Finite Group Databasebeta

← #26104576

Subgroups — #26104576

Order 12032.

630 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 123 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#20243456C212yesmaximal, complemented
2#20241214C212yesmaximal, complemented
3#20241209C212yesmaximal, complemented
4C47×#2297C211yesmaximal, characteristic
5C2×C22⋊(C47⋊C8)—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—
17C22×C47⋊C8—24no—
18C22⋊(C47⋊C8)—44no—
19C2×#41×C47—22no—
20C2×#41×C47—22no—
21C2×#41×C47—42no—
22C2×#41×C47—42no—
23C22×Q8×C47C2×C411yescharacteristic
24C22×D4×C47—21no—
25C22×D4×C47C2×C411yescharacteristic
26C23×C4×C47—21no—
27C23×C4×C47C2×C411yescharacteristic
28C23×C4×C47D412yes—
29#117×C47—42no—
30C2×C4⋊C4×C47C2×C411yescharacteristic
31C2×C4⋊C4×C47C2×C411yescharacteristic
32C2×C22⋊C4×C47—21no—
33C2×C22⋊C4×C47—22no—
34C2×C22⋊C4×C47—22no—
35C2×C42×C47—21no—
36C2×C42×C47—42no—
37C2×C42×C47C2×C411yescharacteristic
38C2×C42×C47D412yes—
39C2×C42×C47C2311yescharacteristic
40#95×C47—22no—
41#95×C47—22no—
42C24×C47—21no—
43C24×C47C22⋊C411yescharacteristic
44#41×C47—48no—
45#41×C47—48no—
46C2×Q8×C47—21no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47M1612yes—
50C2×D4×C47—24no—
51C2×D4×C47—21no—
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47—42no—
55C2×D4×C47—42no—
56C2×D4×C47M1612yes—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—24no—
62C22×C4×C47—24no—
63C22×C4×C47—21no—
64C22×C4×C47—24no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47C22⋊C412yes—
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D412yes—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—44no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—44no—
79C42×C47—41no—
80C42×C47—42no—
81C42×C47—41no—
82C42×C47—42no—
83C2×C47⋊C8—44no—
84C2×C47⋊C8—44no—
85C2×C47⋊C8—44no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—22no—
89C23×C47—22no—
90C23×C47—22no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47#9912yes—
96C23×C47C2×C22⋊C411yescharacteristic
97Q8×C47—42no—
98Q8×C47—42no—
99D4×C47—42no—
100D4×C47—42no—
101D4×C47—48no—
102C2×C4×C47—24no—
103C2×C4×C47—24no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—48no—
115C2×C4×C47—48no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47#10012yes—
123C2×C4×C47C2×M1611yescharacteristic
124C2×C4×C47C2×M1611yescharacteristic
125C2×C4×C47#13514yes—
126C47⋊C8—84no—
127#7645—471nomaximal, Sylow
128C22×C47—22no—
129C22×C47—24no—
130C22×C47—41no—
131C22×C47—41no—
132C22×C47—42no—
133C22×C47—42no—
134C22×C47—44no—
135C22×C47—81no—
136C22×C47—81no—
137C22×C47—82no—
138C22×C47#40711yescharacteristic
139C22×C47C2×#9911yescharacteristic
140C22×C47C2×#10011yescharacteristic
141C22×C47#41612yes—
142C22×C47#42112yes—
143C4×C47—22no—
144C4×C47—21no—
145C4×C47—21no—
146C4×C47—24no—
147C4×C47—42no—
148C4×C47—42no—
149C4×C47—81no—
150C4×C47—81no—
151#2297D4711yescharacteristic
152#1817—472no—
153#1272—472no—
154#1270—472no—
155C2×C47—42no—
156C2×C47—82no—
157C2×C47#146211yescharacteristic
158C2×C47#148812yes—
159C2×C47#151612yes—
160C2×C47#153611yescharacteristic
161C2×C47#154311yescharacteristic
162C22×#41Dic4711yescharacteristic
163C2×#117Dic4711yescharacteristic
164C22×C42C2×D4711yescharacteristic
165C2×C22⋊C8—942no—
166#391Dic4711yescharacteristic
167#391C2×D4711yescharacteristic
168#390Dic4711yescharacteristic
169#390C2×D4711yescharacteristic
170#386—22no—
171C4×C22⋊C4—22no—
172#336—474no—
173#336—942no—
174C47#764511yescharacteristic, complemented, Sylow
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—42no—
178C2×#41—42no—
179C22×Q8C2×Dic4711yescharacteristic
180C22×D4—21no—
181C22×D4C2×Dic4711yescharacteristic
182C23×C4—21no—
183C23×C4C2×Dic4711yescharacteristic
184C23×C4C22⋊D4712yes—
185C22×C8—944no—
186#117—42no—
187C2×C4⋊C4C2×Dic4711yescharacteristic
188C2×C4⋊C4C2×Dic4711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42C2×Dic4711yescharacteristic
195C2×C42C22⋊D4712yes—
196C2×C42C22×D4711yescharacteristic
197C22⋊C8—1884no—
198#95—22no—
199#95—22no—
200C24—21no—
201C24C22⋊Dic4711yescharacteristic
202#41—48no—
203#41—48no—
204C2×Q8—21no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8#1062532912yes—
208C2×D4—24no—
209C2×D4—21no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4#1062532912yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊Dic4712yes—
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22×Dic4711yescharacteristic
231C22×C4C2×C22⋊D4712yes—
232C2×C8—1884no—
233C2×C8—1884no—
234C2×C8—1884no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—41no—
241C42—42no—
242C42—41no—
243C42—42no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23#1309393912yes—
254C23C2×C22⋊Dic4711yescharacteristic
255Q8—42no—
256Q8—42no—
257D4—42no—
258D4—42no—
259D4—48no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4#1309394112yes—
281C2×C4C2×#1062532911yescharacteristic
282C2×C4C2×#1062532911yescharacteristic
283C2×C4#1309404414yes—
284C8—3764no—
285C22—24no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—44no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22#1568908711yescharacteristic
296C22C2×#1309393911yescharacteristic
297C22C2×#1309394111yescharacteristic
298C22#1568911912yes—
299C22#1568914012yes—
300C4—24no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—42no—
306C4—81no—
307C4—81no—
308C2—42no—
309C2—82no—
310C2#2024171511yescharacteristic
311C2#2024182212yes—
312C2#2024194412yes—
313C2#2024202311yescharacteristic
314C2#2024205011yescharacteristic
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#7645
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. #20243456C2semidirect
2. #20241214C2semidirect
3. #20241209C2semidirect
174. C47#7645semidirect

Maximal subgroups

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

subgroupindex
#202434562
#202412142
#202412092
C47×#22972
#764547