Finite Group Databasebeta

← #20280843

Subgroups — #20280843

Order 6784.

598 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133191C211yesmaximal, characteristic, complemented
2#16810553C211yesmaximal, characteristic, complemented
3#16810548C211yesmaximal, characteristic, complemented
4#16810540C211yesmaximal, characteristic, complemented
5#16810538C212yesmaximal, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810531C212yesmaximal, complemented
8#16810523C212yesmaximal, complemented
9#16810513C212yesmaximal, complemented
10C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
11C2×#123×C53C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2212yes—
14C22×C22⋊D53C2212yes—
15#13133191—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×C4×Dic53C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29C22×D4×C53C2211yescharacteristic
30C23×C4×C53C2211yescharacteristic
31#123×C53—24no—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C24×C53C2311yescharacteristic
39C23×D53C2312yes—
40C2×D4×C53—22no—
41C2×D4×C53—42no—
42C22×C4×C53—42no—
43C22×C4×C53D414yescomplemented
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—42no—
54C2×C22⋊D53—42no—
55C22×Dic53C2312yes—
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C2×D212—22no—
59C2×D212—42no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—22no—
62C22⋊C4×C53—22no—
63C22⋊C4×C53—22no—
64C22⋊C4×C53—22no—
65C22⋊Dic53—24no—
66C22⋊Dic53—42no—
67#11719921—24no—
68#11719921—24no—
69#11719921—42no—
70#11719921—42no—
71C4⋊Dic53—22no—
72#11719919—42no—
73C4×Dic53—42no—
74C23×C53—22no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—22no—
78C23×C53—21no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—22no—
82C23×C53—21no—
83C23×C53C2×D412yes—
84C23×C53C2411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92D4×C53—44no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—44no—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D412yes—
105C22⋊D53—48no—
106C22⋊D53—48no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53#4114yes—
116D212—44no—
117C22×C53—22no—
118C22×C53—22no—
119C22×C53—22no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53C22×D411yescharacteristic
128C22×C53C2×#4112yes—
129C22×C53H(5,𝔽2)12yes—
130C22×C53H(5,𝔽2)11yescharacteristic
131C22×C53H(5,𝔽2)11yescharacteristic
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144Dic53—24no—
145Dic53—42no—
146Dic53—42no—
147#2661—531nomaximal, Sylow
148C2×C53—42no—
149C2×C53—81no—
150C2×C53#53211yescharacteristic
151C2×C53#53411yescharacteristic
152C2×C53#53511yescharacteristic
153C2×C53#53812yes—
154C2×C53#54012yes—
155D53—82no—
156C2×#123D5311yescharacteristic, complemented
157C2×#121—531no—
158#398—531no—
159#397—531no—
160#394—531no—
161#394—532no—
162#392—532no—
163#392—531no—
164#390—532no—
165#388—532no—
166C4×C22⋊C4—531no—
167C53#266111yescharacteristic, complemented, Sylow
168C22×D4—532no—
169C22×D4—532no—
170C22×D4—531no—
171C22×D4C2×D5311yescharacteristic
172C23×C4C2×D5311yescharacteristic
173#123—24no—
174#121—1064no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—532no—
177C2×C4⋊C4C2×D5311yescharacteristic
178C2×C4⋊C4C2×D5311yescharacteristic
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4C2×D5311yescharacteristic
186C2×C22⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4C2×D5311yescharacteristic
188C2×C42—532no—
189#95—531no—
190#95—531no—
191#95—532no—
192#95—532no—
193C24—532no—
194C24C22×D5311yescharacteristic
195C24C22×D5311yescharacteristic
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—1064no—
199C2×D4—1064no—
200C2×D4—1062no—
201C2×D4—2122no—
202C2×D4—2122no—
203C2×D4—2122no—
204C2×D4—2122no—
205C2×D4—2122no—
206C22×C4—42no—
207C22×C4—532no—
208C22×C4—532no—
209C22×C4—532no—
210C22×C4C22⋊D5314yescomplemented
211C22×C4C22×D5311yescharacteristic
212C22×C4C22×D5311yescharacteristic
213C22×C4C22×D5311yescharacteristic
214C22×C4C22×D5311yescharacteristic
215C22×C4C22×D5311yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—1062no—
219C4⋊C4—2122no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—1064no—
224C22⋊C4—1064no—
225C22⋊C4—1064no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C42—2122no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—1062no—
240C23—1062no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23C2×C22⋊D5312yes—
247C23C23×D5311yescharacteristic
248D4—44no—
249D4—2124no—
250D4—2128no—
251D4—2128no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—534no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4C2×C22⋊D5312yes—
272C2×C4C2×C22⋊D5312yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—2122no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22C22×C22⋊D5311yescharacteristic
291C22#41×D5312yes—
292C22#1313326212yes—
293C22#1313326311yescharacteristic
294C22#1313326311yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—1064no—
301C4—2122no—
302C4—2122no—
303C2—42no—
304C2—81no—
305C2—4242no—
306C2#1681116911yescharacteristic
307C2#1681118211yescharacteristic
308C2#1681118711yescharacteristic
309C2#1681120912yes—
310C2#1681122512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2661
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13133191C2semidirect
2. #16810553C2semidirect
3. #16810548C2semidirect
4. #16810540C2semidirect
5. #16810538C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810523C2semidirect
9. #16810513C2semidirect
10. C4×C22⋊Dic53C2semidirect
11. C2×#123×C53C2semidirect
43. C22×C4×C53D4semidirect
156. C2×#123D53semidirect
167. C53#2661semidirect
210. C22×C4C22⋊D53semidirect