Finite Group Databasebeta

← #20277883

Subgroups — #20277883

Order 6784.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13133196C211yesmaximal, characteristic
2#16810527C212yesmaximal, complemented
3#16810520C212yesmaximal, complemented
4#16810517C212yesmaximal
5#16810511C212yesmaximal
6#16810510C212yesmaximal, complemented
7#16810508C212yesmaximal, complemented
8#16810483C211yesmaximal, characteristic
9C53×#381C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11#13133196C418yescomplemented
12C2×C4⋊Dic53C2212yes—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2212yes—
20C2×C4×Dic53C2212yes—
21C2×C4×Dic53C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27C23×C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2212yes—
29#95×C53C2212yes—
30#95×C53C2211yescharacteristic
31#95×C53C2211yescharacteristic
32C24×C53C2311yescharacteristic
33C22×C4×C53—42no—
34C22×C4×C53C2×C414yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C2×Dic106—42no—
44C2×Dic106C2×C414yes—
45C22⋊C4×C53—42no—
46C22⋊Dic53—41no—
47C22⋊Dic53—41no—
48C22⋊Dic53C2×C418yes—
49C4⋊Dic53—42no—
50C4⋊Dic53C2×C414yes—
51#11719919—42no—
52#11719919—42no—
53#11719919C2×C418yes—
54C4×Dic53—42no—
55C4×Dic53—42no—
56C23×C53—22no—
57C23×C53—21no—
58C23×C53—22no—
59C23×C53—21no—
60C23×C53C22×C412yes—
61C23×C53C2411yescharacteristic
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—44no—
71C2×C4×C53C22×C412yes—
72C2×C4×C53C22×C412yes—
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×Dic53C22×C414yes—
84C2×Dic53C22×C414yes—
85Dic106—44no—
86C22×C53—22no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53C23×C411yescharacteristic
92C22×C53H(5,𝔽2)12yes—
93C22×C53H(5,𝔽2)11yescharacteristic
94C22×C532−1+412yes—
95C22×C532−1+411yescharacteristic
96C4×C53—22no—
97C4×C53—22no—
98C4×C53—42no—
99C4×C53—42no—
100Dic53—24no—
101Dic53—24no—
102Dic53—42no—
103Dic53—42no—
104#2378—531nomaximal, Sylow
105C2×C53—42no—
106C2×C53#51811yescharacteristic
107C2×C53#51911yescharacteristic
108C2×C53#52011yescharacteristic
109C2×C53#56212yes—
110C2×C53#56312yes—
111C2×C22⋊Q8—531no—
112#391—532no—
113#389—532no—
114#388—532no—
115#388—532no—
116#387—532no—
117#387—532no—
118#381D5311yescharacteristic
119#380—531no—
120C53#237811yescharacteristic, complemented, Sylow
121C22×Q8—531no—
122C23×C4C2×D5311yescharacteristic
123C22⋊Q8—538no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—532no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4—531no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4C2×D5312yes—
133C2×C42—532no—
134C2×C42—532no—
135#95—532no—
136#95—532no—
137#95—532no—
138#95—532no—
139#95—532no—
140#95C2×D5311yescharacteristic
141#95C2×D5312yes—
142#95C2×D5311yescharacteristic
143C24C22×D5311yescharacteristic
144C2×Q8—534no—
145C2×Q8—2122no—
146C22×C4—42no—
147C22×C4—532no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—532no—
151C22×C4C4×D5314yes—
152C22×C4C22×D5312yes—
153C22×C4C22×D5312yes—
154C22×C4C22×D5311yescharacteristic
155C22×C4C22×D5311yescharacteristic
156C4⋊C4—538no—
157C4⋊C4—534no—
158C4⋊C4—2122no—
159C4⋊C4—2122no—
160C4⋊C4—2122no—
161C22⋊C4—42no—
162C22⋊C4—538no—
163C22⋊C4—2121no—
164C22⋊C4—2121no—
165C42—2122no—
166C42—2122no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—21no—
171C23C2×C4×D5312yes—
172C23C23×D5311yescharacteristic
173Q8—2124no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—534no—
184C2×C4—534no—
185C2×C4—1062no—
186C2×C4—1062no—
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×C4C2×C4×D5312yes—
196C2×C4C2×C4×D5312yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C22×C4×D5311yescharacteristic
203C22#1313326212yes—
204C22#1313326311yescharacteristic
205C22#1313326412yes—
206C22#1313326411yescharacteristic
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—1064no—
212C4—1064no—
213C4—2122no—
214C4—2122no—
215C2—42no—
216C2#1681109611yescharacteristic
217C2#1681110011yescharacteristic
218C2#1681110311yescharacteristic
219C2#1681137012yes—
220C2#1681137612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#381C2
socleC23×C53C24
hypercenter#381D53
hypocenterC53#2378
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #16810527C2semidirect
3. #16810520C2semidirect
6. #16810510C2semidirect
7. #16810508C2semidirect
11. #13133196C4semidirect
120. C53#2378semidirect