Finite Group Databasebeta

← #20249131

Subgroups — #20249131

Order 6016.

632 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13094009C212yesmaximal, complemented
2C2×#13093999C211yesmaximal, characteristic
3C2×C4⋊C4×D47C211yesmaximal, characteristic, complemented
4#15689054C211yesmaximal, characteristic, complemented
5#15689047C212yesmaximal, complemented
6#15689045C212yesmaximal, complemented
7#15689041C211yesmaximal, characteristic, complemented
8#15689040C211yesmaximal, characteristic, complemented
9#15689001C212yesmaximal, complemented
10#15688982C211yesmaximal, characteristic, complemented
11C47×#398C211yesmaximal, characteristic, complemented
12C22×D188C2212yes—
13C22×D188C2211yescharacteristic
14C22×C4×D47C2211yescharacteristic
15C22×C4×D47C2211yescharacteristic
16C22×C4×D47C2211yescharacteristic
17#13094009—28no—
18#13093999—24no—
19C2×C4⋊Dic47C2211yescharacteristic
20C2×Dic47⋊C4C2211yescharacteristic
21C2×Dic47⋊C4C2211yescharacteristic
22C2×Dic47⋊C4C2212yes—
23C2×Dic47⋊C4C2212yes—
24C4⋊C4×D47—24no—
25C2×(C2×D47)⋊C4C2212yes—
26C2×(C2×D47)⋊C4C2212yes—
27C2×(C2×D47)⋊C4C2212yes—
28C2×(C2×D47)⋊C4C2212yes—
29C2×(C2×D47)⋊C4C2212yes—
30C2×(C2×D47)⋊C4C2212yes—
31C47⋊#95C2211yescharacteristic
32C47⋊#95C2212yes—
33C2×C4⋊C4×C47C2211yescharacteristic
34C2×C4⋊C4×C47C2212yes—
35C2×C4⋊C4×C47C2211yescharacteristic
36#95×C47C2211yescharacteristic
37#95×C47C2212yes—
38C23×D47C2312yes—
39C23×D47C2311yescharacteristic
40C22×C4×C47C2312yes—
41C22×C4×C47C2312yes—
42C22×C4×C47C2311yescharacteristic
43C22×C4×C47C2311yescharacteristic
44C22×C4×C47C2311yescharacteristic
45C22×Dic47C2312yes—
46C22×Dic47C2311yescharacteristic
47C22×Dic47C2311yescharacteristic
48C22×Dic47C2311yescharacteristic
49C2×D188—24no—
50C2×D188—24no—
51C2×D188—24no—
52C2×D188—22no—
53C2×D188—42no—
54C2×C4×D47—22no—
55C2×C4×D47—22no—
56C2×C4×D47—22no—
57C2×C4×D47—42no—
58C2×C4×D47—42no—
59C2×C4×D47D414yes—
60C2×C4×D47D414yescomplemented
61C4⋊C4×C47—22no—
62C4⋊C4×C47—22no—
63C4⋊C4×C47—24no—
64(C2×D47)⋊C4—24no—
65(C2×D47)⋊C4—24no—
66(C2×D47)⋊C4—24no—
67(C2×D47)⋊C4—24no—
68(C2×D47)⋊C4—42no—
69(C2×D47)⋊C4—42no—
70C4⋊Dic47—22no—
71Dic47⋊C4—22no—
72Dic47⋊C4—22no—
73Dic47⋊C4—24no—
74Dic47⋊C4—42no—
75C23×C47C2411yescharacteristic
76C22×D47—22no—
77C22×D47—22no—
78C22×D47—22no—
79C22×D47—22no—
80C22×D47—22no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47—22no—
84C22×D47—22no—
85C22×D47C2×D412yes—
86C22×D47C2×D412yes—
87C22×D47#4112yes—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—22no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47C2×D412yes—
100C2×C4×C47C2×D412yes—
101C2×Dic47—21no—
102C2×Dic47—21no—
103C2×Dic47—22no—
104C2×Dic47—21no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47C2×D412yes—
110C2×Dic47C2×D412yes—
111D188—44no—
112D188—48no—
113C4×D47—28no—
114C4×D47—44no—
115C4×D47—44no—
116C22×C47C22×D411yescharacteristic
117C22×C47C22×D411yescharacteristic
118C22×C47C2×#4111yescharacteristic
119C22×C47H(5,𝔽2)12yes—
120C22×C472−1+412yes—
121C2×D47—22no—
122C2×D47—22no—
123C2×D47—24no—
124C2×D47—24no—
125C2×D47—42no—
126C2×D47—42no—
127C2×D47—42no—
128C2×D47—42no—
129C2×D47—42no—
130C2×D47—42no—
131C2×D47—42no—
132C2×D47#12114yes—
133C4×C47—22no—
134C4×C47—22no—
135C4×C47—42no—
136C4×C47—41no—
137C4×C47—42no—
138Dic47—22no—
139Dic47—22no—
140Dic47—41no—
