Finite Group Databasebeta

← #20280828

Subgroups — #20280828

Order 6784.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133195C211yesmaximal, characteristic
2C2×#13133193C211yesmaximal, characteristic
3#16810557C212yesmaximal, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810543C212yesmaximal, complemented
7#16810542C211yesmaximal, characteristic, complemented
8#16810482C212yesmaximal
9#16810480C212yesmaximal
10#95×D53C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C22×C4×D53C2212yes—
15#13133195—24no—
16#13133193—24no—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27#13133124C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C23×D53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—42no—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C2×C4×D53—42no—
51C2×C4×D53—42no—
52C2×C4×D53—42no—
53C2×C4×D53D414yes—
54C2×C4×D53Q818yescomplemented
55C2×Dic106—22no—
56C2×Dic106—42no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—42no—
59#11719921—24no—
60#11719921—22no—
61#11719921—22no—
62C4⋊Dic53—22no—
63C4⋊Dic53—42no—
64#11719919—22no—
65#11719919—24no—
66#11719919—22no—
67#11719919—42no—
68#11719919—42no—
69C23×C53C2411yescharacteristic
70C22×D53—22no—
71C22×D53—21no—
72C22×D53—21no—
73C22×D53C2×D412yes—
74C22×D53C2×Q814yes—
75Q8×C53—44no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53C2×D412yes—
88C2×C4×C53C2×Q814yes—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—22no—
100C2×Dic53C2×D412yes—
101C2×Dic53C2×Q814yes—
102C4×D53—44no—
103C4×D53—48no—
104Dic106—44no—
105C22×C53C22×D411yescharacteristic
106C22×C53C22×Q812yes—
107C22×C53H(5,𝔽2)12yes—
108C22×C53H(5,𝔽2)11yescharacteristic
109C22×C532−1+411yescharacteristic
110C2×D53—22no—
111C2×D53—22no—
112C2×D53—24no—
113C2×D53—24no—
114C2×D53C4⋊Q814yes—
115C4×C53—22no—
116C4×C53—24no—
117C4×C53—41no—
118C4×C53—42no—
119C4×C53—41no—
120Dic53—22no—
121Dic53—24no—
122Dic53—41no—
123Dic53—41no—
124Dic53—42no—
125#2659—531nomaximal, Sylow
126C2×C53C2×C4⋊Q811yescharacteristic
127C2×C53#53411yescharacteristic
128C2×C53#53711yescharacteristic
129C2×C53#54312yes—
130C2×C53#54412yes—
131D53—24no—
132C2×C22⋊Q8—531no—
133C2×C22⋊Q8—531no—
134#398—531no—
135#398—531no—
136#398—532no—
137#395—531no—
138#395—532no—
139#395D5311yescharacteristic, complemented
140#380—532no—
141#380—532no—
142C2×#95—531no—
143C53#265911yescharacteristic, complemented, Sylow
144C22×Q8—531no—
145C22×Q8C2×D5311yescharacteristic
146C23×C4—531no—
147C23×C4—532no—
148C22⋊Q8—1064no—
149C22⋊Q8—1064no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4C2×D5311yescharacteristic
158C2×C4⋊C4C2×D5312yes—
159C2×C22⋊C4—531no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—531no—
162#95—531no—
163#95—532no—
164#95—532no—
165#95—532no—
166#95—532no—
167#95C2×D5311yescharacteristic
168#95C2×D5312yes—
169C24—531no—
170C2×Q8—22no—
171C2×Q8—42no—
172C2×Q8—1062no—
173C2×Q8—2122no—
174C22×C4—532no—
175C22×C4—531no—
176C22×C4—534no—
177C22×C4—538no—
178C22×C4—531no—
179C22×C4—532no—
180C22×C4—531no—
181C22×C4—2122no—
182C22×C4—2122no—
183C22×C4—2122no—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5312yes—
186C22×C4C22×D5311yescharacteristic
187C22×C4C22×D5311yescharacteristic
188C22×C4C22×D5312yes—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—1062no—
192C4⋊C4—1064no—
193C4⋊C4—1062no—
194C4⋊C4—1062no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—1062no—
199C22⋊C4—1062no—
200C22⋊C4—1064no—
201C23—532no—
202C23—534no—
203C23—1062no—
204C23—1061no—
205C23—1061no—
206C23C23×D5311yescharacteristic
207Q8—44no—
208Q8—2124no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—532no—
221C2×C4—534no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1061no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—2124no—
234C2×C4—2128no—
235C2×C4D4×D5312yes—
236C2×C4Q8×D5314yes—
237C22—534no—
238C22—1062no—
239C22—1062no—
240C22—1064no—
241C22—1064no—
242C22C2×D4×D5311yescharacteristic
243C22C2×Q8×D5312yes—
244C22#1313326212yes—
245C22#1313326311yescharacteristic
246C22#1313326511yescharacteristic
247C4—22no—
248C4—24no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—1062no—
253C4—1064no—
254C4—2121no—
255C4—2121no—
256C4—2122no—
257C2—1064no—
258C2C4⋊Q8×D5311yescharacteristic
259C2#1681118311yescharacteristic
260C2#1681120311yescharacteristic
261C2#1681124812yes—
262C2#1681125012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2659
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #16810557C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
6. #16810543C2semidirect
7. #16810542C2semidirect
11. C53×#395C2semidirect
54. C2×C4×D53Q8semidirect
139. #395D53semidirect
143. C53#2659semidirect