Finite Group Databasebeta

← #20280456

Subgroups — #20280456

Order 6784.

632 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 143 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×#13133199C212yesmaximal, complemented
2C2×#13133188C211yesmaximal, characteristic
3C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810546C212yesmaximal, complemented
7#16810546C212yesmaximal, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810499C212yesmaximal, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2212yes—
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133199—28no—
17#13133188—24no—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2212yes—
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C4⋊C4×D53—24no—
26C2×C22⋊Dic53C2212yes—
27C2×C22⋊Dic53C2212yes—
28C2×C22⋊Dic53C2212yes—
29C2×#11719921C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C22×D4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2312yes—
39C23×D53C2311yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53—42no—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C22×Dic53C2312yes—
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2312yes—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—42no—
58C2×C4×D53—42no—
59C2×C4×D53D414yes—
60C2×C4×D53D414yescomplemented
61C4⋊C4×C53—22no—
62C22⋊C4×C53—24no—
63C22⋊C4×C53—42no—
64C22⋊Dic53—24no—
65C22⋊Dic53—24no—
66C22⋊Dic53—42no—
67#11719921—24no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—24no—
71#11719919—22no—
72#11719919—24no—
73#11719919—22no—
74#11719919—42no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53C2411yescharacteristic
83C22×D53—22no—
84C22×D53—22no—
85C22×D53C2×D412yes—
86C22×D53C2×D412yes—
87C22×D53#4112yes—
88D4×C53—44no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53C2×D412yes—
98C2×C4×C53C2×D412yes—
99C22⋊D53—48no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53C2×D412yes—
112C2×Dic53C2×D412yes—
113C4×D53—28no—
114C4×D53—44no—
115C4×D53—44no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53C22×D411yescharacteristic
124C22×C53C22×D411yescharacteristic
125C22×C53C2×#4111yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C22×C532−1+412yes—
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53#12114yes—
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—41no—
136C4×C53—42no—
137Dic53—22no—
138Dic53—22no—
139Dic53—42no—
140Dic53—41no—
141Dic53—42no—
142#2618—531nomaximal, Sylow
143C2×C53—82no—
144C2×C53C2×#12111yescharacteristic
145C2×C53#53411yescharacteristic
146C2×C53#53611yescharacteristic
147C2×C53#53712yes—
148C2×C53#53912yes—
149D53—24no—
150C2×#123—532no—
151C2×#121—531no—
152C22×C4⋊C4—531no—
153#398—531no—
154#398—531no—
155#396—532no—
156#396—532no—
157#394—531no—
158#394D5311yescharacteristic, complemented
159#386—532no—
160#380—531no—
161C53#261811yescharacteristic, complemented, Sylow
162C22×D4—532no—
163C22×D4C2×D5311yescharacteristic
164C23×C4—531no—
165C23×C4—531no—
166C23×C4—531no—
167#123—1068no—
168#121—1064no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—532no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4—1064no—
177C2×C4⋊C4C2×D5311yescharacteristic
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4—532no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4C2×D5312yes—
183C2×C22⋊C4C2×D5312yes—
184#95—531no—
185#95—532no—
186#95—532no—
187#95C2×D5311yescharacteristic
188C24—531no—
189C24C22×D5312yes—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—1064no—
193C2×D4—1064no—
194C2×D4—1064no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—534no—
198C22×C4—534no—
199C22×C4—532no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—1062no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—2122no—
206C22×C4—2122no—
207C22×C4C22×D5311yescharacteristic
208C22×C4C22×D5312yes—
209C22×C4C22×D5311yescharacteristic
210C22×C4C22×D5311yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—1064no—
215C4⋊C4—1062no—
216C4⋊C4—1062no—
217C4⋊C4—1064no—
218C4⋊C4—2122no—
219C22⋊C4—24no—
220C22⋊C4—42no—
221C22⋊C4—1064no—
222C22⋊C4—1064no—
223C22⋊C4—1064no—
224C22⋊C4—2122no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—532no—
233C23—532no—
234C23—532no—
235C23—1062no—
236C23—1062no—
237C23C23×D5311yescharacteristic
238D4—44no—
239D4—2128no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—1068no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1061no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—2124no—
263C2×C4—2124no—
264C2×C4D4×D5312yes—
265C2×C4D4×D5312yes—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—534no—
274C22—1062no—
275C22—1062no—
276C22—1064no—
277C22—1064no—
278C22C2×D4×D5311yescharacteristic
279C22C2×D4×D5311yescharacteristic
280C22#41×D5311yescharacteristic
281C22#1313326212yes—
282C22#1313326412yes—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—42no—
287C4—1062no—
288C4—1062no—
289C4—2122no—
290C4—2121no—
291C4—2122no—
292C2—82no—
293C2—1064no—
294C2#121×D5311yescharacteristic
295C2#1681118311yescharacteristic
296C2#1681119311yescharacteristic
297C2#1681120212yes—
298C2#1681121612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2618
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
3. C2×C4⋊C4×D53C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
6. #16810546C2semidirect
7. #16810546C2semidirect
8. #16810537C2semidirect
9. #16810499C2semidirect
10. #16810482C2semidirect
11. C53×#394C2semidirect
60. C2×C4×D53D4semidirect
158. #394D53semidirect
161. C53#2618semidirect