Finite Group Databasebeta

← #20277881

Subgroups — #20277881

Order 6784.

464 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 213 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#16810530C212yesmaximal, complemented
2#16810522C212yesmaximal, complemented
3#16810516C212yesmaximal
4#16810516C212yesmaximal
5#16810509C212yesmaximal, complemented
6#16810509C212yesmaximal, complemented
7#16810486C211yesmaximal, characteristic
8#16810483C211yesmaximal, characteristic
9C2×C22⋊Q8×C53C211yesmaximal, characteristic
10C2×C4⋊Dic53C2212yes—
11C2×C4⋊Dic53C2212yes—
12C2×C4⋊Dic53C2212yes—
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2211yescharacteristic
15C2×C22⋊Dic53C2212yes—
16C2×C4×Dic53C2212yes—
17C2×C4×Dic53C2212yes—
18#13133124C2212yes—
19#13133124C2212yes—
20#13133124C2211yescharacteristic
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2211yescharacteristic
26C22×Q8×C53C2211yescharacteristic
27C23×C4×C53C2211yescharacteristic
28C22⋊Q8×C53C418yescomplemented
29C2×C4⋊C4×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2212yes—
32C24×C53C2311yescharacteristic
33C2×Q8×C53—42no—
34C2×Q8×C53C2×C414yes—
35C22×C4×C53—42no—
36C22×C4×C53C2×C414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C4⋊C4×C53C2×C414yes—
46C4⋊C4×C53C2×C418yes—
47C22⋊C4×C53C2×C418yes—
48C22⋊Dic53—41no—
49C22⋊Dic53—42no—
50C22⋊Dic53—41no—
51C4⋊Dic53—42no—
52C4⋊Dic53—42no—
53C4⋊Dic53—42no—
54C4×Dic53—42no—
55C4×Dic53—42no—
56C23×C53—22no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—22no—
60C23×C53C22×C412yes—
61C23×C53C2411yescharacteristic
62Q8×C53—44no—
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—44no—
70C2×C4×C53C22×C412yes—
71C2×C4×C53C22×C412yes—
72C2×C4×C53C22×C414yes—
73C2×C4×C53C22×C414yes—
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—22no—
85C2×Dic53—22no—
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+411yescharacteristic
95C22×C532−1+412yes—
96C4×C53—22no—
97C4×C53—22no—
98C4×C53—24no—
99C4×C53—24no—
100Dic53—42no—
101Dic53—42no—
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⋊Q8D5311yescharacteristic
112#391—532no—
113#389—532no—
114#388—532no—
115#388—532no—
116#387—532no—
117#387—532no—
118#381—531no—
119#380—531no—
120C53#237811yescharacteristic, complemented, Sylow
121C22×Q8C2×D5311yescharacteristic
122C23×C4C2×D5311yescharacteristic
123C22⋊Q8Dic5318yescomplemented
124C2×C4⋊C4—532no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4C2×D5311yescharacteristic
128C2×C4⋊C4C2×D5312yes—
129C2×C22⋊C4—531no—
130C2×C22⋊C4—531no—
131C2×C22⋊C4—532no—
132C2×C22⋊C4C2×D5312yes—
133C2×C42—532no—
134C2×C42—532no—
135#95—532no—
136#95—531no—
137#95—532no—
138#95—531no—
139#95—532no—
140#95—532no—
141#95—532no—
142#95—532no—
143C24C22×D5311yescharacteristic
144C2×Q8—42no—
145C2×Q8C2×Dic5314yes—
146C22×C4—42no—
147C22×C4—532no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—532no—
151C22×C4C2×Dic5314yes—
152C22×C4C22×D5311yescharacteristic
153C22×C4C22×D5312yes—
154C22×C4C22×D5312yes—
155C22×C4C22×D5311yescharacteristic
156C4⋊C4—2122no—
157C4⋊C4—2122no—
158C4⋊C4—2122no—
159C4⋊C4C2×Dic5314yes—
160C4⋊C4C2×Dic5318yes—
161C22⋊C4—2122no—
162C22⋊C4—2121no—
163C22⋊C4—2121no—
164C22⋊C4C2×Dic5318yes—
165C42—2122no—
166C42—2122no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23C22×Dic5312yes—
172C23C23×D5311yescharacteristic
173Q8—44no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—1062no—
182C2×C4—1062no—
183C2×C4—1062no—
184C2×C4—1062no—
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×C4C22×Dic5314yes—
194C2×C4C22×Dic5314yes—
195C2×C4C22×Dic5312yes—
196C2×C4C22×Dic5312yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C23×Dic5311yescharacteristic
203C22#1313326212yes—
204C22#1313326311yescharacteristic
205C22#1313326411yescharacteristic
206C22#1313326512yes—
207C4—22no—
208C4—22no—
209C4—24no—
210C4—24no—
211C4—2122no—
212C4—2122no—
213C4—2122no—
214C4—2122no—
215C2—42no—
216C2#1681109811yescharacteristic
217C2#1681110111yescharacteristic
218C2#1681110611yescharacteristic
219C2#1681136912yes—
220C2#1681137412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC2×C22⋊Q8×C53C2
socleC23×C53C24
hypercenterC2×C22⋊Q8D53
hypocenterC53#2378
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #16810530C2semidirect
2. #16810522C2semidirect
5. #16810509C2semidirect
6. #16810509C2semidirect
28. C22⋊Q8×C53C4semidirect
120. C53#2378semidirect
123. C22⋊Q8Dic53semidirect