Finite Group Databasebeta

← #20279205

Subgroups — #20279205

Order 6784.

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

normal subgroups only

This group has 596 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
1C2×#13133189C211yesmaximal, characteristic
2C22×#11719919C211yesmaximal, characteristic, complemented
3#16810552C212yesmaximal, complemented
4#16810551C212yesmaximal, complemented
5#16810523C212yesmaximal, complemented
6#16810503C211yesmaximal, characteristic, complemented
7#16810502C211yesmaximal, characteristic, complemented
8#16810500C211yesmaximal, characteristic, complemented
9#16810487C212yesmaximal, complemented
10C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
11C53×#386C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16#13133189—24no—
17C2×#11719919—24no—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×D4×C53C2211yescharacteristic
33C23×C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2212yes—
37C24×C53C2311yescharacteristic
38C24×C53C2311yescharacteristic
39C23×D53C2311yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—42no—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C22×C4×C53C2312yes—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—24no—
53C22×Dic53—24no—
54C22×Dic53—22no—
55C22×Dic53—42no—
56C22×Dic53D414yes—
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2312yes—
59C22×Dic53C2312yes—
60C22×Dic53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22⋊C4×C53—22no—
63C22⋊C4×C53—42no—
64C22⋊Dic53—22no—
65C22⋊Dic53—22no—
66C22⋊Dic53—42no—
67#11719921—22no—
68#11719921—42no—
69#11719921—42no—
70#11719919—24no—
71#11719919—24no—
72#11719919—22no—
73C4×Dic53—24no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—22no—
81C23×C53C2×D412yes—
82C23×C53#4112yes—
83C23×C53#4112yes—
84C23×C53C2411yescharacteristic
85C22×D53—21no—
86C22×D53—22no—
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—22no—
90D4×C53—44no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—44no—
101C22⋊D53—44no—
102C22⋊D53—44no—
103C2×Dic53—22no—
104C2×Dic53—28no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—44no—
115C2×Dic53C2×D412yes—
116C2×Dic53C2×D412yes—
117C2×Dic53#4114yes—
118C22×C53—22no—
119C22×C53—22no—
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53#12414yes—
128C22×C53C22×D411yescharacteristic
129C22×C53C2×#4112yes—
130C22×C53C2×#4111yescharacteristic
131C22×C53C2×#4111yescharacteristic
132C22×C53H(5,𝔽2)12yes—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—41no—
136C2×D53—42no—
137C2×D53—42no—
138C4×C53—41no—
139C4×C53—42no—
140C4×C53—42no—
141Dic53—22no—
142Dic53—41no—
143Dic53—42no—
144Dic53—42no—
145Dic53#12314yes—
146#2504—531nomaximal, Sylow
147C2×C53—24no—
148C2×C53—81no—
149C2×C53C2×#12311yescharacteristic
150C2×C53C2×#12411yescharacteristic
151C2×C53#53511yescharacteristic
152C2×C53#55312yes—
153C2×C53#55312yes—
154D53—81no—
155C2×#121—531no—
156C22×C4⋊C4—531no—
157#397—532no—
158#397—532no—
159#390—532no—
160#386—531no—
161#386—531no—
162#386—531no—
163#386D5311yescharacteristic, complemented
164#382—532no—
165C4×C22⋊C4—531no—
166C53#250411yescharacteristic, complemented, Sylow
167C22×D4—531no—
168C22×D4—531no—
169C22×D4C2×D5311yescharacteristic
170C23×C4—531no—
171C23×C4—531no—
172C23×C4C2×D5311yescharacteristic
173#121—1064no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1064no—
178C2×C22⋊C4—531no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4C2×D5311yescharacteristic
185C2×C22⋊C4C2×D5312yes—
186C2×C42—532no—
187#95—532no—
188#95—532no—
189#95—532no—
190#95—532no—
191#95C2×D5312yes—
192C24—531no—
193C24C22×D5311yescharacteristic
194C24C22×D5311yescharacteristic
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—1062no—
199C2×D4—1064no—
200C2×D4—1062no—
201C2×D4—1062no—
202C2×D4—1062no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—531no—
206C22×C4—534no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—532no—
210C22×C4—532no—
211C22×C4—1064no—
212C22×C4—1062no—
213C22×C4—2122no—
214C22×C4C22×D5311yescharacteristic
215C22×C4C22×D5312yes—
216C22×C4C22×D5312yes—
217C4⋊C4—1064no—
218C4⋊C4—1064no—
219C4⋊C4—1062no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—1062no—
223C22⋊C4—1062no—
224C22⋊C4—1062no—
225C22⋊C4—2122no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C42—1064no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—1061no—
237C23—1062no—
238C23—1061no—
239C23—1061no—
240C23—1062no—
241C23#1171994512yes—
242C23D4×D5312yes—
243C23#1171994712yes—
244C23C23×D5311yescharacteristic
245D4—44no—
246D4—2124no—
247D4—2124no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—532no—
259C2×C4—534no—
260C2×C4—532no—
261C2×C4—1068no—
262C2×C4—1062no—
263C2×C4—1061no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1061no—
267C2×C4—1062no—
268C2×C4—1061no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—2124no—
273C22—24no—
274C22—24no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—2122no—
283C22—2122no—
284C22—2121no—
285C22—2121no—
286C22—2121no—
287C22#1313320314yes—
288C22C2×D4×D5311yescharacteristic
289C22#41×D5312yes—
290C22C2×#1171994711yescharacteristic
291C22C2×#1171994511yescharacteristic
292C22#1313326212yes—
293C4—42no—
294C4—41no—
295C4—42no—
296C4—534no—
297C4—1062no—
298C4—2122no—
299C4—2121no—
300C4—2122no—
301C2—24no—
302C2—81no—
303C2—4241no—
304C2#123×D5311yescharacteristic
305C2C2×#1313320311yescharacteristic
306C2#1681118811yescharacteristic
307C2#1681131612yes—
308C2#1681132212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2504
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#11719919C2semidirect
3. #16810552C2semidirect
4. #16810551C2semidirect
5. #16810523C2semidirect
6. #16810503C2semidirect
7. #16810502C2semidirect
8. #16810500C2semidirect
9. #16810487C2semidirect
10. C22⋊C4×Dic53C2semidirect
11. C53×#386C2semidirect
163. #386D53semidirect
166. C53#2504semidirect