Finite Group Databasebeta

← #26108462

Subgroups — #26108462

Order 12032.

1002 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 139 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#20243306C211yesmaximal, characteristic, complemented
2#20243300C211yesmaximal, characteristic, complemented
3#20241944C214yesmaximal, complemented
4C47×#3487C211yesmaximal, characteristic
5C2×#13093939—22no—
6C2×C22⋊(C47⋊C8)—22no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688986C2212yes—
9#15688860—24no—
10#15688857—24no—
11C2×2−1+4×C47C411yescharacteristic, complemented
12C22×#41×C47C2211yescharacteristic
13C47×#550—24no—
14C47×#548—24no—
15C47×#535—22no—
16C47×#535C412yescomplemented
17C47×#525—22no—
18C47×#525C2212yes—
19C47×#519C411yescharacteristic, complemented
20C22×C47⋊C8—22no—
21C2×C22⋊Dic47—22no—
22#13093939—44no—
23C22⋊(C47⋊C8)—24no—
24C22⋊(C47⋊C8)—44no—
25C47⋊#95—24no—
262−1+4×C47—44no—
27C2×#41×C47—22no—
28C2×#41×C47—22no—
29C2×#41×C47—24no—
30C2×#41×C47—44no—
31C2×#41×C47C2×C412yes—
32C22×Q8×C47—22no—
33C22×Q8×C47C2×C411yescharacteristic
34C22×D4×C47C2×C411yescharacteristic
35C22×D4×C47D412yes—
36C23×C4×C47D412yes—
37C23×C4×C47C2311yescharacteristic
38#127×C47—24no—
39#127×C47—22no—
40#124×C47—24no—
41#124×C47—22no—
42#123×C47—22no—
43C22⋊Q8×C47—24no—
44C22⋊Q8×C47D414yes—
45#121×C47—24no—
46#121×C47—28no—
47#121×C47—24no—
48#121×C47D414yes—
49#120×C47—24no—
50#120×C47—22no—
51C4×Q8×C47—24no—
52C4×D4×C47—28no—
53C4×D4×C47—24no—
54#117×C47—22no—
55#117×C47C2×C412yes—
56C24×C47C22⋊C412yes—
57C24×C47C2×D411yescharacteristic
58#41×C47—44no—
59#41×C47—44no—
60#41×C47—416no—
61#41×C47—48no—
62#41×C47—44no—
63C2×Q8×C47—22no—
64C2×Q8×C47—24no—
65C2×Q8×C47—22no—
66C2×Q8×C47—48no—
67C2×Q8×C47C22⋊C412yes—
68C2×D4×C47—22no—
69C2×D4×C47—22no—
70C2×D4×C47—24no—
71C2×D4×C47—28no—
72C2×D4×C47—24no—
73C2×D4×C47—24no—
74C2×D4×C47—22no—
75C2×D4×C47—42no—
76C2×D4×C47—42no—
77C2×D4×C47C22⋊C412yes—
78C22×C4×C47—24no—
79C22×C4×C47—24no—
80C22×C4×C47—22no—
81C22×C4×C47—24no—
82C22×C4×C47—42no—
83C22×C4×C47—42no—
84C22×C4×C47C22⋊C412yes—
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D414yes—
88C22×Dic47—42no—
89C4⋊C4×C47—24no—
90C4⋊C4×C47—24no—
91C4⋊C4×C47C22⋊C414yes—
92C22⋊C4×C47—28no—
93C22⋊C4×C47—22no—
94C22⋊C4×C47—24no—
95C22⋊C4×C47—24no—
96C42×C47—24no—
97C42×C47—22no—
98C22⋊Dic47—44no—
99C2×C47⋊C8—44no—
100C2×C47⋊C8—42no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—42no—
107C23×C47—44no—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#12012yes—
110Q8×C47—28no—
111Q8×C47—42no—
112Q8×C47—44no—
113D4×C47—48no—
114D4×C47—42no—
115D4×C47—44no—
116D4×C47—44no—
117D4×C47—44no—
118D4×C47—84no—
119D4×C47—88no—
120C2×C4×C47—22no—
121C2×C4×C47—22no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—44no—
127C2×C4×C47—48no—
128C2×C4×C47—48no—
129C2×C4×C47—44no—
130C2×C4×C47#9914yes—
131C2×C4×C47C2×C22⋊C412yes—
132C2×C4×C47#12012yes—
133C2×Dic47—82no—
134C2×Dic47—84no—
135C47⋊C8—82no—
136#8464—471nomaximal, Sylow
137C22×C47—22no—
138C22×C47—24no—
139C22×C47—22no—
140C22×C47—44no—
141C22×C47—44no—
142C22×C47—44no—
143C22×C47—42no—
144C22×C47—42no—
145C22×C47—82no—
146C22×C47—82no—
147C22×C47#37911yescharacteristic
148C22×C47C2×#9912yes—
149C4×C47—22no—
150C4×C47—24no—
151C4×C47—22no—
152C4×C47—44no—
153C4×C47—42no—
154Dic47—162no—
155#3487D4711yescharacteristic
156#1782—471no—
