescharacteristic, complemented, Sylow
180C23×C4—941no—
181C23×C4—941no—
182C23×C4C4×D4711yescharacteristic
183C23×C4D18811yescharacteristic
184C23×C4C22⋊D4711yescharacteristic
185C2×M16—41no—
186C2×M16—1881no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4Dic9411yescharacteristic
190C2×C4⋊C4C4×D4711yescharacteristic
191C2×C4⋊C4C2×Dic4711yescharacteristic
192C2×C4⋊C4C22⋊D4711yescharacteristic
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—942no—
196C22⋊C8—1881no—
197#95—1881no—
198#95—1881no—
199#95—1882no—
200#95—1882no—
201C24(C2×D47)⋊C411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—942no—
215C22×C4—942no—
216C22×C4—1881no—
217C22×C4—1881no—
218C22×C4—1881no—
219C22×C4—1881no—
220C22×C4—1881no—
221C22×C4—1881no—
222C22×C4—1881no—
223C22×C4—1881no—
224C22×C4—1881no—
225C22×C4—1881no—
226C22×C4—1881no—
227C22×C4—1881no—
228C22×C4C4×Dic4711yescharacteristic
229C22×C4Dic47⋊C411yescharacteristic
230C22×C4Dic47⋊C411yescharacteristic
231C22×C4C4⋊Dic4711yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4C22⋊Dic4711yescharacteristic
234M16—81no—
235M16—81no—
236M16—3761no—
237M16—3761no—
238C2×C8—42no—
239C2×C8—1882no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23C47⋊#9511yescharacteristic
251C23#1309393811yescharacteristic
252C23#1309393811yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—3761no—
276C2×C4—3761no—
277C2×C4—3761no—
278C2×C4—3761no—
279C2×C4—3761no—
280C2×C4—3761no—
281C2×C4—3761no—
282C2×C4—3761no—
283C2×C4—1882no—
284C2×C4—1882no—
285C2×C4—1882no—
286C2×C4—1882no—
287C2×C4—1882no—
288C2×C4—1882no—
289C8—82no—
290C8—3762no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1568888911yescharacteristic
306C22#1568889411yescharacteristic
307C22#1568889511yescharacteristic
308C22#1568891511yescharacteristic
309C22#1568891711yescharacteristic
310C22#1568891911yescharacteristic
311C22#1568892111yescharacteristic
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3762no—
319C4—3762no—
320C2—21no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2#2024129711yescharacteristic
327C2#2024151211yescharacteristic
328C2#2024151411yescharacteristic
subgroupG/H
centerC22#15688889
commutatorC22×C4×C47C42
FrattiniC22×C4⋊C4C2×D47
FittingC47×#1463C2
socleC22×C47#342
prime radicalC24×C47C22⋊C4
cosocleC22×C4⋊C4×C47C22
hypercenter#1463D47
hypocenterC47#7326
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
10. #15688849C4semidirect
14. C47×#324C4semidirect
178. #324Dic47semidirect
179. C47#7326semidirect

Maximal subgroups

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

subgroupindex
#202417182
#202416562
C47×#14632
#732647