Finite Group Databasebeta

← #20282727

Subgroups — #20282727

Order 6784.

478 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133202C211yesmaximal, characteristic
2C2×#13133193C211yesmaximal, characteristic, complemented
3#16810554C212yesmaximal, complemented
4#16810553C211yesmaximal, characteristic, complemented
5#16810552C212yesmaximal, complemented
6#16810550C211yesmaximal, characteristic, complemented
7#16810547C212yesmaximal, complemented
8#16810540C212yesmaximal
9#16810534C211yesmaximal, characteristic, complemented
10#16810482C211yesmaximal, characteristic, complemented
11C53×#395C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14#13133202—24no—
15#13133193—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C2×Q8×C53—22no—
40C2×Q8×C53—42no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C2×D212—22no—
51C2×D212—42no—
52C2×C4×D53—42no—
53C2×C4×D53D414yes—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—42no—
56#11719921—24no—
57#11719921—22no—
58#11719921—22no—
59#11719921—22no—
60#11719921—42no—
61#11719921—42no—
62#11719921—42no—
63C4⋊Dic53—22no—
64#11719919—24no—
65#11719919—22no—
66#11719919—42no—
67#11719919—42no—
68C23×C53C2411yescharacteristic
69C22×D53—22no—
70C22×D53—21no—
71C22×D53—22no—
72C22×D53—21no—
73C22×D53—21no—
74C22×D53—22no—
75C22×D53—21no—
76C22×D53—22no—
77C22×D53—21no—
78C22×D53C2×D412yes—
79Q8×C53—44no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53C2×D412yes—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53C2×D412yes—
103D212—44no—
104C4×D53—44no—
105C22×C53C22×D411yescharacteristic
106C22×C53H(5,𝔽2)12yes—
107C22×C53H(5,𝔽2)11yescharacteristic
108C22×C532−1+412yes—
109C22×C532−1+411yescharacteristic
110C2×D53—22no—
111C2×D53—22no—
112C2×D53—22no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—41no—
122C4×C53—42no—
123C4×C53—41no—
124C4×C53—42no—
125Dic53—22no—
126Dic53—41no—
127Dic53—42no—
128Dic53—42no—
129#2814—531nomaximal, Sylow
130C2×C53#53411yescharacteristic
131C2×C53#53611yescharacteristic
132C2×C53#53711yescharacteristic
133C2×C53#56212yes—
134C2×C53#56312yes—
135D53—42no—
136D53—81no—
137C2×#123—531no—
138C2×C22⋊Q8—531no—
139#398—531no—
140#398—532no—
141#397—531no—
142#397—532no—
143#396—532no—
144#395D5311yescharacteristic, complemented
145#394—532no—
146#393—531no—
147#380—531no—
148C53#281411yescharacteristic, complemented, Sylow
149C22×Q8C2×D5311yescharacteristic
150C22×D4—531no—
151C23×C4—531no—
152#123—1064no—
153C22⋊Q8—1064no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4C2×D5311yescharacteristic
160C2×C4⋊C4C2×D5312yes—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—531no—
164C2×C22⋊C4—531no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—532no—
168#95—531no—
169#95—532no—
170#95—532no—
171#95—532no—
172#95C2×D5311yescharacteristic
173#95C2×D5312yes—
174C24—531no—
175C24—531no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—1062no—
179C2×D4—2122no—
180C22×C4—531no—
181C22×C4—534no—
182C22×C4—531no—
183C22×C4—532no—
184C22×C4—532no—
185C22×C4—2122no—
186C22×C4C22×D5311yescharacteristic
187C22×C4C22×D5312yes—
188C22×C4C22×D5312yes—
189C22×C4C22×D5311yescharacteristic
190C22×C4C22×D5311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—1064no—
194C4⋊C4—1062no—
195C4⋊C4—1062no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—1062no—
199C22⋊C4—1064no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C22⋊C4—2122no—
203C22⋊C4—2122no—
204C22⋊C4—2122no—
205C23—532no—
206C23—1062no—
207C23—1061no—
208C23—1062no—
209C23—1061no—
210C23—1061no—
211C23—1062no—
212C23—1061no—
213C23—1062no—
214C23—1061no—
215C23C23×D5311yescharacteristic
216Q8—44no—
217D4—2124no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—532no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—2124no—
241C2×C4D4×D5312yes—
242C22—1062no—
243C22—1062no—
244C22—1062no—
245C22—2122no—
246C22—2122no—
247C22—2121no—
248C22—2121no—
249C22—2121no—
250C22—2122no—
251C22—2122no—
252C22C2×D4×D5311yescharacteristic
253C22#1313326212yes—
254C22#1313326311yescharacteristic
255C22#1313326512yes—
256C22#1313326511yescharacteristic
257C4—22no—
258C4—42no—
259C4—41no—
260C4—42no—
261C4—41no—
262C4—1062no—
263C4—2122no—
264C4—2121no—
265C4—2122no—
266C2—4241no—
267C2—2122no—
268C2#1681118311yescharacteristic
269C2#1681119811yescharacteristic
270C2#1681120311yescharacteristic
271C2#1681136912yes—
272C2#1681137212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2814
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#13133193C2semidirect
3. #16810554C2semidirect
4. #16810553C2semidirect
5. #16810552C2semidirect
6. #16810550C2semidirect
7. #16810547C2semidirect
9. #16810534C2semidirect
10. #16810482C2semidirect
11. C53×#395C2semidirect
144. #395D53semidirect
148. C53#2814semidirect