Finite Group Databasebeta

← #20280543

Subgroups — #20280543

Order 6784.

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

normal subgroups only

This group has 488 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×#13133175C211yesmaximal, characteristic
3#16810562C211yesmaximal, characteristic, complemented
4#16810560C211yesmaximal, characteristic
5#16810558C211yesmaximal, characteristic
6#16810544C212yesmaximal, complemented
7#16810536C212yesmaximal
8#16810520C212yesmaximal, complemented
9#16810487C211yesmaximal, characteristic, complemented
10#16810482C212yesmaximal
11C53×#400C211yesmaximal, characteristic, complemented
12C22×Dic106C2212yes—
13C22×C4×D53C2211yescharacteristic
14#13133216—24no—
15#13133175—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53Q814yescomplemented
51C2×Dic106—24no—
52C2×Dic106—42no—
53C2×Dic106—42no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56#11719921—22no—
57#11719921—22no—
58#11719921—42no—
59#11719921—42no—
60C4⋊Dic53—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53—42no—
63#11719919—24no—
64#11719919—24no—
65#11719919—42no—
66C4×Dic53—22no—
67C4×Dic53D414yes—
68C23×C53C2411yescharacteristic
69C22×D53—21no—
70C22×D53—21no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53C2×Q812yes—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
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×C53—22no—
85C2×C4×C53C2×D412yes—
86C2×C4×C53C2×Q812yes—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53C2×D412yes—
97C2×Dic53C2×D412yes—
98C2×Dic53C2×Q812yes—
99C2×Dic53#4112yes—
100C4×D53—28no—
101Dic106—48no—
102C22×C53C22×D411yescharacteristic
103C22×C53C22×Q811yescharacteristic
104C22×C53C2×#4111yescharacteristic
105C22×C53H(5,𝔽2)12yes—
106C22×C532−1+412yes—
107C2×D53—22no—
108C2×D53—24no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—42no—
112C4×C53—22no—
113C4×C53—22no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53—42no—
117Dic53—22no—
118Dic53—42no—
119Dic53—41no—
120Dic53—42no—
121Dic53C22⋊Q814yes—
122#2628—531nomaximal, Sylow
123C2×C53C2×C22⋊Q811yescharacteristic
124C2×C53#53511yescharacteristic
125C2×C53#53611yescharacteristic
126C2×C53#54212yes—
127C2×C53#54412yes—
128D53—42no—
129C2×C4⋊Q8—531no—
130C2×#117—531no—
131#400—531no—
132#400D5311yescharacteristic, complemented
133#399—531no—
134#399—531no—
135#395—532no—
136#393—532no—
137#389—532no—
138#382—531no—
139#380—532no—
140C53#262811yescharacteristic, complemented, Sylow
141C22×Q8—532no—
142C23×C4—531no—
143C4⋊Q8—1064no—
144#117—1064no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C4⋊C4C2×D5311yescharacteristic
153C2×C22⋊C4—531no—
154C2×C22⋊C4—531no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C42—531no—
158C2×C42—531no—
159#95—532no—
160#95—532no—
161#95—531no—
162#95—532no—
163#95C2×D5312yes—
164#95C2×D5311yescharacteristic
165#95C2×D5312yes—
166C24—531no—
167C2×Q8—1064no—
168C2×Q8—2122no—
169C2×Q8—2122no—
170C22×C4—531no—
171C22×C4—531no—
172C22×C4—534no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4—531no—
176C22×C4—1062no—
177C22×C4—1062no—
178C22×C4C22×D5311yescharacteristic
179C22×C4C22×D5312yes—
180C22×C4C22×D5312yes—
181C22×C4C22×D5311yescharacteristic
182C22×C4C22×D5311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—1062no—
186C4⋊C4—1064no—
187C4⋊C4—1064no—
188C4⋊C4—1062no—
189C4⋊C4—2122no—
190C4⋊C4—2122no—
191C22⋊C4—1062no—
192C22⋊C4—1062no—
193C22⋊C4—2122no—
194C22⋊C4—2122no—
195C42—534no—
196C42—1062no—
197C23—532no—
198C23—1061no—
199C23—1061no—
200C23—1062no—
201C23—1062no—
202C23C23×D5311yescharacteristic
203Q8—2128no—
204C2×C4—21no—
205C2×C4—21no—
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—1068no—
220C2×C4—1062no—
221C2×C4—1061no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4D4×D5312yes—
230C2×C4Q8×D5312yes—
231C22—1062no—
232C22—1064no—
233C22—2121no—
234C22—2121no—
235C22—2122no—
236C22C2×D4×D5311yescharacteristic
237C22C2×Q8×D5311yescharacteristic
238C22#41×D5311yescharacteristic
239C22#1313326312yes—
240C22#1313326412yes—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—41no—
245C4—42no—
246C4—534no—
247C4—1062no—
248C4—2121no—
249C4—2122no—
250C4—2122no—
251C2—2122no—
252C2C22⋊Q8×D5311yescharacteristic
253C2#1681119111yescharacteristic
254C2#1681119511yescharacteristic
255C2#1681123912yes—
256C2#1681125312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2628
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
3. #16810562C2semidirect
6. #16810544C2semidirect
8. #16810520C2semidirect
9. #16810487C2semidirect
11. C53×#400C2semidirect
50. C2×C4×D53Q8semidirect
132. #400D53semidirect
140. C53#2628semidirect