Finite Group Databasebeta

← #20282749

Subgroups — #20282749

Order 6784.

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

normal subgroups only

This group has 444 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×#13133195C211yesmaximal, characteristic
2C2×#13133193C211yesmaximal, characteristic
3#16810564C212yesmaximal, complemented
4#16810563C211yesmaximal, characteristic, complemented
5#16810562C211yesmaximal, characteristic, complemented
6#16810552C212yesmaximal
7#16810549C212yesmaximal
8#16810543C212yesmaximal, complemented
9#16810542C211yesmaximal, characteristic, complemented
10#16810485C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14#13133195—24no—
15#13133193—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719921C2211yescharacteristic
22C2×#11719921C2212yes—
23C2×#11719921C2211yescharacteristic
24C2×#11719921C2212yes—
25#13133124C2212yes—
26#13133124C2211yescharacteristic
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C23×D53C2311yescharacteristic
38C2×Q8×C53—22no—
39C2×Q8×C53—42no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×C4×D53—42no—
51C2×C4×D53D414yes—
52C2×Dic106—22no—
53C2×Dic106—42no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—42no—
56#11719921—24no—
57#11719921—22no—
58#11719921—22no—
59#11719921—42no—
60C4⋊Dic53—22no—
61#11719919—22no—
62#11719919—24no—
63#11719919—22no—
64#11719919—42no—
65C23×C53C2411yescharacteristic
66C22×D53—22no—
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—22no—
70C22×D53C2×D412yes—
71Q8×C53—44no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53C2×D412yes—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—24no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53C2×D412yes—
97C4×D53—44no—
98Dic106—44no—
99C22×C53C22×D411yescharacteristic
100C22×C53H(5,𝔽2)12yes—
101C22×C532−1+411yescharacteristic
102C22×C532−1+411yescharacteristic
103C22×C532−1+412yes—
104C2×D53—22no—
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—42no—
108C2×D53—42no—
109C4×C53—22no—
110C4×C53—41no—
111C4×C53—42no—
112C4×C53—41no—
113C4×C53—42no—
114Dic53—22no—
115Dic53—41no—
116Dic53—41no—
117Dic53—42no—
118Dic53—42no—
119#2817—531nomaximal, Sylow
120C2×C53#53411yescharacteristic
121C2×C53#53611yescharacteristic
122C2×C53#53611yescharacteristic
123C2×C53#56312yes—
124C2×C53#56312yes—
125D53—42no—
126C2×C22⋊Q8—531no—
127C2×C22⋊Q8—531no—
128#400—531no—
129#400—531no—
130#400—532no—
131#397—532no—
132#397—532no—
133#395—531no—
134#395—532no—
135#395D5311yescharacteristic, complemented
136#381—531no—
137C53#281711yescharacteristic, complemented, Sylow
138C22×Q8—531no—
139C22×Q8C2×D5311yescharacteristic
140C23×C4—531no—
141C22⋊Q8—1064no—
142C22⋊Q8—1064no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—531no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4C2×D5311yescharacteristic
149C2×C4⋊C4C2×D5312yes—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—532no—
152C2×C22⋊C4—531no—
153C2×C22⋊C4—532no—
154#95—532no—
155#95—531no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95—532no—
160#95—532no—
161#95C2×D5311yescharacteristic
162#95C2×D5312yes—
163C24—531no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—1062no—
167C2×Q8—2122no—
168C22×C4—531no—
169C22×C4—534no—
170C22×C4—532no—
171C22×C4—531no—
172C22×C4—532no—
173C22×C4—531no—
174C22×C4—2122no—
175C22×C4C22×D5312yes—
176C22×C4C22×D5311yescharacteristic
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5312yes—
179C22×C4C22×D5311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—1062no—
183C4⋊C4—1064no—
184C4⋊C4—1062no—
185C4⋊C4—1062no—
186C4⋊C4—2122no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C22⋊C4—1064no—
190C22⋊C4—2122no—
191C23—532no—
192C23—1062no—
193C23—1061no—
194C23—1061no—
195C23—1062no—
196C23C23×D5311yescharacteristic
197Q8—44no—
198Q8—2124no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—532no—
212C2×C4—1062no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1061no—
222C2×C4—1064no—
223C2×C4—2124no—
224C2×C4D4×D5312yes—
225C22—1062no—
226C22—1062no—
227C22—1062no—
228C22—2122no—
229C22—2122no—
230C22C2×D4×D5311yescharacteristic
231C22#1313326212yes—
232C22#1313326411yescharacteristic
233C22#1313326511yescharacteristic
234C22#1313326512yes—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4—1062no—
241C4—2122no—
242C4—2121no—
243C4—2121no—
244C4—2122no—
245C2—2122no—
246C2#1681118311yescharacteristic
247C2#1681119611yescharacteristic
248C2#1681119811yescharacteristic
249C2#1681137212yes—
250C2#1681137412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2817
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #16810564C2semidirect
4. #16810563C2semidirect
5. #16810562C2semidirect
8. #16810543C2semidirect
9. #16810542C2semidirect
11. C53×#395C2semidirect
135. #395D53semidirect
137. C53#2817semidirect