Finite Group Databasebeta

← #20250115

Subgroups — #20250115

Order 6016.

612 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#13094005C211yesmaximal, characteristic, complemented
2#15689037C212yesmaximal, complemented
3#15689037C212yesmaximal, complemented
4#15689037C212yesmaximal, complemented
5#15689007C212yesmaximal, complemented
6#15689003C212yesmaximal, complemented
7#15689003C212yesmaximal, complemented
8#15689003C211yesmaximal, characteristic, complemented
9C2×#127×C47C211yesmaximal, characteristic
10C23×Dic47C2211yescharacteristic
11C23×Dic47C2211yescharacteristic
12#13094005—24no—
13C2×Dic47⋊C4C2212yes—
14C2×Dic47⋊C4C2212yes—
15C2×C22⋊Dic47C2212yes—
16C2×C22⋊Dic47C2211yescharacteristic
17C2×C22⋊Dic47C2212yes—
18C2×C22⋊Dic47C2212yes—
19C2×C22⋊Dic47C2212yes—
20C2×C22⋊Dic47C2211yescharacteristic
21C2×C22⋊Dic47C2212yes—
22C2×C22⋊Dic47C2212yes—
23C47⋊#95C2212yes—
24C47⋊#95C2212yes—
25C47⋊#95C2212yes—
26C47⋊#95C2212yes—
27C22×D4×C47C2212yes—
28C22×D4×C47C2212yes—
29C22×D4×C47C2211yescharacteristic
30C22×D4×C47C2211yescharacteristic
31#127×C47—24no—
32C2×C42×C47C2211yescharacteristic
33C24×C47C2312yes—
34C24×C47C2311yescharacteristic
35C24×C47C2311yescharacteristic
36C2×D4×C47—24no—
37C2×D4×C47—24no—
38C2×D4×C47—22no—
39C2×D4×C47—24no—
40C2×D4×C47—24no—
41C2×D4×C47—41no—
42C2×D4×C47—41no—
43C2×D4×C47—42no—
44C2×D4×C47—42no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47D414yes—
48C22×C4×C47C2311yescharacteristic
49C22×C4×C47C2312yes—
50C22×Dic47—22no—
51C22×Dic47—41no—
52C22×Dic47—41no—
53C22×Dic47—41no—
54C22×Dic47—42no—
55C22×Dic47C2312yes—
56C22×Dic47C2312yes—
57C22×Dic47C2312yes—
58C22×Dic47C2312yes—
59C42×C47—22no—
60C22⋊Dic47—22no—
61C22⋊Dic47—24no—
62C22⋊Dic47—41no—
63C22⋊Dic47—42no—
64C22⋊Dic47—42no—
65C22⋊Dic47—42no—
66C22⋊Dic47—42no—
67C22⋊Dic47—42no—
68Dic47⋊C4—24no—
69Dic47⋊C4—24no—
70C23×C47—22no—
71C23×C47—22no—
72C23×C47—21no—
73C23×C47—22no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—22no—
78C23×C47—22no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—22no—
83C23×C47—22no—
84C23×C47—22no—
85C23×C47C2×D412yes—
86C23×C47C2×D412yes—
87C23×C47C2411yescharacteristic
88D4×C47—44no—
89D4×C47—48no—
90D4×C47—48no—
91D4×C47—44no—
92C2×C4×C47—22no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—22no—
96C2×C4×C47C2×D412yes—
97C2×C4×C47#4114yes—
98C2×Dic47—22no—
99C2×Dic47—22no—
100C2×Dic47—22no—
101C2×Dic47—22no—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—44no—
111C2×Dic47—44no—
112C22×C47—22no—
113C22×C47—22no—
114C22×C47—24no—
115C22×C47—24no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—41no—
119C22×C47—41no—
120C22×C47—41no—
121C22×C47—41no—
122C22×C47—42no—
123C22×C47—42no—
124C22×C47—42no—
125C22×C47—42no—
126C22×C47—42no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47C22×D411yescharacteristic
130C22×C47C2×#4112yes—
131C22×C47H(5,𝔽2)11yescharacteristic
