Finite Group Databasebeta

← #20251390

Subgroups — #20251390

Order 6016.

578 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13093998C211yesmaximal, characteristic, complemented
2#15689062C211yesmaximal, characteristic, complemented
3#15689059C211yesmaximal, characteristic, complemented
4#15689045C212yesmaximal, complemented
5#15689040C211yesmaximal, characteristic, complemented
6#15689039C212yesmaximal, complemented
7#15689038C212yesmaximal, complemented
8#15689032C211yesmaximal, characteristic, complemented
9#15689031C212yesmaximal, complemented
10#15688983C211yesmaximal, characteristic, complemented
11C2×#123×C47C211yesmaximal, characteristic, complemented
12C22×D188C2211yescharacteristic
13C22×C22⋊D47C2212yes—
14C22×C22⋊D47C2212yes—
15#13093998—24no—
16C2×C4⋊Dic47C2211yescharacteristic
17C2×C4⋊Dic47C2212yes—
18C2×Dic47⋊C4C2212yes—
19C2×C22⋊Dic47C2212yes—
20C2×C22⋊Dic47C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C2×(C2×D47)⋊C4C2212yes—
23C2×(C2×D47)⋊C4C2212yes—
24C2×(C2×D47)⋊C4C2212yes—
25C47⋊#95C2211yescharacteristic
26C47⋊#95C2211yescharacteristic
27C47⋊#95C2212yes—
28C47⋊#95C2212yes—
29C22×D4×C47C2211yescharacteristic
30C23×C4×C47C2211yescharacteristic
31#123×C47—24no—
32C2×C4⋊C4×C47C2211yescharacteristic
33C2×C4⋊C4×C47C2211yescharacteristic
34C2×C22⋊C4×C47C2211yescharacteristic
35C2×C22⋊C4×C47C2211yescharacteristic
36C2×C22⋊C4×C47C2211yescharacteristic
37C24×C47C2311yescharacteristic
38C24×C47C2311yescharacteristic
39C23×D47C2312yes—
40C2×D4×C47—22no—
41C2×D4×C47—42no—
42C22×C4×C47—42no—
43C22×C4×C47D414yescomplemented
44C22×C4×C47C2311yescharacteristic
45C22×C4×C47C2311yescharacteristic
46C22×C4×C47C2311yescharacteristic
47C22×C4×C47C2311yescharacteristic
48C22×C4×C47C2311yescharacteristic
49C2×C22⋊D47—24no—
50C2×C22⋊D47—42no—
51C2×C22⋊D47—42no—
52C2×C22⋊D47—42no—
53C2×C22⋊D47—42no—
54C2×C22⋊D47—42no—
55C22×Dic47C2312yes—
56C22×Dic47C2312yes—
57C22×Dic47C2312yes—
58C2×D188—22no—
59C2×D188—42no—
60C4⋊C4×C47—22no—
61C4⋊C4×C47—22no—
62C22⋊C4×C47—22no—
63C22⋊C4×C47—22no—
64C22⋊C4×C47—22no—
65C22⋊Dic47—42no—
66C22⋊Dic47—42no—
67(C2×D47)⋊C4—24no—
68(C2×D47)⋊C4—42no—
69(C2×D47)⋊C4—42no—
70(C2×D47)⋊C4—42no—
71C4⋊Dic47—22no—
72C4⋊Dic47—42no—
73Dic47⋊C4—42no—
74C23×C47—21no—
75C23×C47—22no—
76C23×C47—22no—
77C23×C47—21no—
78C23×C47—21no—
79C23×C47—22no—
80C23×C47—21no—
81C23×C47—22no—
82C23×C47—21no—
83C23×C47C2×D412yes—
84C23×C47C2411yescharacteristic
85C22×D47—22no—
86C22×D47—22no—
87C22×D47—22no—
88C22×D47—22no—
89C22×D47—22no—
90C22×D47—22no—
91C22×D47—22no—
92D4×C47—44no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—44no—
103C2×C4×C47C2×D412yes—
104C2×C4×C47C2×D412yes—
105C22⋊D47—48no—
106C22⋊D47—48no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—22no—
111C2×Dic47—22no—
112C2×Dic47—22no—
113C2×Dic47—22no—
114C2×Dic47—22no—
115C2×Dic47—22no—
116D188—44no—
117C22×C47—22no—
118C22×C47—22no—
119C22×C47—22no—
120C22×C47—42no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47—41no—
124C22×C47—41no—
125C22×C47—42no—
126C22×C47—42no—
127C22×C47C22×D411yescharacteristic
128C22×C47H(5,𝔽2)12yes—
129C22×C47H(5,𝔽2)12yes—
130C22×C47H(5,𝔽2)11yescharacteristic
131C22×C472−1+411yescharacteristic
132C2×D47—42no—
133C2×D47—42no—
134C2×D47—42no—
135C2×D47—42no—
136C2×D47—42no—
137C2×D47—42no—
138C2×D47—42no—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—41no—
142C4×C47—41no—
143C4×C47—41no—
144Dic47—42no—
145Dic47—42no—
