Finite Group Databasebeta

← #26104522

Subgroups — #26104522

Order 12032.

730 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 131 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#20243082C212yesmaximal, complemented
2#20241214C212yesmaximal, complemented
3#20241209C212yesmaximal, complemented
4C47×#2297C211yesmaximal, characteristic
5C22×#10625329—22no—
6#15688878—22no—
7#15688878C2214yes—
8C22×#41×C47C411yescharacteristic
9C2×#117×C47C411yescharacteristic
10C22×C42×C47C2211yescharacteristic
11C47×#391C411yescharacteristic
12C47×#391C2211yescharacteristic
13C47×#390C411yescharacteristic
14C47×#390C2211yescharacteristic
15C47×#386—22no—
16C4×C22⋊C4×C47—22no—
17C2×#10625329—28no—
18C2×#10625329—44no—
19C2×#10625329—44no—
20C22×C47⋊C8—24no—
21C2×#41×C47—22no—
22C2×#41×C47—22no—
23C2×#41×C47—42no—
24C2×#41×C47—42no—
25C22×Q8×C47C2×C411yescharacteristic
26C22×D4×C47—21no—
27C22×D4×C47C2×C411yescharacteristic
28C23×C4×C47—21no—
29C23×C4×C47C2×C411yescharacteristic
30C23×C4×C47D412yes—
31#117×C47—42no—
32C2×C4⋊C4×C47C2×C411yescharacteristic
33C2×C4⋊C4×C47C2×C411yescharacteristic
34C2×C22⋊C4×C47—21no—
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47—22no—
37C2×C42×C47—21no—
38C2×C42×C47—42no—
39C2×C42×C47C2×C411yescharacteristic
40C2×C42×C47D412yes—
41C2×C42×C47C2311yescharacteristic
42#95×C47—22no—
43#95×C47—22no—
44C24×C47—21no—
45C24×C47C22⋊C411yescharacteristic
46#41×C47—48no—
47#41×C47—48no—
48C2×Q8×C47—21no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47M1612yescomplemented
52C2×D4×C47—24no—
53C2×D4×C47—21no—
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47—42no—
57C2×D4×C47—42no—
58C2×D4×C47M1612yescomplemented
59C22×C4×C47—22no—
60C22×C4×C47—24no—
61C22×C4×C47—22no—
62C22×C4×C47—24no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—24no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—42no—
72C22×C4×C47—42no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C412yes—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D412yes—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C22⋊C4×C47—42no—
80C22⋊C4×C47—44no—
81C22⋊C4×C47—42no—
82C22⋊C4×C47—42no—
83C42×C47—41no—
84C42×C47—42no—
85C42×C47—41no—
86C42×C47—42no—
87#10625329—416no—
88C2×C47⋊C8—28no—
89C2×C47⋊C8—44no—
90C2×C47⋊C8—44no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47#9912yes—
101C23×C47C2×C22⋊C411yescharacteristic
102Q8×C47—42no—
103Q8×C47—42no—
104D4×C47—42no—
105D4×C47—42no—
106D4×C47—48no—
107C2×C4×C47—28no—
108C2×C4×C47—28no—
109C2×C4×C47—24no—
110C2×C4×C47—24no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—48no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47#10012yes—
128C2×C4×C47#10414yes—
129C2×C4×C47C2×M1611yescharacteristic
130C2×C4×C47C2×M1611yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)12yes—
132C2×C4×C47#13712yes—
133C47⋊C8—48no—
134#7631—471nomaximal, Sylow
135C22×C47—22no—
136C22×C47—24no—
137C22×C47—41no—
138C22×C47—42no—
139C22×C47—41no—
140C22×C47—42no—
141C22×C47—44no—
142C22×C47—81no—
143C22×C47—82no—
144C22×C47—81no—
145C22×C47#40711yescharacteristic
146C22×C47C2×#9911yescharacteristic
147C22×C47C2×#10011yescharacteristic
148C22×C47#41712yes—
149C22×C47C2×#10412yes—
150C4×C47—22no—
151C4×C47—21no—
152C4×C47—21no—
153C4×C47—22no—
154C4×C47—24no—
155C4×C47—22no—
156C4×C47—42no—
157C4×C47—81no—
