Finite Group Databasebeta

← #20280865

Subgroups — #20280865

Order 6784.

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

normal subgroups only

This group has 498 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×#13133196C211yesmaximal, characteristic, complemented
2#16810562C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810551C212yesmaximal, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810537C212yesmaximal
7#16810535C211yesmaximal, characteristic, complemented
8#16810518C212yesmaximal, complemented
9#16810517C212yesmaximal, complemented
10C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
11C2×#123×C53C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13#13133196—24no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2212yes—
20C2×C22⋊Dic53C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C4×Dic53C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
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
39C2×D4×C53—22no—
40C2×D4×C53—42no—
41C22×C4×C53—42no—
42C22×C4×C53D414yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×Dic53C2312yes—
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2312yes—
52C2×Dic106—22no—
53C2×Dic106—42no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—22no—
57C22⋊C4×C53—22no—
58C22⋊C4×C53—22no—
59C22⋊Dic53—24no—
60C22⋊Dic53—24no—
61C22⋊Dic53—24no—
62C22⋊Dic53—42no—
63C4⋊Dic53—22no—
64#11719919—24no—
65#11719919—24no—
66#11719919—42no—
67C4×Dic53—42no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53C2×D412yes—
78C23×C53C2411yescharacteristic
79D4×C53—44no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—44no—
90C2×C4×C53C2×D412yes—
91C2×C4×C53C2×D412yes—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53#4114yes—
104Dic106—44no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53C22×D411yescharacteristic
116C22×C53C2×#4112yes—
117C22×C53H(5,𝔽2)12yes—
118C22×C532−1+411yescharacteristic
119C22×C532−1+411yescharacteristic
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—41no—
123C4×C53—41no—
124C4×C53—41no—
125Dic53—24no—
126Dic53—42no—
127Dic53—42no—
128Dic53—42no—
129#2663—531nomaximal, Sylow
130C2×C53—42no—
131C2×C53—81no—
132C2×C53#53211yescharacteristic
133C2×C53#53411yescharacteristic
134C2×C53#53611yescharacteristic
135C2×C53#53912yes—
136C2×C53#54212yes—
137C2×#123D5311yescharacteristic
138C2×C22⋊Q8—531no—
139#400—531no—
140#399—531no—
141#397—532no—
142#395—531no—
143#394—532no—
144#393—531no—
145#389—532no—
146#388—532no—
147C4×C22⋊C4—531no—
148C53#266311yescharacteristic, complemented, Sylow
149C22×Q8—531no—
150C22×D4C2×D5311yescharacteristic
151C23×C4C2×D5311yescharacteristic
152#123—24no—
153C22⋊Q8—1064no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4C2×D5311yescharacteristic
159C2×C4⋊C4C2×D5311yescharacteristic
160C2×C22⋊C4—532no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4C2×D5311yescharacteristic
165C2×C22⋊C4C2×D5311yescharacteristic
166C2×C22⋊C4C2×D5311yescharacteristic
167C2×C42—532no—
168#95—531no—
169#95—531no—
170#95—532no—
171#95—532no—
172#95—532no—
173#95—532no—
174C24C22×D5311yescharacteristic
175C24C22×D5311yescharacteristic
176C2×Q8—1062no—
177C2×Q8—2122no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—42no—
181C22×C4—532no—
182C22×C4—532no—
183C22×C4—532no—
184C22×C4—532no—
185C22×C4C22⋊D5314yes—
186C22×C4C22×D5311yescharacteristic
187C22×C4C22×D5311yescharacteristic
188C22×C4C22×D5311yescharacteristic
189C22×C4C22×D5311yescharacteristic
190C22×C4C22×D5311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—1064no—
194C4⋊C4—1064no—
195C4⋊C4—1062no—
196C4⋊C4—2122no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—1064no—
201C22⋊C4—1064no—
202C22⋊C4—1064no—
203C22⋊C4—2122no—
204C42—2122no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23C2×C22⋊D5312yes—
215C23C23×D5311yescharacteristic
216Q8—2124no—
217D4—44no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—44no—
228C2×C4—534no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4C2×C22⋊D5312yes—
241C2×C4C2×C22⋊D5312yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22C22×C22⋊D5311yescharacteristic
253C22#41×D5312yes—
254C22#1313326212yes—
255C22#1313326411yescharacteristic
256C22#1313326411yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—1064no—
263C4—2122no—
264C4—2122no—
265C4—2122no—
266C2—42no—
267C2—81no—
268C2#1681116911yescharacteristic
269C2#1681118211yescharacteristic
270C2#1681119411yescharacteristic
271C2#1681121612yes—
272C2#1681124112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2663
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13133196C2semidirect
2. #16810562C2semidirect
3. #16810559C2semidirect
4. #16810551C2semidirect
5. #16810543C2semidirect
7. #16810535C2semidirect
8. #16810518C2semidirect
9. #16810517C2semidirect
10. C4×C22⋊Dic53C2semidirect
148. C53#2663semidirect