Finite Group Databasebeta

← #20278995

Subgroups — #20278995

Order 6784.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#11719921C211yesmaximal, characteristic
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3#16810552C212yesmaximal, complemented
4#16810549C212yesmaximal, complemented
5#16810496C211yesmaximal, characteristic, complemented
6#16810494C212yesmaximal, complemented
7#16810485C212yesmaximal, complemented
8#16810484C211yesmaximal, characteristic, complemented
9#16810484C211yesmaximal, characteristic, complemented
10#16810478C211yesmaximal, characteristic
11C53×#385C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16C2×#11719919C2212yescomplemented
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2211yescharacteristic
19C2×#11719921—24no—
20C2×#11719921C2211yescharacteristic
21C2×#11719921C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C22⋊C4×D53—24no—
27#13133124C2212yescomplemented
28#13133124C2212yescomplemented
29#13133124C2212yescomplemented
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32C23×C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yescomplemented
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36#95×C53C2212yescomplemented
37C24×C53C2311yescharacteristic
38C23×D53—22no—
39C23×D53—22no—
40C23×D53—24no—
41C23×D53—42no—
42C23×D53D414yes—
43C23×D53C2311yescharacteristic
44C23×D53C2311yescharacteristic
45C22×C4×C53—22no—
46C22×C4×C53—42no—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2312yes—
49C22×C4×C53C2312yes—
50C22×C4×C53C2311yescharacteristic
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53C2312yes—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—24no—
61C2×C4×D53—42no—
62C4⋊C4×C53—42no—
63C22⋊C4×C53—22no—
64C22⋊C4×C53—22no—
65C22⋊Dic53—22no—
66C22⋊Dic53—22no—
67#11719921—22no—
68#11719921—22no—
69#11719921—24no—
70#11719921—24no—
71#11719921—24no—
72#11719921—42no—
73#11719919—42no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53C2×D412yes—
77C23×C53#4112yes—
78C23×C53#4112yes—
79C23×C53C2411yescharacteristic
80C22×D53—21no—
81C22×D53—21no—
82C22×D53—22no—
83C22×D53—28no—
84C22×D53—22no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—22no—
88C22×D53—28no—
89C22×D53—44no—
90C22×D53—48no—
91C22×D53—48no—
92C22×D53C2×D412yes—
93C22×D53C2×D412yes—
94C22×D53#4114yes—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—44no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—44no—
117C4×D53—44no—
118C4×D53—44no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53—24no—
122C22×C53—24no—
123C22×C53#12314yes—
124C22×C53C22×D411yescharacteristic
125C22×C53C2×#4112yes—
126C22×C53C2×#4111yescharacteristic
127C22×C53C2×#4111yescharacteristic
128C22×C53H(5,𝔽2)12yes—
129C2×D53—22no—
130C2×D53—22no—
131C2×D53—24no—
132C2×D53—24no—
133C2×D53—22no—
134C2×D53—22no—
135C2×D53—22no—
136C2×D53—416no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53#12314yes—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—42no—
143C4×C53—42no—
144Dic53—41no—
145Dic53—42no—
146Dic53—41no—
147Dic53—42no—
148#2483—531nomaximal, Sylow
149C2×C53—24no—
150C2×C53C2×#12311yescharacteristic
151C2×C53C2×#12311yescharacteristic
152C2×C53#53411yescharacteristic
153C2×C53#55112yes—
154C2×C53#55112yes—
155D53—24no—
156D53—42no—
157C22×C22⋊C4—531no—
158C22×C22⋊C4—531no—
159#397—532no—
160#397—532no—
161#385—531no—
162#385—532no—
163#385D5311yescharacteristic, complemented
164#381—531no—
165#381—531no—
166#381—532no—
167#379—531no—
168C53#248311yescharacteristic, complemented, Sylow
169C25—531no—
170C23×C4—531no—
171C23×C4—531no—
172C23×C4—531no—
173C23×C4C2×D5311yescharacteristic
174C2×C4⋊C4—532no—
175C2×C4⋊C4C2×D5312yescomplemented
176C2×C22⋊C4—531no—
177C2×C22⋊C4—531no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—531no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—1064no—
185C2×C22⋊C4—1064no—
186C2×C22⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4C2×D5311yescharacteristic
188#95—532no—
189#95—532no—
190#95—532no—
191#95—532no—
192#95—532no—
193#95C2×D5312yescomplemented
194C24—531no—
195C24—531no—
196C24—534no—
197C24—1064no—
198C24—1062no—
199C24—1062no—
200C24—2122no—
201C24C22×D5311yescharacteristic
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—531no—
205C22×C4—532no—
206C22×C4—532no—
207C22×C4—531no—
208C22×C4—1062no—
209C22×C4—1062no—
210C22×C4—1064no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—2122no—
215C22×C4C22×D5311yescharacteristic
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5312yes—
219C4⋊C4—42no—
220C4⋊C4—2122no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—1062no—
224C22⋊C4—1062no—
225C22⋊C4—1064no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—1062no—
229C22⋊C4—1062no—
230C22⋊C4—2122no—
231C23—22no—
232C23—22no—
233C23—532no—
234C23—534no—
235C23—532no—
236C23—1068no—
237C23—1061no—
238C23—1061no—
239C23—1062no—
240C23—1068no—
241C23—1062no—
242C23—1061no—
243C23—1061no—
244C23—1062no—
245C23—2124no—
246C23—2128no—
247C23—2128no—
248C23#1171994512yes—
249C23D4×D5312yes—
250C23#1171994912yes—
251C23C23×D5311yescharacteristic
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—2124no—
274C2×C4—2124no—
275C2×C4—2124no—
276C22—24no—
277C22—24no—
278C22—22no—
279C22—22no—
280C22—534no—
281C22—1064no—
282C22—1064no—
283C22—1062no—
284C22—1062no—
285C22—1062no—
286C22—1062no—
287C22—1062no—
288C22—21216no—
289C22—2122no—
290C22—2122no—
291C22#1313320214yes—
292C22C2×D4×D5311yescharacteristic
293C22#41×D5312yes—
294C22C2×#1171994511yescharacteristic
295C22C2×#1171994911yescharacteristic
296C22#1313326212yes—
297C4—42no—
298C4—41no—
299C4—42no—
300C4—41no—
301C4—2122no—
302C4—2121no—
303C4—2122no—
304C4—2121no—
305C2—24no—
306C2—1064no—
307C2—2122no—
308C2#123×D5311yescharacteristic
309C2C2×#1313320211yescharacteristic
310C2#1681118311yescharacteristic
311C2#1681130012yes—
312C2#1681130212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2483
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #16810552C2semidirect
4. #16810549C2semidirect
5. #16810496C2semidirect
6. #16810494C2semidirect
7. #16810485C2semidirect
8. #16810484C2semidirect
9. #16810484C2semidirect
11. C53×#385C2semidirect
16. C2×#11719919C22semidirect
27. #13133124C22semidirect
28. #13133124C22semidirect
29. #13133124C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
33. C2×C4⋊C4×C53C22semidirect
36. #95×C53C22semidirect
163. #385D53semidirect
168. C53#2483semidirect
175. C2×C4⋊C4C2×D53semidirect
193. #95C2×D53semidirect