Finite Group Databasebeta

← #20282753

Subgroups — #20282753

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×#13133194C212yesmaximal
2#16810565C212yesmaximal, complemented
3#16810564C212yesmaximal, complemented
4#16810551C212yesmaximal
5#16810548C211yesmaximal, characteristic
6#16810544C212yesmaximal, complemented
7#16810543C212yesmaximal, complemented
8#16810484C211yesmaximal, characteristic
9C53×#397C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C23×Dic53C2211yescharacteristic
12#13133194—28no—
13C2×C4⋊Dic53C2212yes—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×C22⋊Dic53C2212yes—
19C2×C22⋊Dic53C2211yescharacteristic
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29C2×C22⋊C4×C53C2211yescharacteristic
30#95×C53C2212yes—
31#95×C53C2211yescharacteristic
32C24×C53C2311yescharacteristic
33C22×C4×C53C2311yescharacteristic
34C22×C4×C53C2312yes—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2312yes—
37C22×Dic53—41no—
38C22×Dic53—41no—
39C22×Dic53D414yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×Dic106—24no—
45C2×Dic106—42no—
46C2×Dic106—42no—
47C4⋊C4×C53—41no—
48C22⋊C4×C53—24no—
49C22⋊C4×C53—41no—
50C22⋊Dic53—24no—
51C22⋊Dic53—41no—
52C4⋊Dic53—24no—
53C4⋊Dic53—41no—
54#11719919—24no—
55#11719919—24no—
56#11719919—42no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—22no—
60C23×C53—22no—
61C23×C53C2×D412yes—
62C23×C53C2411yescharacteristic
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
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—44no—
85C2×Dic53C2×D414yes—
86Dic106—48no—
87C22×C53—22no—
88C22×C53—24no—
89C22×C53—41no—
90C22×C53—42no—
91C22×C53—41no—
92C22×C53C22×D411yescharacteristic
93C22×C53H(5,𝔽2)11yescharacteristic
94C22×C53H(5,𝔽2)11yescharacteristic
95C22×C532−1+412yes—
96C22×C532−1+412yes—
97C4×C53—42no—
98C4×C53—42no—
99C4×C53—41no—
100C4×C53—41no—
101Dic53—24no—
102Dic53—42no—
103Dic53—42no—
104Dic53—42no—
105#2817—531nomaximal, Sylow
106C2×C53—42no—
107C2×C53#53411yescharacteristic
108C2×C53#53612yes—
109C2×C53#56311yescharacteristic
110C2×C53#56312yes—
111C2×C53#56311yescharacteristic
112C2×C22⋊Q8—532no—
113#400—532no—
114#400—532no—
115#397—531no—
116#397—532no—
117#397D5311yescharacteristic
118#395—532no—
119#395—532no—
120#381—531no—
121C53#281711yescharacteristic, complemented, Sylow
122C22×Q8—532no—
123C23×C4—531no—
124C22⋊Q8—1068no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4C2×D5311yescharacteristic
131C2×C22⋊C4—532no—
132C2×C22⋊C4—531no—
133C2×C22⋊C4C2×D5312yes—
134C2×C22⋊C4C2×D5311yescharacteristic
135#95—532no—
136#95—532no—
137#95—532no—
138#95—532no—
139#95—532no—
140#95—532no—
141#95—531no—
142#95C2×D5312yes—
143#95C2×D5311yescharacteristic
144C24C22×D5311yescharacteristic
145C2×Q8—1064no—
146C2×Q8—2122no—
147C2×Q8—2122no—
148C22×C4—532no—
149C22×C4—534no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—2121no—
154C22×C4—2121no—
155C22×C4C22×D5311yescharacteristic
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5312yes—
158C22×C4C22×D5312yes—
159C4⋊C4—41no—
160C4⋊C4—1064no—
161C4⋊C4—1064no—
162C4⋊C4—1064no—
163C4⋊C4—2122no—
164C4⋊C4—2121no—
165C22⋊C4—24no—
166C22⋊C4—41no—
167C22⋊C4—1064no—
168C22⋊C4—2121no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23D4×D5312yes—
174C23C23×D5311yescharacteristic
175Q8—2128no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—534no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—2124no—
199C22—22no—
200C22—24no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22C2×D4×D5311yescharacteristic
205C22#1313326211yescharacteristic
206C22#1313326311yescharacteristic
207C22#1313326412yes—
208C22#1313326412yes—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—42no—
213C4—1064no—
214C4—2122no—
215C4—2122no—
216C4—2122no—
217C2—42no—
218C2#1681118111yescharacteristic
219C2#1681119512yes—
220C2#1681137311yescharacteristic
221C2#1681137512yes—
222C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2817
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #16810565C2semidirect
3. #16810564C2semidirect
6. #16810544C2semidirect
7. #16810543C2semidirect
121. C53#2817semidirect