158C4×C47—81no—
159#2297D4711yescharacteristic
160#1728—472no—
161#1272—472no—
162#1270—472no—
163C2×C47—42no—
164C2×C47—82no—
165C2×C47#146211yescharacteristic
166C2×C47#148912yes—
167C2×C47#150412yes—
168C2×C47#152511yescharacteristic
169C2×C47#156011yescharacteristic
170C22×#41Dic4711yescharacteristic
171C22×M16—942no—
172C2×#117Dic4711yescharacteristic
173C22×C42C2×D4711yescharacteristic
174#391Dic4711yescharacteristic
175#391C2×D4711yescharacteristic
176#390Dic4711yescharacteristic
177#390C2×D4711yescharacteristic
178#386—22no—
179C4×C22⋊C4—22no—
180#336—474no—
181#336—942no—
182C47#763111yescharacteristic, complemented, Sylow
183C2×#41—22no—
184C2×#41—22no—
185C2×#41—42no—
186C2×#41—42no—
187C22×Q8C2×Dic4711yescharacteristic
188C22×D4—21no—
189C22×D4C2×Dic4711yescharacteristic
190C23×C4—21no—
191C23×C4C2×Dic4711yescharacteristic
192C23×C4C22⋊D4712yes—
193C2×M16—948no—
194C2×M16—1884no—
195C2×M16—1884no—
196C22×C8—944no—
197#117—42no—
198C2×C4⋊C4C2×Dic4711yescharacteristic
199C2×C4⋊C4C2×Dic4711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42C2×Dic4711yescharacteristic
206C2×C42C22⋊D4712yes—
207C2×C42C22×D4711yescharacteristic
208#95—22no—
209#95—22no—
210C24—21no—
211C24C22⋊Dic4711yescharacteristic
212#41—48no—
213#41—48no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8#1062532912yescomplemented
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4#1062532912yescomplemented
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic4711yescharacteristic
240C22×C4C22⋊Dic4712yes—
241C22×C4C22×Dic4711yescharacteristic
242C22×C4C2×C22⋊D4712yes—
243M16—18816no—
244C2×C8—948no—
245C2×C8—1884no—
246C2×C8—1884no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C42—42no—
255C42—41no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—41no—
263C23—41no—
264C23—42no—
265C23—42no—
266C23#1309393912yes—
267C23C2×C22⋊Dic4711yescharacteristic
268Q8—42no—
269Q8—42no—
270D4—42no—
271D4—42no—
272D4—48no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—48no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4#1309394112yes—
294C2×C4#1309395114yes—
295C2×C4C2×#1062532911yescharacteristic
296C2×C4C2×#1062532911yescharacteristic
297C2×C4#1309404912yes—
298C2×C4#1309405412yes—
299C8—1888no—
300C22—22no—
301C22—24no—
302C22—41no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—44no—
307C22—81no—
308C22—81no—
309C22—82no—
310C22#1568908711yescharacteristic
311C22C2×#1309393911yescharacteristic
312C22C2×#1309394111yescharacteristic
313C22#1568912412yes—
314C22C2×#1309395112yes—
315C4—22no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—24no—
320C4—22no—
321C4—42no—
322C4—81no—
323C4—81no—
324C2—42no—
325C2—82no—
326C2#2024171511yescharacteristic
327C2#2024182712yes—
328C2#2024189212yes—
329C2#2024198011yescharacteristic
330C2#2024212311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2297C2
socleC23×C47C2×C22⋊C4
prime radicalC22×#41×C47C4
cosocleC2×C42×C47C23
hypercenter#2297D47
hypocenterC47#7631
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
1. #20243082C2semidirect
2. #20241214C2semidirect
3. #20241209C2semidirect
51. C2×Q8×C47M16semidirect
58. C2×D4×C47M16semidirect
182. C47#7631semidirect
217. C2×Q8#10625329semidirect
224. C2×D4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202430822
#202412142
#202412092
C47×#22972
#763147