Finite Group Databasebeta

← #20248954

Subgroups — #20248954

Order 6016.

604 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 145 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×C4×D188C211yesmaximal, characteristic, complemented
2#15689062C211yesmaximal, characteristic, complemented
3#15689059C211yesmaximal, characteristic, complemented
4#15689040C211yesmaximal, characteristic, complemented
5#15689032C211yesmaximal, characteristic
6#15689014C212yesmaximal, complemented
7#15689002C212yesmaximal, complemented
8#15688999C212yesmaximal, complemented
9#15688994C212yesmaximal, complemented
10#15688991C211yesmaximal, characteristic, complemented
11C2×#117×C47C211yesmaximal, characteristic, complemented
12C22×D188C2211yescharacteristic
13C22×C22⋊D47C2212yes—
14C22×C4×D47C2212yes—
15C4×D188—24no—
16C2×C4⋊Dic47C2211yescharacteristic
17C2×Dic47⋊C4C2212yes—
18C2×C22⋊Dic47C2212yes—
19C2×(C2×D47)⋊C4C2212yes—
20C2×(C2×D47)⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C2×(C2×D47)⋊C4C2212yes—
23C47⋊#95C2212yes—
24C47⋊#95C2212yes—
25C47⋊#95C2211yescharacteristic
26C47⋊#95C2211yescharacteristic
27C47⋊#95C2212yes—
28C47⋊#95C2212yes—
29C23×C4×C47C2211yescharacteristic
30#117×C47—24no—
31C2×C4⋊C4×C47C2211yescharacteristic
32C2×C4⋊C4×C47C2211yescharacteristic
33C2×C22⋊C4×C47C2211yescharacteristic
34C2×C22⋊C4×C47C2211yescharacteristic
35C2×C42×C47C2211yescharacteristic
36C2×C42×C47C2211yescharacteristic
37C24×C47C2311yescharacteristic
38C23×D47C2312yes—
39C22×C4×C47—22no—
40C22×C4×C47—22no—
41C22×C4×C47—22no—
42C22×C4×C47C2311yescharacteristic
43C22×C4×C47C2311yescharacteristic
44C22×C4×C47C2311yescharacteristic
45C22×C4×C47C2311yescharacteristic
46C22×C4×C47C2311yescharacteristic
47C22×C4×C47C2311yescharacteristic
48C2×C22⋊D47—24no—
49C2×C22⋊D47—24no—
50C2×C22⋊D47—42no—
51C22×Dic47C2312yes—
52C22×Dic47C2312yes—
53C22×Dic47C2312yes—
54C2×D188—24no—
55C2×D188—22no—
56C2×C4×D47—24no—
57C2×C4×D47—42no—
58C2×C4×D47—42no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—22no—
61C22⋊C4×C47—22no—
62C22⋊C4×C47—22no—
63C42×C47—22no—
64C42×C47D414yescomplemented
65C22⋊Dic47—42no—
66(C2×D47)⋊C4—24no—
67(C2×D47)⋊C4—24no—
68(C2×D47)⋊C4—24no—
69(C2×D47)⋊C4—42no—
70C4⋊Dic47—22no—
71Dic47⋊C4—42no—
72C23×C47—21no—
73C23×C47—22no—
74C23×C47—21no—
75C23×C47—22no—
76C23×C47#4112yes—
77C23×C47C2411yescharacteristic
78C22×D47—22no—
79C22×D47—22no—
80C22×D47—22no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47—22no—
84C22×D47#4114yes—
85C2×C4×C47—28no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47C2×D412yes—
97C2×C4×C47C2×D412yes—
98C2×C4×C47C2×D412yes—
99C2×C4×C47#4112yes—
100C22⋊D47—48no—
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—
110D188—28no—
111C4×D47—48no—
112C22×C47—22no—
113C22×C47—24no—
114C22×C47—41no—
115C22×C47—41no—
116C22×C47—42no—
117C22×C47C22×D411yescharacteristic
118C22×C47C2×#4112yes—
119C22×C47C2×#4111yescharacteristic
120C22×C47C2×#4111yescharacteristic
121C22×C47H(5,𝔽2)12yes—
122C2×D47—24no—
123C2×D47—24no—
124C2×D47—24no—
125C2×D47—42no—
126C2×D47—42no—
127C2×D47—42no—
128C2×D47—42no—
129C4×C47—22no—
130C4×C47—22no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—41no—
134C4×C47—41no—
135Dic47—42no—
136Dic47—42no—
137Dic47—42no—
138#2598—471nomaximal, Sylow
