Finite Group Databasebeta

← #20281300

Subgroups — #20281300

Order 6784.

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

normal subgroups only

This group has 612 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×#13133189C212yesmaximal, complemented
2#16810553C211yesmaximal, characteristic, complemented
3#16810540C211yesmaximal, characteristic, complemented
4#16810539C212yesmaximal, complemented
5#16810532C212yesmaximal, complemented
6#16810514C212yesmaximal, complemented
7#16810500C212yesmaximal, complemented
8#16810484C212yesmaximal, complemented
9C53×#390C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×C22⋊D53C2212yes—
12C22×C22⋊D53C2212yes—
13C23×Dic53C2212yes—
14#13133189—28no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22#13133124C2211yescharacteristic
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27C22×D4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2212yes—
31C2×C42×C53C2211yescharacteristic
32C24×C53C2312yes—
33C23×D53C2312yes—
34C2×D4×C53—41no—
35C2×D4×C53—41no—
36C2×D4×C53D414yescomplemented
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C2×C22⋊D53—24no—
42C2×C22⋊D53—24no—
43C2×C22⋊D53—24no—
44C2×C22⋊D53—42no—
45C2×C22⋊D53—42no—
46C2×C22⋊D53—42no—
47C22×Dic53—24no—
48C22×Dic53—24no—
49C22×Dic53—42no—
50C22×Dic53C2312yes—
51C22×Dic53C2312yes—
52C22×Dic53C2312yes—
53C2×D212—42no—
54C2×D212—41no—
55C4⋊C4×C53—41no—
56C22⋊C4×C53—42no—
57C22⋊C4×C53—42no—
58C42×C53—41no—
59C22⋊Dic53—24no—
60#11719921—24no—
61#11719921—24no—
62#11719921—42no—
63#11719921—42no—
64C4⋊Dic53—41no—
65#11719919—24no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C2×D414yes—
73C23×C53C2411yescharacteristic
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—22no—
80C22×D53—22no—
81D4×C53—44no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53C2×D412yes—
92C22⋊D53—48no—
93C22⋊D53—48no—
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—48no—
103C2×Dic53#4114yes—
104D212—44no—
105C22×C53—24no—
106C22×C53—24no—
107C22×C53—24no—
108C22×C53—42no—
109C22×C53—42no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53C22×D411yescharacteristic
113C22×C53C2×#4112yes—
114C22×C53H(5,𝔽2)11yescharacteristic
115C22×C53H(5,𝔽2)11yescharacteristic
116C22×C53H(5,𝔽2)12yes—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C2×D53—42no—
121C2×D53—42no—
122C2×D53—42no—
123C2×D53—42no—
124C4×C53—22no—
125C4×C53—42no—
126C4×C53—41no—
127C4×C53—41no—
128Dic53—24no—
129Dic53—42no—
130Dic53—42no—
131#2699—531nomaximal, Sylow
132C2×C53—44no—
133C2×C53#53411yescharacteristic
134C2×C53#54012yes—
135C2×C53#54612yes—
136C2×C53#55511yescharacteristic
137C2×C53#56011yescharacteristic
138D53—82no—
139C2×#121—532no—
140#398—531no—
141#394—531no—
142#394—532no—
143#392—532no—
144#390D5311yescharacteristic, complemented
145#388—532no—
146#386—532no—
147#381—532no—
148C53#269911yescharacteristic, complemented, Sylow
149C22×D4—532no—
150C22×D4—532no—
151C22×D4—531no—
152C22×D4C2×D5311yescharacteristic
153C23×C4—532no—
154#121—1068no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4C2×D5311yescharacteristic
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4C2×D5312yes—
164C2×C22⋊C4C2×D5312yes—
165C2×C42C2×D5311yescharacteristic
166#95—531no—
167#95—532no—
168#95—532no—
169#95—532no—
170#95—531no—
171C24—532no—
172C24C22×D5312yes—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—1064no—
176C2×D4—1064no—
177C2×D4—1064no—
178C2×D4—2122no—
179C2×D4—2121no—
180C2×D4—2122no—
181C2×D4—2122no—
182C2×D4—2122no—
183C2×D4D21214yescomplemented
184C22×C4—532no—
185C22×C4—532no—
186C22×C4—532no—
187C22×C4—1064no—
188C22×C4—1064no—
189C22×C4—2122no—
190C22×C4C22×D5311yescharacteristic
191C22×C4C22×D5312yes—
192C22×C4C22×D5311yescharacteristic
193C22×C4C22×D5311yescharacteristic
194C4⋊C4—41no—
195C4⋊C4—1064no—
196C4⋊C4—2121no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—1064no—
200C22⋊C4—1064no—
201C22⋊C4—1064no—
202C22⋊C4—2122no—
203C22⋊C4—2122no—
204C42—41no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—1062no—
212C23—1062no—
213C23—1062no—
214C23—1062no—
215C23—1062no—
216C23—1062no—
217C23—1062no—
218C23C2×D21214yes—
219C23C23×D5311yescharacteristic
220D4—44no—
221D4—2124no—
222D4—2128no—
223D4—2128no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—534no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—2128no—
243C2×C4C2×D21212yes—
244C22—24no—
245C22—24no—
246C22—24no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—2122no—
252C22—2122no—
253C22—2122no—
254C22—2122no—
255C22—2122no—
256C22—2122no—
257C22—2122no—
258C22C22×D21211yescharacteristic
259C22#41×D5312yes—
260C22#1313326212yes—
261C22#1313326211yescharacteristic
262C22#1313326211yescharacteristic
263C4—22no—
264C4—41no—
265C4—41no—
266C4—42no—
267C4—1064no—
268C4—2122no—
269C4—2122no—
270C2—44no—
271C2—4242no—
272C2#1681118511yescharacteristic
273C2#1681122312yes—
274C2#1681126812yes—
275C2#1681133511yescharacteristic
276C2#1681136011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2699
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133189C2semidirect
2. #16810553C2semidirect
3. #16810540C2semidirect
4. #16810539C2semidirect
5. #16810532C2semidirect
6. #16810514C2semidirect
7. #16810500C2semidirect
8. #16810484C2semidirect
9. C53×#390C2semidirect
36. C2×D4×C53D4semidirect
144. #390D53semidirect
148. C53#2699semidirect
183. C2×D4D212semidirect