141Dic47—42no—
142#2618—471nomaximal, Sylow
143C2×C47C2×#12111yescharacteristic
144C2×C47#53411yescharacteristic
145C2×C47#53611yescharacteristic
146C2×C47#53712yes—
147C2×C47#53912yes—
148D47—24no—
149D47—82no—
150C2×#123—472no—
151C2×#121—471no—
152C22×C4⋊C4—471no—
153#398—471no—
154#398D4711yescharacteristic, complemented
155#396—472no—
156#396—472no—
157#394—471no—
158#394—471no—
159#386—472no—
160#380—471no—
161C47#261811yescharacteristic, complemented, Sylow
162C22×D4—472no—
163C22×D4—471no—
164C23×C4—471no—
165C23×C4—471no—
166C23×C4—471no—
167#123—948no—
168#121—944no—
169C2×C4⋊C4—471no—
170C2×C4⋊C4—471no—
171C2×C4⋊C4—471no—
172C2×C4⋊C4—472no—
173C2×C4⋊C4—472no—
174C2×C4⋊C4—944no—
175C2×C4⋊C4C2×D4711yescharacteristic
176C2×C4⋊C4C2×D4711yescharacteristic
177C2×C4⋊C4C2×D4712yes—
178C2×C22⋊C4—472no—
179C2×C22⋊C4—472no—
180C2×C22⋊C4—472no—
181C2×C22⋊C4—472no—
182C2×C22⋊C4—472no—
183C2×C22⋊C4—472no—
184#95—471no—
185#95—472no—
186#95C2×D4711yescharacteristic
187#95C2×D4712yes—
188C24—471no—
189C24—472no—
190C2×D4—944no—
191C2×D4—944no—
192C2×D4—944no—
193C2×D4—942no—
194C2×D4—1882no—
195C22×C4—471no—
196C22×C4—471no—
197C22×C4—474no—
198C22×C4—474no—
199C22×C4—472no—
200C22×C4—471no—
201C22×C4—942no—
202C22×C4—942no—
203C22×C4—942no—
204C22×C4—1882no—
205C22×C4—1882no—
206C22×C4C22×D4712yes—
207C22×C4C22×D4712yes—
208C22×C4C22×D4711yescharacteristic
209C22×C4C22×D4711yescharacteristic
210C22×C4C22×D4711yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—24no—
214C4⋊C4—942no—
215C4⋊C4—944no—
216C4⋊C4—942no—
217C4⋊C4—942no—
218C4⋊C4—1882no—
219C22⋊C4—944no—
220C22⋊C4—944no—
221C22⋊C4—944no—
222C22⋊C4—944no—
223C22⋊C4—1882no—
224C22⋊C4—1882no—
225C23—472no—
226C23—472no—
227C23—472no—
228C23—942no—
229C23—942no—
230C23—942no—
231C23—942no—
232C23—942no—
233C23—942no—
234C23—942no—
235C23—942no—
236C23—942no—
237C23C23×D4711yescharacteristic
238D4—1884no—
239D4—1888no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—472no—
252C2×C4—472no—
253C2×C4—948no—
254C2×C4—941no—
255C2×C4—941no—
256C2×C4—942no—
257C2×C4—942no—
258C2×C4—941no—
259C2×C4—942no—
260C2×C4—942no—
261C2×C4—942no—
262C2×C4—1884no—
263C2×C4—1884no—
264C2×C4D4×D4712yes—
265C2×C4D4×D4712yes—
266C22—474no—
267C22—942no—
268C22—942no—
269C22—944no—
270C22—944no—
271C22—1882no—
272C22—1882no—
273C22—1882no—
274C22—1882no—
275C22—1882no—
276C22—1882no—
277C22—1882no—
278C22C2×D4×D4711yescharacteristic
279C22C2×D4×D4711yescharacteristic
280C22#41×D4711yescharacteristic
281C22#1309407012yes—
282C22#1309407212yes—
283C4—22no—
284C4—22no—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—942no—
289C4—942no—
290C4—1881no—
291C4—1882no—
292C2—944no—
293C2—3762no—
294C2#121×D4711yescharacteristic
295C2#1568968411yescharacteristic
296C2#1568969811yescharacteristic
297C2#1568970312yes—
298C2#1568972112yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC47×#398C2
hypercenter#398D47
hypocenterC47#2618
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13094009C2semidirect
3. C2×C4⋊C4×D47C2semidirect
4. #15689054C2semidirect
5. #15689047C2semidirect
6. #15689045C2semidirect
7. #15689041C2semidirect
8. #15689040C2semidirect
9. #15689001C2semidirect
10. #15688982C2semidirect
11. C47×#398C2semidirect
60. C2×C4×D47D4semidirect
154. #398D47semidirect
161. C47#2618semidirect