139C2×C47—42no—
140C2×C47#52511yescharacteristic
141C2×C47#52511yescharacteristic
142C2×C47#53511yescharacteristic
143C2×C47#55112yes—
144C2×C47#55312yes—
145D47—44no—
146C2×C4×D4—471no—
147C2×#117D4711yescharacteristic, complemented
148#400—471no—
149#399—471no—
150#394—471no—
151#392—471no—
152#388—472no—
153#386—472no—
154#386—472no—
155#385—472no—
156#383—471no—
157C47#259811yescharacteristic, complemented, Sylow
158C22×D4—472no—
159C22×D4—471no—
160C23×C4—472no—
161C23×C4C2×D4711yescharacteristic
162C4×D4—944no—
163#117—24no—
164C2×C4⋊C4—471no—
165C2×C4⋊C4—472no—
166C2×C4⋊C4C2×D4711yescharacteristic
167C2×C4⋊C4C2×D4711yescharacteristic
168C2×C22⋊C4—472no—
169C2×C22⋊C4—472no—
170C2×C22⋊C4—472no—
171C2×C22⋊C4—472no—
172C2×C22⋊C4—472no—
173C2×C22⋊C4C2×D4711yescharacteristic
174C2×C22⋊C4C2×D4711yescharacteristic
175C2×C42C2×D4711yescharacteristic
176C2×C42C2×D4711yescharacteristic
177#95—472no—
178#95—472no—
179#95—471no—
180#95—471no—
181#95—472no—
182#95—472no—
183C24—472no—
184C24C22×D4711yescharacteristic
185C2×D4—944no—
186C2×D4—944no—
187C2×D4—944no—
188C2×D4—942no—
189C2×D4—1882no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—472no—
194C22×C4—472no—
195C22×C4—472no—
196C22×C4—944no—
197C22×C4—1882no—
198C22×C4—1882no—
199C22×C4C22×D4711yescharacteristic
200C22×C4C22×D4711yescharacteristic
201C22×C4C22×D4711yescharacteristic
202C22×C4C22×D4711yescharacteristic
203C22×C4C22×D4711yescharacteristic
204C22×C4C22×D4711yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—942no—
208C4⋊C4—1882no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—944no—
212C22⋊C4—944no—
213C22⋊C4—944no—
214C22⋊C4—1882no—
215C22⋊C4—1882no—
216C42—22no—
217C42C22⋊D4714yescomplemented
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—474no—
223C23—942no—
224C23—942no—
225C23—942no—
226C23—942no—
227C23—942no—
228C23—942no—
229C23#1062535012yes—
230C23C23×D4711yescharacteristic
231D4—948no—
232D4—1888no—
233C2×C4—28no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—942no—
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—1888no—
254C2×C4#1062535012yes—
255C2×C4C2×C22⋊D4712yes—
256C2×C4C2×C22⋊D4712yes—
257C2×C4C2×C22⋊D4712yes—
258C22—24no—
259C22—22no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—944no—
264C22—944no—
265C22—944no—
266C22—1882no—
267C22—1882no—
268C22—1882no—
269C22—1882no—
270C22C22×C22⋊D4711yescharacteristic
271C22#41×D4712yes—
272C22C2×#1062535011yescharacteristic
273C22C2×#1062535011yescharacteristic
274C22#1309407012yes—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—1882no—
282C4—1882no—
283C4—1882no—
284C2—42no—
285C2—1884no—
286C2#1568963011yescharacteristic
287C2#1568963411yescharacteristic
288C2#1568968711yescharacteristic
289C2#1568979712yes—
290C2#1568981712yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#117×C47C2
hypercenterC2×#117D47
hypocenterC47#2598
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C4×D188C2semidirect
2. #15689062C2semidirect
3. #15689059C2semidirect
4. #15689040C2semidirect
6. #15689014C2semidirect
7. #15689002C2semidirect
8. #15688999C2semidirect
9. #15688994C2semidirect
10. #15688991C2semidirect
11. C2×#117×C47C2semidirect
64. C42×C47D4semidirect
147. C2×#117D47semidirect
157. C47#2598semidirect
217. C42C22⋊D47semidirect