Finite Group Databasebeta

← #20249134

Subgroups — #20249134

Order 6016.

632 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 157 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×#13094007C211yesmaximal, characteristic, complemented
2C2×#13093996C211yesmaximal, characteristic
3C22×C4⋊Dic47C211yesmaximal, characteristic, complemented
4#15689056C212yesmaximal, complemented
5#15689046C212yesmaximal, complemented
6#15689045C212yesmaximal, complemented
7#15689039C212yesmaximal, complemented
8#15689003C211yesmaximal, characteristic, complemented
9#15689000C211yesmaximal, characteristic, complemented
10#15688981C211yesmaximal, characteristic, complemented
11C2×#123×C47C211yesmaximal, characteristic, complemented
12C22×C22⋊D47C2211yescharacteristic
13C22×C22⋊D47C2211yescharacteristic
14C23×Dic47C2211yescharacteristic
15C23×Dic47C2211yescharacteristic
16#13094007—24no—
17#13093996—24no—
18C2×C4⋊Dic47—24no—
19C2×C4⋊Dic47C2212yes—
20C2×C4⋊Dic47C2212yes—
21C2×Dic47⋊C4C2212yes—
22C2×Dic47⋊C4C2211yescharacteristic
23C2×Dic47⋊C4C2212yes—
24C2×C22⋊Dic47C2211yescharacteristic
25C2×C22⋊Dic47C2211yescharacteristic
26C2×C22⋊Dic47C2212yes—
27C2×(C2×D47)⋊C4C2212yes—
28C2×(C2×D47)⋊C4C2211yescharacteristic
29C2×(C2×D47)⋊C4C2212yes—
30C47⋊#95C2212yes—
31C47⋊#95C2212yes—
32C47⋊#95C2212yes—
33C22×D4×C47C2211yescharacteristic
34C23×C4×C47C2211yescharacteristic
35#123×C47—24no—
36C2×C4⋊C4×C47C2212yes—
37C2×C22⋊C4×C47C2211yescharacteristic
38C2×C22⋊C4×C47C2212yes—
39C24×C47C2311yescharacteristic
40C24×C47C2311yescharacteristic
41C23×D47C2311yescharacteristic
42C2×D4×C47—22no—
43C2×D4×C47—22no—
44C2×D4×C47—22no—
45C22×C4×C47—22no—
46C22×C4×C47—22no—
47C22×C4×C47D414yescomplemented
48C22×C4×C47C2312yes—
49C22×C4×C47C2311yescharacteristic
50C22×C4×C47C2312yes—
51C2×C22⋊D47—22no—
52C2×C22⋊D47—22no—
53C2×C22⋊D47—22no—
54C2×C22⋊D47—22no—
55C2×C22⋊D47—42no—
56C22×Dic47—22no—
57C22×Dic47—42no—
58C22×Dic47—42no—
59C22×Dic47D414yes—
60C22×Dic47C2312yes—
61C22×Dic47C2312yes—
62C22×Dic47C2311yescharacteristic
63C22×Dic47C2311yescharacteristic
64C22×Dic47C2311yescharacteristic
65C4⋊C4×C47—24no—
66C22⋊C4×C47—22no—
67C22⋊C4×C47—24no—
68C22⋊Dic47—22no—
69C22⋊Dic47—22no—
70C22⋊Dic47—42no—
71(C2×D47)⋊C4—24no—
72(C2×D47)⋊C4—22no—
73(C2×D47)⋊C4—42no—
74C4⋊Dic47—24no—
75C4⋊Dic47—24no—
76Dic47⋊C4—24no—
77Dic47⋊C4—22no—
78Dic47⋊C4—42no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—21no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47—22no—
85C23×C47—22no—
86C23×C47C2×D412yes—
87C23×C47C2×D412yes—
88C23×C47#4112yes—
89C23×C47C2411yescharacteristic
90C22×D47—21no—
91C22×D47—21no—
92C22×D47—21no—
93C22×D47—22no—
94C22×D47—22no—
95D4×C47—44no—
96C2×C4×C47—28no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47C2×D414yes—
106C22⋊D47—44no—
107C22⋊D47—44no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—21no—
111C2×Dic47—22no—
112C2×Dic47—21no—
113C2×Dic47—22no—
114C2×Dic47—21no—
115C2×Dic47—22no—
116C2×Dic47—22no—
117C2×Dic47—22no—
118C2×Dic47—22no—
119C2×Dic47—44no—
120C2×Dic47—44no—
121C2×Dic47C2×D412yes—
122C2×Dic47C2×D412yes—
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—24no—
126C22×C47—24no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47—42no—
131C22×C47—42no—
132C22×C47#12114yes—
133C22×C47C22×D411yescharacteristic
134C22×C47C22×D411yescharacteristic
135C22×C47C2×#4111yescharacteristic
136C22×C47H(5,𝔽2)12yes—
137C22×C472−1+412yes—
138C2×D47—41no—
139C2×D47—41no—
140C2×D47—41no—
141C2×D47—42no—
142C2×D47—42no—
143C4×C47—24no—
144C4×C47—41no—
145C4×C47—42no—
