Finite Group Databasebeta

← #20279958

Subgroups — #20279958

Order 6784.

476 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 129 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#16810562C211yesmaximal, characteristic, complemented
2#16810535C212yesmaximal, complemented
3#16810528C211yesmaximal, characteristic, complemented
4#16810516C212yesmaximal, complemented
5#16810503C212yesmaximal
6#16810490C212yesmaximal, complemented
7#16810484C212yesmaximal, complemented
8C22⋊C4×Dic53C212yesmaximal, complemented
9C53×#394C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C23×Dic53C2212yes—
12C2×C4⋊Dic53C2211yescharacteristic
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2212yes—
15C2×C22⋊Dic53C2212yes—
16C2×C4×Dic53C2212yes—
17C2×C4×Dic53C2211yescharacteristic
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26C22×D4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2212yes—
30#95×C53C2211yescharacteristic
31C24×C53C2312yes—
32C2×D4×C53—24no—
33C2×D4×C53—41no—
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53—24no—
39C22×Dic53—24no—
40C22×Dic53—42no—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C2×Dic106—22no—
46C2×Dic106—41no—
47C2×Dic106—41no—
48C4⋊C4×C53—41no—
49C22⋊C4×C53—24no—
50C22⋊C4×C53—24no—
51C22⋊Dic53—24no—
52C22⋊Dic53—42no—
53C22⋊Dic53—42no—
54C4⋊Dic53—41no—
55C4×Dic53—24no—
56C4×Dic53—41no—
57C23×C53—22no—
58C23×C53—22no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53#4114yes—
64C23×C53C2411yescharacteristic
65D4×C53—44no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53#4112yes—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—48no—
88C2×Dic53#4114yes—
89Dic106—44no—
90C22×C53—24no—
91C22×C53—24no—
92C22×C53—24no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53C2×#4112yes—
98C22×C53C2×#4111yescharacteristic
99C22×C53C2×#4112yes—
100C22×C53H(5,𝔽2)11yescharacteristic
101C22×C53H(5,𝔽2)11yescharacteristic
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—24no—
107Dic53—42no—
108Dic53—42no—
109Dic53—42no—
110#2572—531nomaximal, Sylow
111C2×C53—44no—
112C2×C53#52912yes—
113C2×C53#53811yescharacteristic
114C2×C53#55112yes—
115C2×C53#55111yescharacteristic
116C2×C53#55511yescharacteristic
117#400—531no—
118#394D5311yescharacteristic
119#393—532no—
120#391—531no—
121#388—532no—
122#386—532no—
123#383—532no—
124#381—532no—
125C4×C22⋊C4—532no—
126C53#257211yescharacteristic, complemented, Sylow
127C22×Q8—531no—
128C22×D4C2×D5311yescharacteristic
129C23×C4—532no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C22⋊C4—532no—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—532no—
135C2×C22⋊C4C2×D5312yes—
136C2×C22⋊C4C2×D5312yes—
137C2×C42—532no—
138C2×C42—531no—
139#95—532no—
140#95—532no—
141#95—531no—
142#95—532no—
143#95—532no—
144#95—532no—
145#95—532no—
146#95—532no—
147#95C2×D5311yescharacteristic
148C24C22×D5312yes—
149C2×Q8—1062no—
150C2×Q8—2121no—
151C2×Q8—2121no—
152C2×D4—24no—
153C2×D4—41no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—1064no—
159C22×C4—1064no—
160C22×C4—2122no—
161C22×C4C22×D5311yescharacteristic
162C22×C4C22×D5312yes—
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5311yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—2121no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—1064no—
170C22⋊C4—2122no—
171C22⋊C4—2122no—
172C42—1064no—
173C42—2121no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23#1171994714yes—
181C23C23×D5311yescharacteristic
182Q8—2124no—
183D4—44no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—534no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—2128no—
206C2×C4#1171994712yes—
207C22—24no—
208C22—24no—
209C22—24no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22#41×D5312yes—
215C22C2×#1171994712yes—
216C22C2×#1171994711yescharacteristic
217C22#1313326211yescharacteristic
218C22#1313326311yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—1064no—
224C4—2122no—
225C4—2122no—
226C4—2122no—
227C2—44no—
228C2#1681115412yes—
229C2#1681121011yescharacteristic
230C2#1681129612yes—
231C2#1681130411yescharacteristic
232C2#1681133511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2572
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #16810562C2semidirect
2. #16810535C2semidirect
3. #16810528C2semidirect
4. #16810516C2semidirect
6. #16810490C2semidirect
7. #16810484C2semidirect
8. C22⋊C4×Dic53C2semidirect
126. C53#2572semidirect