157#1780—471no—
158#1516—474no—
159C2×C47—22no—
160C2×C47—42no—
161C2×C47—82no—
162C2×C47—82no—
163C2×C47#178111yescharacteristic
164C2×C47#179412yes—
165C2×2−1+4Dic4711yescharacteristic, complemented
166C22×#41C2×D4711yescharacteristic
167#550—24no—
168#548—24no—
169#535—22no—
170#535Dic4712yescomplemented
171#525—22no—
172#525C2×D4712yes—
173#519Dic4711yescharacteristic, complemented
174C2×#99—942no—
175C2×C22⋊C8—472no—
176C2×C22⋊C8—942no—
177#381—472no—
178#328—944no—
179#327—944no—
180C47#846411yescharacteristic, complemented, Sylow
1812−1+4—44no—
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—24no—
185C2×#41—44no—
186C2×#41C2×Dic4712yes—
187C22×Q8—22no—
188C22×Q8C2×Dic4711yescharacteristic
189C22×D4C2×Dic4711yescharacteristic
190C22×D4C22⋊D4712yes—
191C23×C4C22⋊D4712yes—
192C23×C4C22×D4711yescharacteristic
193C22×C8—942no—
194#127—24no—
195#127—22no—
196#124—24no—
197#124—22no—
198#123—22no—
199C22⋊Q8—24no—
200C22⋊Q8C22⋊D4714yes—
201#121—24no—
202#121—28no—
203#121—24no—
204#121C22⋊D4714yes—
205#120—24no—
206#120—22no—
207C4×Q8—24no—
208C4×D4—28no—
209C4×D4—24no—
210#117—22no—
211#117C2×Dic4712yes—
212C2×C22⋊C4—942no—
213#99—1884no—
214C22⋊C8—944no—
215C22⋊C8—1884no—
216#95—944no—
217C24C22⋊Dic4712yes—
218C24C2×C22⋊D4711yescharacteristic
219#41—44no—
220#41—44no—
221#41—416no—
222#41—48no—
223#41—44no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—48no—
228C2×Q8C22⋊Dic4712yes—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—28no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4C22⋊Dic4712yes—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1882no—
246C22×C4C22⋊Dic4712yes—
247C22×C4C22×Dic4711yescharacteristic
248C22×C4C2×C22⋊D4711yescharacteristic
249C22×C4C2×C22⋊D4714yes—
250C2×C8—1884no—
251C2×C8—1882no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4C22⋊Dic4714yes—
255C22⋊C4—28no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—1884no—
260C42—24no—
261C42—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23C2×C22⋊Dic4711yescharacteristic
270C23#1309399312yes—
271Q8—28no—
272Q8—42no—
273Q8—44no—
274D4—48no—
275D4—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—84no—
280D4—88no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—3762no—
292C2×C4—3764no—
293C2×C4#1309393914yes—
294C2×C4C2×C22⋊Dic4712yes—
295C2×C4#1309399312yes—
296C8—3762no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—82no—
306C22—82no—
307C22#1568897911yescharacteristic
308C22C2×#1309393912yes—
309C4—22no—
310C4—24no—
311C4—22no—
312C4—42no—
313C4—44no—
314C4—7522no—
315C2—22no—
316C2—42no—
317C2—82no—
318C2—82no—
319C2#2024330311yescharacteristic
320C2#2024335212yes—
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
Fitting, prime radicalC47×#3487C2
socleC22×C47#379
cosocleC23×C4×C47C23
hypercenter#3487D47
hypocenterC47#8464
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3487

Complements

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

NH ≅ G/Ntype
1. #20243306C2semidirect
2. #20243300C2semidirect
3. #20241944C2semidirect
11. C2×2−1+4×C47C4semidirect
16. C47×#535C4semidirect
19. C47×#519C4semidirect
165. C2×2−1+4Dic47semidirect
170. #535Dic47semidirect
173. #519Dic47semidirect
180. C47#8464semidirect

Maximal subgroups

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

subgroupindex
#202433062
#202433002
#202419442
C47×#34872
#846447