Finite Group Databasebeta

← #20282839

Subgroups — #20282839

Order 6784.

508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#16810552C212yesmaximal, complemented
2#16810541C212yesmaximal, complemented
3#16810540C211yesmaximal, characteristic, complemented
4#16810540C212yesmaximal, complemented
5#16810533C212yesmaximal, complemented
6#16810533C212yesmaximal, complemented
7#16810524C211yesmaximal, characteristic, complemented
8#16810512C212yesmaximal, complemented
9C53×#400C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×D212C2212yes—
12C22×D212C2211yescharacteristic
13C22×D212C2212yes—
14C2×#11719919C2212yes—
15C2×#11719921C2212yes—
16C2×#11719921C2212yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28#95×C53C2212yes—
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31C23×D53C2312yes—
32C23×D53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53C2312yes—
39C22×Dic53C2312yes—
40C2×D212—22no—
41C2×D212—41no—
42C2×D212—41no—
43C2×D212—42no—
44C2×D212—42no—
45C2×D212—42no—
46C2×D212—42no—
47C2×D212—42no—
48C2×D212—42no—
49C2×D212—41no—
50C2×D212—42no—
51C4⋊C4×C53—41no—
52C4⋊C4×C53—41no—
53#11719921—24no—
54#11719921—42no—
55#11719921—42no—
56#11719921—42no—
57#11719921—42no—
58#11719921—42no—
59#11719921—42no—
60#11719921—42no—
61#11719919—42no—
62C4×Dic53—41no—
63C23×C53C2411yescharacteristic
64C22×D53—22no—
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53#4112yes—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97D212—44no—
98D212—44no—
99D212—48no—
100D212—48no—
101C22×C53C2×#4111yescharacteristic
102C22×C53H(5,𝔽2)11yescharacteristic
103C22×C53H(5,𝔽2)11yescharacteristic
104C22×C53H(5,𝔽2)12yes—
105C22×C53H(5,𝔽2)12yes—
106C2×D53—42no—
107C2×D53—42no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C4×C53—22no—
121C4×C53—41no—
122C4×C53—42no—
123C4×C53—41no—
124C4×C53—42no—
125Dic53—42no—
126Dic53—42no—
127#2828—531nomaximal, Sylow
128C2×C53#53811yescharacteristic
129C2×C53#54012yes—
130C2×C53#56011yescharacteristic
131C2×C53#56012yes—
132C2×C53H(3,𝔽4)11yescharacteristic
133D53—82no—
134D53—82no—
135#400D5311yescharacteristic, complemented
136#397—532no—
137#394—532no—
138#394—531no—
139#394—532no—
140#392—532no—
141#392—532no—
142#390—531no—
143#388—532no—
144C53#282811yescharacteristic, complemented, Sylow
145C22×D4—531no—
146C22×D4—532no—
147C22×D4—531no—
148C22×D4—532no—
149C2×C4⋊C4—532no—
150C2×C4⋊C4C2×D5311yescharacteristic
151C2×C4⋊C4C2×D5311yescharacteristic
152C2×C22⋊C4—532no—
153C2×C22⋊C4—532no—
154C2×C22⋊C4—532no—
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C22⋊C4—532no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C42—531no—
161#95—532no—
162#95—531no—
163#95C2×D5312yes—
164#95C2×D5312yes—
165#95C2×D5311yescharacteristic
166C24—532no—
167C24—532no—
168C2×D4—1062no—
169C2×D4—2121no—
170C2×D4—2121no—
171C2×D4—2121no—
172C2×D4—2122no—
173C2×D4—2122no—
174C2×D4—2122no—
175C2×D4—2122no—
176C2×D4—2122no—
177C2×D4—2122no—
178C2×D4—2122no—
179C22×C4—532no—
180C22×C4—532no—
181C22×C4C22×D5312yes—
182C22×C4C22×D5311yescharacteristic
183C22×C4C22×D5312yes—
184C22×C4C22×D5311yescharacteristic
185C22×C4C22×D5311yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—2122no—
189C22⋊C4—1064no—
190C22⋊C4—2122no—
191C22⋊C4—2122no—
192C22⋊C4—2122no—
193C22⋊C4—2122no—
194C22⋊C4—2122no—
195C22⋊C4—2122no—
196C22⋊C4—2122no—
197C42—2121no—
198C23—1062no—
199C23—1062no—
200C23—1062no—
201C23—1062no—
202C23—1062no—
203C23—1062no—
204C23—1062no—
205C23—1062no—
206C23—1062no—
207C23—1062no—
208C23—1062no—
209C23—1062no—
210C23—1062no—
211C23—1062no—
212C23C23×D5311yescharacteristic
213D4—2124no—
214D4—2124no—
215D4—2128no—
216D4—2128no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4#1171994912yes—
236C22—2122no—
237C22—2122no—
238C22—2122no—
239C22—2122no—
240C22—2122no—
241C22—2122no—
242C22—2122no—
243C22—2122no—
244C22—2122no—
245C22—2122no—
246C22—2122no—
247C22—2122no—
248C22—2122no—
249C22—2122no—
250C22C2×#1171994911yescharacteristic
251C22#1313326211yescharacteristic
252C22#1313326312yes—
253C22#1313326311yescharacteristic
254C22#1313326312yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—42no—
260C4—2122no—
261C4—2122no—
262C2—4242no—
263C2—4242no—
264C2#1681121311yescharacteristic
265C2#1681122912yes—
266C2#1681136311yescharacteristic
267C2#1681136412yes—
268C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2828
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #16810552C2semidirect
2. #16810541C2semidirect
3. #16810540C2semidirect
4. #16810540C2semidirect
5. #16810533C2semidirect
6. #16810533C2semidirect
7. #16810524C2semidirect
8. #16810512C2semidirect
9. C53×#400C2semidirect
135. #400D53semidirect
144. C53#2828semidirect