Finite Group Databasebeta

← #20280480

Subgroups — #20280480

Order 6784.

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

normal subgroups only

This group has 528 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×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133194C212yesmaximal
3C2×#13133173C211yesmaximal, characteristic
4#16810557C212yesmaximal, complemented
5#16810555C211yesmaximal, characteristic, complemented
6#16810554C211yesmaximal, characteristic, complemented
7#16810551C212yesmaximal
8#16810548C211yesmaximal, characteristic
9#16810527C212yesmaximal, complemented
10#16810487C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic
12C22×Dic106C2212yes—
13C23×Dic53C2211yescharacteristic
14#13133216—24no—
15#13133194—28no—
16#13133173—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C22⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36#95×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×Dic53—22no—
44C22×Dic53—22no—
45C22×Dic53D414yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2311yescharacteristic
51C22×Dic53C2311yescharacteristic
52C2×Dic106—24no—
53C2×Dic106—24no—
54C2×Dic106—24no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—24no—
57C22⋊C4×C53—22no—
58C22⋊Dic53—24no—
59C22⋊Dic53—22no—
60C4⋊Dic53—22no—
61C4⋊Dic53—24no—
62#11719919—22no—
63#11719919—22no—
64#11719919—24no—
65#11719919—24no—
66#11719919—24no—
67C4×Dic53—22no—
68C4×Dic53D414yes—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53C2×D412yes—
74C23×C53C2411yescharacteristic
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—22no—
88C2×Dic53—28no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53C2×D412yes—
98C2×Dic53C2×D412yes—
99C2×Dic53C2×D412yes—
100C2×Dic53C2×D412yes—
101C2×Dic53#4112yes—
102Dic106—48no—
103C22×C53—22no—
104C22×C53—24no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—42no—
108C22×C53C22×D411yescharacteristic
109C22×C53C22×D411yescharacteristic
110C22×C53C2×#4111yescharacteristic
111C22×C532−1+412yes—
112C22×C532−1+412yes—
113C4×C53—22no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53—42no—
117Dic53—22no—
118Dic53—22no—
119Dic53—42no—
120Dic53—41no—
121Dic53—42no—
122Dic53#12114yes—
123#2621—531nomaximal, Sylow
124C2×C53—42no—
125C2×C53C2×#12111yescharacteristic
126C2×C53#53611yescharacteristic
127C2×C53#53611yescharacteristic
128C2×C53#53612yes—
129C2×C53#54112yes—
130C2×C4⋊Q8—531no—
131C2×C22⋊Q8—532no—
132C2×#117—531no—
133#398—531no—
134#398—531no—
135#398—532no—
136#397—531no—
137#397—532no—
138#397D5311yescharacteristic
139#391—532no—
140#382—531no—
141C53#262111yescharacteristic, complemented, Sylow
142C22×Q8—532no—
143C23×C4—531no—
144C4⋊Q8—1064no—
145C22⋊Q8—1068no—
146#117—1064no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4C2×D5311yescharacteristic
155C2×C22⋊C4—531no—
156C2×C22⋊C4—532no—
157C2×C22⋊C4C2×D5311yescharacteristic
158C2×C22⋊C4C2×D5312yes—
159C2×C42—531no—
160C2×C42—531no—
161#95—532no—
162#95—531no—
163#95—532no—
164#95—532no—
165#95—532no—
166#95C2×D5311yescharacteristic
167#95C2×D5312yes—
168C24C22×D5311yescharacteristic
169C2×Q8—1064no—
170C2×Q8—1064no—
171C2×Q8—1064no—
172C22×C4—531no—
173C22×C4—534no—
174C22×C4—531no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—532no—
178C22×C4—532no—
179C22×C4—1062no—
180C22×C4—1062no—
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5312yes—
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—1062no—
187C4⋊C4—1064no—
188C4⋊C4—1064no—
189C4⋊C4—1064no—
190C4⋊C4—1064no—
191C4⋊C4—1062no—
192C4⋊C4—1062no—
193C22⋊C4—24no—
194C22⋊C4—22no—
195C22⋊C4—1062no—
196C22⋊C4—1064no—
197C42—534no—
198C42—1062no—
199C23—21no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23D4×D5312yes—
204C23C23×D5311yescharacteristic
205Q8—2128no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—532no—
216C2×C4—532no—
217C2×C4—532no—
218C2×C4—532no—
219C2×C4—532no—
220C2×C4—1068no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4D4×D5312yes—
233C22—22no—
234C22—24no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22C2×D4×D5311yescharacteristic
239C22C2×D4×D5311yescharacteristic
240C22#41×D5311yescharacteristic
241C22#1313326412yes—
242C22#1313326412yes—
243C4—22no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—534no—
248C4—1062no—
249C4—1062no—
250C4—2122no—
251C4—2121no—
252C4—2122no—
253C2—42no—
254C2#121×D5311yescharacteristic
255C2#1681119311yescharacteristic
256C2#1681119511yescharacteristic
257C2#1681119512yes—
258C2#1681123012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2621
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
4. #16810557C2semidirect
5. #16810555C2semidirect
6. #16810554C2semidirect
9. #16810527C2semidirect
10. #16810487C2semidirect
141. C53#2621semidirect