Finite Group Databasebeta

← #20280443

Subgroups — #20280443

Order 6784.

816 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 153 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×#11719947C211yesmaximal, characteristic
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×#13133194C212yesmaximal, complemented
4#16810544C211yesmaximal, characteristic, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810539C212yesmaximal, complemented
7#16810538C212yesmaximal, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810482C211yesmaximal, characteristic, complemented
10#16810481C212yesmaximal, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—24no—
14C22×Dic106C2211yescharacteristic
15C22×Dic106C2211yescharacteristic
16C22×C22⋊D53C2212yes—
17C23×Dic53C2212yes—
18C22×C4×D53C2211yescharacteristic
19#13133195—24no—
20#13133194—28no—
21C2×C4⋊Dic53C2212yes—
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2212yes—
27C2×C22⋊Dic53C2212yes—
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2212yes—
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33#13133124C2212yes—
34C22×D4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2212yes—
38#95×C53C2211yescharacteristic
39C24×C53C2312yes—
40C23×D53C2311yescharacteristic
41C2×D4×C53—24no—
42C2×D4×C53—22no—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C22×Dic53—24no—
51C22×Dic53—24no—
52C22×Dic53D418yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2312yes—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58#11719947—232no—
59C2×C4×D53—24no—
60C2×C4×D53D414yes—
61C2×Dic106—22no—
62C2×Dic106—24no—
63C2×Dic106—24no—
64C2×Dic106—22no—
65C4⋊C4×C53—22no—
66C22⋊C4×C53—24no—
67C22⋊C4×C53—42no—
68C22⋊Dic53—24no—
69C22⋊Dic53—42no—
70#11719921—22no—
71#11719921—22no—
72#11719921—42no—
73C4⋊Dic53—24no—
74#11719919—22no—
75#11719919—24no—
76#11719919—24no—
77#11719919—22no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53C2×D414yes—
85C23×C53C2411yescharacteristic
86C22×D53—22no—
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—21no—
90C22×D53C2×D412yes—
91D4×C53—28no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53C2×D412yes—
102C22⋊D53—216no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—216no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53C2×D414yes—
115C2×Dic53C2×D414yes—
116C2×Dic53C2×D412yes—
117C4×D53—28no—
118Dic106—28no—
119Dic106—44no—
120C22×C53—24no—
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53C22×D411yescharacteristic
128C22×C53C22×D412yes—
129C22×C53H(5,𝔽2)11yescharacteristic
130C22×C532−1+412yes—
131C22×C532−1+411yescharacteristic
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—22no—
135C2×D53—42no—
136C2×D53—42no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—42no—
140C4×C53—41no—
141Dic53—24no—
142Dic53—41no—
143Dic53—42no—
144Dic53—41no—
145Dic53#12014yes—
146#2616—531nomaximal, Sylow
147C2×C53—44no—
148C2×C53C2×#12011yescharacteristic
149C2×C53#53611yescharacteristic
150C2×C53#53612yes—
151C2×C53#53712yes—
152C2×C53#53711yescharacteristic
153D53—42no—
154C22×#41—531no—
155C2×C22⋊Q8—531no—
156C2×C22⋊Q8—532no—
157#395—531no—
158#395—531no—
159#394—531no—
160#394—532no—
161#394—532no—
162#394D5311yescharacteristic, complemented
163#380—532no—
164#380—531no—
165C53#261611yescharacteristic, complemented, Sylow
166C2×#41—1068no—
167C2×#41—1064no—
168C22×Q8—531no—
169C22×Q8—531no—
170C22×D4—532no—
171C22×D4C2×D5311yescharacteristic
172C23×C4—532no—
173C23×C4—531no—
174C22⋊Q8—1068no—
175C22⋊Q8—1064no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—532no—
178C2×C4⋊C4—532no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—532no—
181C2×C4⋊C4C2×D5311yescharacteristic
182C2×C22⋊C4—532no—
183C2×C22⋊C4—531no—
184C2×C22⋊C4—531no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4C2×D5312yes—
188C2×C22⋊C4C2×D5312yes—
189#95—532no—
190#95—532no—
191#95—531no—
192#95C2×D5311yescharacteristic
193C24—531no—
194C24C22×D5312yes—
195#41—10632no—
196C2×Q8—1064no—
197C2×Q8—1064no—
198C2×Q8—1062no—
199C2×Q8—1062no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—1064no—
203C2×D4—1064no—
204C2×D4—1064no—
205C22×C4—531no—
206C22×C4—538no—
207C22×C4—532no—
208C22×C4—534no—
209C22×C4—532no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—1064no—
213C22×C4—1064no—
214C22×C4—1064no—
215C22×C4C22×D5311yescharacteristic
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—1064no—
221C4⋊C4—1064no—
222C4⋊C4—1062no—
223C4⋊C4—1064no—
224C4⋊C4—1062no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1064no—
230C22⋊C4—2122no—
231C22⋊C4—2122no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—532no—
239C23—1062no—
240C23—1061no—
241C23—1062no—
242C23—1061no—
243C23D4×D5314yes—
244C23C23×D5311yescharacteristic
245Q8—1068no—
246Q8—2124no—
247D4—28no—
248D4—10616no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—532no—
259C2×C4—534no—
260C2×C4—534no—
261C2×C4—1068no—
262C2×C4—10616no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1061no—
271C2×C4—1062no—
272C2×C4—1061no—
273C2×C4D4×D5312yes—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—1062no—
282C22—1062no—
283C22—1062no—
284C22—2122no—
285C22—2122no—
286C22C2×D4×D5311yescharacteristic
287C22C2×D4×D5312yes—
288C22#1313326211yescharacteristic
289C22#1313326412yes—
290C22#1313326411yescharacteristic
291C4—22no—
292C4—41no—
293C4—42no—
294C4—41no—
295C4—534no—
296C4—1064no—
297C4—2121no—
298C4—2122no—
299C4—2121no—
300C2—44no—
301C2—2122no—
302C2#120×D5311yescharacteristic
303C2#1681119311yescharacteristic
304C2#1681119512yes—
305C2#1681120012yes—
306C2#1681120211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2616
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133195C2semidirect
3. C2×#13133194C2semidirect
4. #16810544C2semidirect
5. #16810543C2semidirect
6. #16810539C2semidirect
7. #16810538C2semidirect
8. #16810537C2semidirect
9. #16810482C2semidirect
10. #16810481C2semidirect
11. C53×#394C2semidirect
162. #394D53semidirect
165. C53#2616semidirect