Finite Group Databasebeta

← #20281276

Subgroups — #20281276

Order 6784.

680 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133198C211yesmaximal, characteristic, complemented
2#16810551C212yesmaximal, complemented
3#16810546C212yesmaximal, complemented
4#16810537C212yesmaximal, complemented
5#16810526C211yesmaximal, characteristic, complemented
6#16810517C212yesmaximal, complemented
7#16810503C212yesmaximal, complemented
8#16810479C212yesmaximal, complemented
9C2×#120×C53C211yesmaximal, characteristic
10C23×Dic53C2211yescharacteristic, complemented
11#13133198—24no—
12C2×C4⋊Dic53C2211yescharacteristic, complemented
13C2×#11719919C2212yescomplemented
14C2×#11719919C2212yescomplemented
15C2×C22⋊Dic53C2212yes—
16C2×C22⋊Dic53C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2211yescharacteristic, complemented
22C2×C22⋊Dic53C2212yescomplemented
23C2×C4×Dic53C2211yescharacteristic, complemented
24#13133124C2212yescomplemented
25#13133124C2212yescomplemented
26#13133124C2212yescomplemented
27C25×C53C2211yescharacteristic
28C22×D4×C53C2212yes—
29C22×D4×C53C2211yescharacteristic
30#120×C53—24no—
31C2×C22⋊C4×C53C2212yes—
32C2×C22⋊C4×C53C2211yescharacteristic
33C24×C53—22no—
34C24×C53—24no—
35C24×C53—41no—
36C24×C53—42no—
37C24×C53—41no—
38C24×C53C2311yescharacteristic
39C24×C53C2311yescharacteristic
40C24×C53C2312yes—
41C2×D4×C53—24no—
42C2×D4×C53—24no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53—42no—
46C2×D4×C53D414yes—
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C22×Dic53—22no—
50C22×Dic53—41no—
51C22×Dic53—41no—
52C22×Dic53C2312yes—
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2312yes—
56C22⋊C4×C53—22no—
57C22⋊C4×C53—24no—
58C22⋊Dic53—24no—
59C22⋊Dic53—22no—
60C22⋊Dic53—24no—
61C22⋊Dic53—42no—
62C22⋊Dic53—42no—
63C22⋊Dic53—42no—
64C22⋊Dic53—42no—
65C22⋊Dic53—42no—
66C4⋊Dic53—41no—
67#11719919—24no—
68#11719919—42no—
69C4×Dic53—41no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—48no—
85C23×C53—44no—
86C23×C53—44no—
87C23×C53—48no—
88C23×C53—44no—
89C23×C53C2×D412yes—
90C23×C53C2×D412yes—
91C23×C53#4114yes—
92C23×C53C2411yescharacteristic
93D4×C53—44no—
94D4×C53—48no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53C2×D412yes—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—44no—
114C22×C53—22no—
115C22×C53—22no—
116C22×C53—24no—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53—24no—
120C22×C53—24no—
121C22×C53—416no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—42no—
132C22×C53C22×D411yescharacteristic
133C22×C53C2×#4112yes—
134C22×C53H(5,𝔽2)11yescharacteristic
135C22×C53H(5,𝔽2)11yescharacteristic
136C22×C53H(5,𝔽2)12yes—
137C4×C53—22no—
138C4×C53—42no—
139Dic53—42no—
140Dic53—42no—
141Dic53—42no—
142Dic53—42no—
143#2697—531nomaximal, Sylow
144C2×C53—42no—
145C2×C53—42no—
146C2×C53—44no—
147C2×C53#53411yescharacteristic
148C2×C53#53812yes—
149C2×C53#54612yes—
150C2×C53#55111yescharacteristic
151C2×C53#56011yescharacteristic
152C2×#123—531no—
153C2×#120D5311yescharacteristic
154#397—532no—
155#396—532no—
156#394—532no—
157#390—531no—
158#388—532no—
159#386—532no—
160#379—532no—
161C53#269711yescharacteristic, complemented, Sylow
162C25C2×D5311yescharacteristic
163C22×D4C2×D5312yes—
164C22×D4C2×D5311yescharacteristic
165C23×C4—531no—
166#123—1064no—
167#120—24no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—531no—
171C2×C22⋊C4—532no—
172C2×C22⋊C4—531no—
173C2×C22⋊C4—532no—
174C2×C22⋊C4—532no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4C2×D5312yes—
180C2×C22⋊C4C2×D5311yescharacteristic
181C2×C42—531no—
182#95—532no—
183#95—532no—
184#95—532no—
185C24—24no—
186C24—22no—
187C24—42no—
188C24—41no—
189C24—41no—
190C24C22×D5312yes—
191C24C22×D5311yescharacteristic
192C24C22×D5311yescharacteristic
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—42no—
198C2×D4C22⋊D5314yes—
199C22×C4—532no—
200C22×C4—532no—
201C22×C4—532no—
202C22×C4—532no—
203C22×C4—1062no—
204C22×C4—2121no—
205C22×C4—2121no—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C4⋊C4—1064no—
209C4⋊C4—2122no—
210C4⋊C4—2121no—
211C22⋊C4—22no—
212C22⋊C4—24no—
213C22⋊C4—1064no—
214C22⋊C4—1062no—
215C22⋊C4—1064no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C42—2121no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—48no—
237C23—44no—
238C23—44no—
239C23—48no—
240C23—44no—
241C23#1171994714yes—
242C23C2×C22⋊D5312yes—
243C23C2×C22⋊D5312yes—
244C23C23×D5311yescharacteristic
245D4—44no—
246D4—48no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—2124no—
265C2×C4C2×C22⋊D5312yes—
266C22—24no—
267C22—24no—
268C22—24no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—416no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22C22×C22⋊D5311yescharacteristic
285C22C2×#1171994712yes—
286C22#1313326212yes—
287C22#1313326211yescharacteristic
288C22#1313326211yescharacteristic
289C4—22no—
290C4—42no—
291C4—2122no—
292C4—2122no—
293C4—2122no—
294C4—2122no—
295C2—44no—
296C2—42no—
297C2—42no—
298C2#1681118211yescharacteristic
299C2#1681120812yes—
300C2#1681126712yes—
301C2#1681130411yescharacteristic
302C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2697
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#13133198C2semidirect
2. #16810551C2semidirect
3. #16810546C2semidirect
4. #16810537C2semidirect
5. #16810526C2semidirect
6. #16810517C2semidirect
7. #16810503C2semidirect
8. #16810479C2semidirect
10. C23×Dic53C22semidirect
12. C2×C4⋊Dic53C22semidirect
13. C2×#11719919C22semidirect
14. C2×#11719919C22semidirect
21. C2×C22⋊Dic53C22semidirect
22. C2×C22⋊Dic53C22semidirect
23. C2×C4×Dic53C22semidirect
24. #13133124C22semidirect
25. #13133124C22semidirect
26. #13133124C22semidirect
161. C53#2697semidirect