146Dic47—22no—
147Dic47—22no—
148Dic47—41no—
149Dic47—42no—
150Dic47—42no—
151#2618—471nomaximal, Sylow
152C2×C47—24no—
153C2×C47—81no—
154C2×C47C2×#12111yescharacteristic
155C2×C47#53411yescharacteristic
156C2×C47#53611yescharacteristic
157C2×C47#53712yes—
158C2×C47#53912yes—
159D47—81no—
160C2×#123—471no—
161C2×#123D4711yescharacteristic, complemented
162C2×#121—471no—
163C22×C4⋊C4—471no—
164#398—472no—
165#396—472no—
166#396—472no—
167#394—472no—
168#386—471no—
169#386—471no—
170#380—471no—
171C47#261811yescharacteristic, complemented, Sylow
172C22×D4—471no—
173C22×D4—471no—
174C22×D4C2×D4711yescharacteristic
175C23×C4—471no—
176C23×C4—471no—
177C23×C4C2×D4711yescharacteristic
178#123—24no—
179#123—944no—
180#121—944no—
181C2×C4⋊C4—472no—
182C2×C4⋊C4—472no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—472no—
185C2×C4⋊C4—472no—
186C2×C4⋊C4—944no—
187C2×C4⋊C4C2×D4712yes—
188C2×C22⋊C4—471no—
189C2×C22⋊C4—471no—
190C2×C22⋊C4—471no—
191C2×C22⋊C4—472no—
192C2×C22⋊C4—472no—
193C2×C22⋊C4—472no—
194C2×C22⋊C4C2×D4711yescharacteristic
195C2×C22⋊C4C2×D4712yes—
196#95—472no—
197#95—472no—
198#95—472no—
199C24—471no—
200C24C22×D4711yescharacteristic
201C24C22×D4711yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—942no—
206C2×D4—942no—
207C2×D4—942no—
208C2×D4—942no—
209C2×D4—1882no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—471no—
213C22×C4—471no—
214C22×C4—474no—
215C22×C4—472no—
216C22×C4—472no—
217C22×C4—471no—
218C22×C4—942no—
219C22×C4—1882no—
220C22×C4—1882no—
221C22×C4C22⋊D4714yescomplemented
222C22×C4C22×D4712yes—
223C22×C4C22×D4711yescharacteristic
224C22×C4C22×D4712yes—
225C4⋊C4—24no—
226C4⋊C4—942no—
227C4⋊C4—944no—
228C4⋊C4—944no—
229C4⋊C4—944no—
230C4⋊C4—1882no—
231C22⋊C4—22no—
232C22⋊C4—24no—
233C22⋊C4—942no—
234C22⋊C4—942no—
235C22⋊C4—942no—
236C22⋊C4—944no—
237C22⋊C4—1882no—
238C22⋊C4—1882no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—941no—
247C23—941no—
248C23—941no—
249C23—942no—
250C23—942no—
251C23D4×D4712yes—
252C23#1062535212yes—
253C23C2×C22⋊D4712yes—
254C23C23×D4711yescharacteristic
255D4—44no—
256D4—1884no—
257D4—1884no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—472no—
268C2×C4—472no—
269C2×C4—942no—
270C2×C4—942no—
271C2×C4—941no—
272C2×C4—942no—
273C2×C4—942no—
274C2×C4—941no—
275C2×C4—941no—
276C2×C4—942no—
277C2×C4—942no—
278C2×C4—942no—
279C2×C4—942no—
280C2×C4—1884no—
281C2×C4—1884no—
282C2×C4C2×C22⋊D4714yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—1881no—
293C22—1881no—
294C22—1881no—
295C22—1882no—
296C22—1882no—
297C22#1309399614yes—
298C22C2×D4×D4711yescharacteristic
299C22C22×C22⋊D4711yescharacteristic
300C22C2×#1062535211yescharacteristic
301C22#1309407012yes—
302C22#1309407112yes—
303C4—24no—
304C4—41no—
305C4—42no—
306C4—942no—
307C4—942no—
308C4—1881no—
309C4—1882no—
310C4—1882no—
311C2—24no—
312C2—81no—
313C2—3761no—
314C2C2×#1309399611yescharacteristic
315C2#1568968411yescharacteristic
316C2#1568969511yescharacteristic
317C2#1568970112yes—
318C2#1568971712yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#123×C47C2
hypercenterC2×#123D47
hypocenterC47#2618
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13094007C2semidirect
3. C22×C4⋊Dic47C2semidirect
4. #15689056C2semidirect
5. #15689046C2semidirect
6. #15689045C2semidirect
7. #15689039C2semidirect
8. #15689003C2semidirect
9. #15689000C2semidirect
10. #15688981C2semidirect
11. C2×#123×C47C2semidirect
47. C22×C4×C47D4semidirect
161. C2×#123D47semidirect
171. C47#2618semidirect
221. C22×C4C22⋊D47semidirect