Finite Group Databasebeta

← #26127184

Subgroups — #26127184

Order 12032.

376 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 79 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#20241702C211yesmaximal, characteristic
3#20241417C212yesmaximal, complemented
4#20241393C212yesmaximal, complemented
5C47×#1822C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic
7#15689176—21no—
8#15689053C2211yescharacteristic
9#15689048C411yescharacteristic
10#15689043C2211yescharacteristic
11#15689035C411yescharacteristic
12#15689011—21no—
13#15689007—21no—
14#15689004—21no—
15#15688996—21no—
16#15688991C411yescharacteristic
17#15688849C2212yes—
18#15688847—21no—
19C2×#117×C47C2211yescharacteristic
20C47×#407—21no—
21C47×#343—21no—
22C47×#343C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C4×Dic47—21no—
33C4⋊(C47⋊C8)—41no—
34C22⋊(C47⋊C8)—22no—
35C4×C47⋊C8—41no—
36C47⋊#95—21no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C2×M16×C47—22no—
46#117×C47—41no—
47#117×C47—41no—
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47C2311yescharacteristic
50C2×C22⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C22⋊C8×C47—41no—
55C22⋊C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—41no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C2×Dic94—41no—
72C2×Dic94—42no—
73M16×C47—42no—
74M16×C47—42no—
75C2×C8×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—41no—
87Dic47⋊C4—42no—
88Dic47⋊C4—41no—
89Dic47⋊C4—41no—
90Dic47⋊C4—42no—
91C4×Dic47—42no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47C2×C22⋊C411yescharacteristic
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47#9911yescharacteristic
114C2×C4×C47#9911yescharacteristic
115C2×C4×C47#13512yes—
116C2×Dic47—22no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128Dic94—42no—
129Dic94—42no—
130C8×C47—82no—
131C47⋊C8—82no—
132#11849—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—42no—
136C22×C47—41no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—81no—
140C22×C47C2×#9911yescharacteristic
141C22×C47#41611yescharacteristic
142C22×C47#42111yescharacteristic
143C4×C47—22no—
144C4×C47—21no—
145C4×C47—21no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150Dic47—42no—
151Dic47—81no—
152Dic47—81no—
153Dic47—81no—
154#2599—471no—
155#1822D4711yescharacteristic
156#1458—471no—
157#1354—472no—
158#1346—472no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47—81no—
162C2×C47#151611yescharacteristic
163C2×C47#212011yescharacteristic
164C2×C47#213511yescharacteristic
165C2×C4×Q8—471no—
166C2×#117C2×D4711yescharacteristic
167#431—941no—
168#407—21no—
169#398—471no—
170#397—471no—
171#395—471no—
172#393—471no—
173#388—941no—
174#387—941no—
175#387—941no—
176#385—941no—
177#383—471no—
178#343—21no—
179#343C2×D4712yes—
180#324—472no—
181#323—941no—
182C47#1184911yescharacteristic, complemented, Sylow
183C22×Q8—471no—
184C23×C4C22⋊D4711yescharacteristic
185C2×M16—22no—
186C4×Q8—1881no—
187C4×Q8—1881no—
188#117—41no—
189#117—41no—
190C2×C4⋊C4—471no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4C22⋊D4711yescharacteristic
195C2×C4⋊C4C22×D4711yescharacteristic
196C2×C22⋊C4—941no—
197C2×C22⋊C4—941no—
198C2×C22⋊C4C4×D4711yescharacteristic
199C2×C22⋊C4D18811yescharacteristic
200C2×C42—941no—
201C2×C42C4×D4711yescharacteristic
202C2×C42D18811yescharacteristic
203C4⋊C8—1881no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—942no—
207C4×C8—1881no—
208#95—471no—
209#95—471no—
210#95—941no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216C24(C2×D47)⋊C411yescharacteristic
217C2×Q8—1881no—
218C2×Q8—1882no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4—941no—
226C22×C4—941no—
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4C2×C4×D4711yescharacteristic
231C22×C4C2×D18811yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—1882no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—1881no—
240C4⋊C4—1882no—
241C4⋊C4—1881no—
242C4⋊C4—1881no—
243C4⋊C4—1882no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—1882no—
247C22⋊C4—1882no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—1882no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×(C2×D47)⋊C411yescharacteristic
259Q8—1882no—
260Q8—1882no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—942no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1881no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4#1309393811yescharacteristic
288C2×C4#1309393811yescharacteristic
289C2×C4#1309404412yes—
290C8—82no—
291C8—3762no—
292C22—21no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22C2×#1309393811yescharacteristic
300C22#1568911711yescharacteristic
301C22#1568914111yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—3761no—
310C4—3761no—
311C4—3761no—
312C4—1882no—
313C2—21no—
314C2—21no—
315C2—81no—
316C2#2024194211yescharacteristic
317C2#2024503011yescharacteristic
318C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1822C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1822D47
hypocenterC47#11849
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #20241417C2semidirect
4. #20241393C2semidirect
182. C47#11849semidirect

Maximal subgroups

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

subgroupindex
#202489652
#202417022
#202414172
#202413932
C47×#18222
#1184947