Finite Group Databasebeta

← #26114253

Subgroups — #26114253

Order 12032.

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

normal subgroups only

This group has 522 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#20252429C211yesmaximal, characteristic
2#20243468C212yesmaximal, complemented
3#20241726C211yesmaximal, characteristic
4#20241294C212yesmaximal, complemented
5C47×#2487C211yesmaximal, characteristic
6#15689201—24no—
7#15689176—21no—
8#15689087—21no—
9#15689087C2212yes—
10#15689087C2212yes—
11#15688892—24no—
12#15688849—21no—
13#15688847—21no—
14C2×C4×D4×C47C2211yescharacteristic
15C22×C4⋊C4×C47C2211yescharacteristic
16C47×#400—21no—
17C47×#399—21no—
18C47×#389—21no—
19C47×#386C412yes—
20C47×#385C412yes—
21C47×#380—21no—
22C47×#380C2211yescharacteristic
23C2×#10625329—22no—
24C2×#10625329—22no—
25C2×#10625329D414yes—
26C4⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C4×C47⋊C8—22no—
33C22×D4×C47C2×C411yescharacteristic
34C23×C4×C47C2×C412yes—
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C4×D4×C47—22no—
38C4×D4×C47—22no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—22no—
44C2×C4⋊C4×C47—22no—
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C411yescharacteristic
48C2×C22⋊C4×C47C2×C412yes—
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51#95×C47—22no—
52#95×C47—22no—
53#95×C47—21no—
54C24×C47C22⋊C411yescharacteristic
55C24×C47C22×C411yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—22no—
58C2×D4×C47—41no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—24no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47—82no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C2×D412yes—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4112yes—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—82no—
80C22⋊C4×C47—22no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—42no—
83C22⋊C4×C47—82no—
84C42×C47—21no—
85C42×C47—21no—
86#10625329—44no—
87#10625329—44no—
88#10625329—82no—
89C2×C47⋊C8—24no—
90C2×C47⋊C8—42no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47C2×C22⋊C411yescharacteristic
101C23×C47C4×D412yes—
102D4×C47—44no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—24no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—81no—
121C2×C4×C47—84no—
122C2×C4×C47#10112yes—
123C2×C4×C47#12011yescharacteristic
124C2×C4×C47#12112yes—
125C2×C4×C47#12311yescharacteristic
126C47⋊C8—44no—
127C47⋊C8—82no—
128#9510—471nomaximal, Sylow
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—22no—
132C22×C47—22no—
133C22×C47—22no—
134C22×C47—42no—
135C22×C47—42no—
136C22×C47—42no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—81no—
141C22×C47#38611yescharacteristic
142C22×C47C2×#10111yescharacteristic
143C22×C47#41311yescharacteristic
144C4×C47—22no—
145C4×C47—21no—
146C4×C47—22no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—82no—
150C4×C47—82no—
151C4×C47#45812yes—
152#2975—471no—
153#2487D4711yescharacteristic
154#1821—472no—
155#1465—471no—
156#1305—472no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47—42no—
160C2×C47—81no—
161C2×C47#189011yescharacteristic
162C2×C47#190111yescharacteristic
163C2×C47#191411yescharacteristic
164C2×C4×D4C2×D4711yescharacteristic
165C22×C4⋊C4C2×D4711yescharacteristic
166#436—944no—
167#431—941no—
168#407—472no—
169#407—472no—
170#407—941no—
171#400—21no—
172#399—21no—
173#389—21no—
174#386Dic4712yes—
175#385Dic4712yes—
176#380—21no—
177#380C2×D4711yescharacteristic
178#343—944no—
179#324—941no—
180#323—941no—
181C47#951011yescharacteristic, complemented, Sylow
182C22×D4C2×Dic4711yescharacteristic
183C23×C4C2×Dic4712yes—
184C23×C4C22⋊D4711yescharacteristic
185C23×C4C22×D4711yescharacteristic
186C2×M16—474no—
187C2×M16—942no—
188C2×M16—942no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic4711yescharacteristic
200C2×C22⋊C4C2×Dic4712yes—
201C2×C22⋊C4C22⋊D4711yescharacteristic
202C2×C42C22⋊D4711yescharacteristic
203C4⋊C8—942no—
204C22⋊C8—942no—
205C22⋊C8—942no—
206C22⋊C8—942no—
207C22⋊C8—942no—
208C22⋊C8—942no—
209C4×C8—942no—
210#95—22no—
211#95—22no—
212#95—21no—
213C24C22⋊Dic4711yescharacteristic
214C24C22×Dic4711yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4C22⋊Dic4711yescharacteristic
227C22×C4C22⋊Dic4711yescharacteristic
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4D4×D4712yes—
230C22×C4#1062535212yes—
231C22×C4C2×C22⋊D4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—3762no—
234M16—1884no—
235M16—1884no—
236C2×C8—944no—
237C2×C8—1882no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C2×C22⋊Dic4711yescharacteristic
260C23D4×Dic4712yes—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—84no—
281C2×C4#1309394312yes—
282C2×C4#1309399211yescharacteristic
283C2×C4#1309399612yes—
284C2×C4#1309400511yescharacteristic
285C8—3762no—
286C8—1884no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1568900311yescharacteristic
300C22C2×#1309394311yescharacteristic
301C22#1568910711yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4#1568931212yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2024380211yescharacteristic
315C2#2024385711yescharacteristic
316C2#2024393611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2487C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#2487D47
hypocenterC47#9510
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #20243468C2semidirect
4. #20241294C2semidirect
181. C47#9510semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202434682
#202417262
#202412942
C47×#24872
#951047