146Dic47—42no—
147#2813—471nomaximal, Sylow
148C2×C47—42no—
149C2×C47—81no—
150C2×C47#53411yescharacteristic
151C2×C47#53511yescharacteristic
152C2×C47#53711yescharacteristic
153C2×C47#56012yes—
154C2×C47#56212yes—
155D47—82no—
156C2×#123D4711yescharacteristic, complemented
157C2×#121—471no—
158#400—471no—
159#399—471no—
160#396—472no—
161#394—471no—
162#394—472no—
163#394—472no—
164#392—472no—
165#392—471no—
166#380—471no—
167C47#281311yescharacteristic, complemented, Sylow
168C22×D4—472no—
169C22×D4—472no—
170C22×D4—471no—
171C22×D4C2×D4711yescharacteristic
172C23×C4C2×D4711yescharacteristic
173#123—24no—
174#121—944no—
175C2×C4⋊C4—471no—
176C2×C4⋊C4—472no—
177C2×C4⋊C4—472no—
178C2×C4⋊C4C2×D4711yescharacteristic
179C2×C4⋊C4C2×D4711yescharacteristic
180C2×C22⋊C4—472no—
181C2×C22⋊C4—472no—
182C2×C22⋊C4—472no—
183C2×C22⋊C4—472no—
184C2×C22⋊C4—472no—
185C2×C22⋊C4—472no—
186C2×C22⋊C4C2×D4711yescharacteristic
187C2×C22⋊C4C2×D4711yescharacteristic
188C2×C22⋊C4C2×D4711yescharacteristic
189#95—471no—
190#95—471no—
191#95—472no—
192#95—472no—
193C24—472no—
194C24C22×D4711yescharacteristic
195C24C22×D4711yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—944no—
199C2×D4—942no—
200C2×D4—1882no—
201C2×D4—1882no—
202C2×D4—1882no—
203C2×D4—1882no—
204C2×D4—1882no—
205C2×D4—1882no—
206C22×C4—42no—
207C22×C4—472no—
208C22×C4—472no—
209C22×C4—472no—
210C22×C4C22⋊D4714yescomplemented
211C22×C4C22×D4711yescharacteristic
212C22×C4C22×D4711yescharacteristic
213C22×C4C22×D4711yescharacteristic
214C22×C4C22×D4711yescharacteristic
215C22×C4C22×D4711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—942no—
219C4⋊C4—1882no—
220C4⋊C4—1882no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—944no—
225C22⋊C4—1882no—
226C22⋊C4—1882no—
227C22⋊C4—1882no—
228C22⋊C4—1882no—
229C22⋊C4—1882no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—942no—
240C23—942no—
241C23—942no—
242C23—942no—
243C23—942no—
244C23—942no—
245C23—942no—
246C23C2×C22⋊D4712yes—
247C23C23×D4711yescharacteristic
248D4—44no—
249D4—1884no—
250D4—1888no—
251D4—1888no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—942no—
263C2×C4—942no—
264C2×C4—942no—
265C2×C4—942no—
266C2×C4—942no—
267C2×C4—942no—
268C2×C4—942no—
269C2×C4—942no—
270C2×C4—942no—
271C2×C4C2×C22⋊D4712yes—
272C2×C4C2×C22⋊D4712yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—1882no—
284C22—1882no—
285C22—1882no—
286C22—1882no—
287C22—1882no—
288C22—1882no—
289C22—1882no—
290C22C22×C22⋊D4711yescharacteristic
291C22#1309406912yes—
292C22#1309407011yescharacteristic
293C22#1309407012yes—
294C22#1309407111yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—1882no—
301C4—1882no—
302C4—1882no—
303C2—42no—
304C2—81no—
305C2—3762no—
306C2#1568968211yescharacteristic
307C2#1568968711yescharacteristic
308C2#1568970111yescharacteristic
309C2#1568986312yes—
310C2#1568987012yes—
subgroupG/H
center, FrattiniC23C23×D47
commutator, socleC23×C47C24
FittingC2×#123×C47C2
hypercenterC2×#123D47
hypocenterC47#2813
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13093998C2semidirect
2. #15689062C2semidirect
3. #15689059C2semidirect
4. #15689045C2semidirect
5. #15689040C2semidirect
6. #15689039C2semidirect
7. #15689038C2semidirect
8. #15689032C2semidirect
9. #15689031C2semidirect
10. #15688983C2semidirect
11. C2×#123×C47C2semidirect
43. C22×C4×C47D4semidirect
156. C2×#123D47semidirect
167. C47#2813semidirect
210. C22×C4C22⋊D47semidirect