Finite Group Databasebeta

← #26104827

Subgroups — #26104827

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243368C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241214C212yesmaximal, complemented
4#20241211C212yesmaximal, complemented
5C47×#2855C211yesmaximal, characteristic
6C2×C4⋊(C47⋊C8)—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)—21no—
9C2×#13093935—21no—
10#15688878C2212yes—
11#15688878C2212yes—
12C2×C22⋊Q8×C47C411yescharacteristic
13C22×C42×C47C2211yescharacteristic
14C47×#391C411yescharacteristic
15C47×#391C2211yescharacteristic
16C47×#389C411yescharacteristic
17C47×#389C2211yescharacteristic
18C47×#387—21no—
19C47×#387—21no—
20C47×#385—21no—
21C47×#381—21no—
22C47×#380C411yescharacteristic
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—42no—
26C22⋊(C47⋊C8)—24no—
27C22⋊(C47⋊C8)—42no—
28#13093935—42no—
29C22×Q8×C47C2×C411yescharacteristic
30C23×C4×C47C2×C411yescharacteristic
31C23×C4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C22⋊Q8×C47—42no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47C2×C411yescharacteristic
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47—21no—
42C2×C42×C47—42no—
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47C2311yescharacteristic
47#95×C47—21no—
48#95×C47—21no—
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54C24×C47C22⋊C411yescharacteristic
55C2×Q8×C47—22no—
56C2×Q8×C47—41no—
57C2×Q8×C47—41no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—22no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C42×C47—22no—
88C42×C47—22no—
89C42×C47—41no—
90C42×C47—41no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47#10412yes—
104C23×C47C2×C22⋊C411yescharacteristic
105Q8×C47—82no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—44no—
121C2×C4×C47—44no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47#9912yes—
137C2×C4×C47#13112yes—
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C47#13711yescharacteristic
140C2×C4×C47#13712yes—
141C47⋊C8—44no—
142C47⋊C8—82no—
143#7701—471nomaximal, Sylow
144C22×C47—22no—
145C22×C47—22no—
146C22×C47—22no—
147C22×C47—41no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47#40811yescharacteristic
152C22×C47C2×#9911yescharacteristic
153C22×C47#41311yescharacteristic
154C22×C47#41711yescharacteristic
155C22×C47#41911yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C22×C47#42111yescharacteristic
158C4×C47—22no—
159C4×C47—22no—
160C4×C47—22no—
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—81no—
166C4×C47—81no—
167C4×C47—81no—
168C4×C47—81no—
169#2855D4711yescharacteristic
170#1797—471no—
171#1749—471no—
172#1272—472no—
173#1271—472no—
174C2×C47—42no—
175C2×C47#146811yescharacteristic
176C2×C47#148311yescharacteristic
177C2×C47#149711yescharacteristic
178C2×C47#150811yescharacteristic
179C2×C47#151311yescharacteristic
180C2×C47#153211yescharacteristic
181C2×C47#155911yescharacteristic
182C2×C22⋊Q8Dic4711yescharacteristic
183C22×C42C2×D4711yescharacteristic
184C2×C4⋊C8—941no—
185C2×C22⋊C8—941no—
186C2×C22⋊C8—941no—
187C2×#97—941no—
188#391Dic4711yescharacteristic
189#391C2×D4711yescharacteristic
190#389Dic4711yescharacteristic
191#389C2×D4711yescharacteristic
192#387—21no—
193#387—21no—
194#385—21no—
195#381—21no—
196#380Dic4711yescharacteristic
197#336—472no—
198#336—472no—
199C47#770111yescharacteristic, complemented, Sylow
200C22×Q8C2×Dic4711yescharacteristic
201C23×C4C2×Dic4711yescharacteristic
202C23×C4C22⋊D4711yescharacteristic
203C23×C4C22⋊D4711yescharacteristic
204C22×C8—942no—
205C22×C8—942no—
206C22⋊Q8—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C2×Dic4711yescharacteristic
211C2×C4⋊C4C2×Dic4711yescharacteristic
212C2×C4⋊C4C2×Dic4711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42—42no—
216C2×C42C2×Dic4711yescharacteristic
217C2×C42C22⋊D4711yescharacteristic
218C2×C42C22⋊D4711yescharacteristic
219C2×C42C22×D4711yescharacteristic
220C4⋊C8—1882no—
221C22⋊C8—944no—
222C22⋊C8—1882no—
223#97—1882no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24C22⋊Dic4711yescharacteristic
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C22⋊Dic4711yescharacteristic
250C22×C4C22⋊Dic4711yescharacteristic
251C22×C4C22⋊Dic4711yescharacteristic
252C22×C4C22×Dic4711yescharacteristic
253C22×C4C2×C22⋊D4711yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255C2×C8—944no—
256C2×C8—1882no—
257C2×C8—1882no—
258C2×C8—1882no—
259C2×C8—1882no—
260C2×C8—1882no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C42—22no—
271C42—22no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1309395112yes—
281C23C2×C22⋊Dic4711yescharacteristic
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4#1309393912yes—
314C2×C4#1309403112yes—
315C2×C4#1309404911yescharacteristic
316C2×C4#1309405411yescharacteristic
317C2×C4#1309405512yes—
318C8—3762no—
319C8—1884no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1568909011yescharacteristic
328C22C2×#1309393911yescharacteristic
329C22#1568910711yescharacteristic
330C22#1568912411yescharacteristic
331C22#1568913211yescharacteristic
332C22C2×#1309395111yescharacteristic
333C22#1568914011yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C2—42no—
346C2#2024173611yescharacteristic
347C2#2024180111yescharacteristic
348C2#2024186111yescharacteristic
349C2#2024190811yescharacteristic
350C2#2024193111yescharacteristic
351C2#2024200811yescharacteristic
352C2#2024211811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2855C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2855D47
hypocenterC47#7701
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #20241214C2semidirect
4. #20241211C2semidirect
199. C47#7701semidirect

Maximal subgroups

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

subgroupindex
#202433682
#202431502
#202412142
#202412112
C47×#28552
#770147