132C22×C47H(5,𝔽2)11yescharacteristic
133C22×C47H(5,𝔽2)12yes—
134C4×C47—22no—
135C4×C47—24no—
136Dic47—42no—
137Dic47—42no—
138Dic47—42no—
139Dic47—42no—
140#2709—471nomaximal, Sylow
141C2×C47—42no—
142C2×C47—42no—
143C2×C47—82no—
144C2×C47#53411yescharacteristic
145C2×C47#54012yes—
146C2×C47#54612yes—
147C2×C47#55911yescharacteristic
148C2×C47#56011yescharacteristic
149C2×#127D4711yescharacteristic
150C2×#123—471no—
151#394—472no—
152#394—472no—
153#394—472no—
154#387—472no—
155#386—472no—
156#386—472no—
157#386—471no—
158C47#270911yescharacteristic, complemented, Sylow
159C22×D4C2×D4712yes—
160C22×D4C2×D4712yes—
161C22×D4C2×D4711yescharacteristic
162C22×D4C2×D4711yescharacteristic
163C23×C4—471no—
164C23×C4—471no—
165#127—24no—
166#123—944no—
167C2×C4⋊C4—472no—
168C2×C4⋊C4—472no—
169C2×C22⋊C4—472no—
170C2×C22⋊C4—471no—
171C2×C22⋊C4—472no—
172C2×C22⋊C4—472no—
173C2×C22⋊C4—472no—
174C2×C22⋊C4—471no—
175C2×C22⋊C4—472no—
176C2×C22⋊C4—472no—
177C2×C42C2×D4711yescharacteristic
178#95—472no—
179#95—472no—
180#95—472no—
181#95—472no—
182C24C22×D4712yes—
183C24C22×D4711yescharacteristic
184C24C22×D4711yescharacteristic
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4C22⋊D4714yes—
197C22×C4—472no—
198C22×C4—472no—
199C22×C4—472no—
200C22×C4—472no—
201C22×C4—942no—
202C22×C4—1881no—
203C22×C4—1881no—
204C22×C4—1882no—
205C22×C4—1881no—
206C22×C4C22×D4712yes—
207C22×C4C22×D4711yescharacteristic
208C4⋊C4—944no—
209C4⋊C4—944no—
210C22⋊C4—942no—
211C22⋊C4—944no—
212C22⋊C4—1881no—
213C22⋊C4—1882no—
214C22⋊C4—1882no—
215C22⋊C4—1882no—
216C22⋊C4—1882no—
217C22⋊C4—1882no—
218C42—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23C2×C22⋊D4712yes—
235C23C2×C22⋊D4712yes—
236C23C23×D4711yescharacteristic
237D4—44no—
238D4—48no—
239D4—48no—
240D4—44no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—942no—
248C2×C4—942no—
249C2×C4—942no—
250C2×C4—942no—
251C2×C4—942no—
252C2×C4—942no—
253C2×C4—942no—
254C2×C4—942no—
255C2×C4—942no—
256C2×C4—942no—
257C2×C4—1884no—
258C2×C4—1884no—
259C2×C4#1062535214yes—
260C2×C4C2×C22⋊D4712yes—
261C22—24no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22C22×C22⋊D4711yescharacteristic
279C22C2×#1062535212yes—
280C22#1309406912yes—
281C22#1309406911yescharacteristic
282C22#1309406911yescharacteristic
283C4—24no—
284C4—22no—
285C4—1882no—
286C4—1882no—
287C4—1882no—
288C4—1882no—
289C2—42no—
290C2—42no—
291C2—82no—
292C2#1568968211yescharacteristic
293C2#1568972712yes—
294C2#1568976712yes—
295C2#1568985911yescharacteristic
296C2#1568986011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#127×C47C2
hypercenterC2×#127D47
hypocenterC47#2709
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#13094005C2semidirect
2. #15689037C2semidirect
3. #15689037C2semidirect
4. #15689037C2semidirect
5. #15689007C2semidirect
6. #15689003C2semidirect
7. #15689003C2semidirect
8. #15689003C2semidirect
158. C47#2709semidirect