Finite Group Databasebeta

← #20282977

Subgroups — #20282977

Order 6784.

396 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 396 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
1#16810564C212yesmaximal, complemented
2#16810556C212yesmaximal, complemented
3#16810553C212yesmaximal, complemented
4#16810546C212yesmaximal
5#16810543C212yesmaximal, complemented
6#16810535C211yesmaximal, characteristic
7#16810484C211yesmaximal, characteristic
8#16810481C212yesmaximal
9C53×#399C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C23×Dic53C2211yescharacteristic
12C2×C4⋊Dic53C2212yes—
13C2×#11719919C2212yes—
14C2×#11719919C2212yes—
15C2×#11719919C2212yes—
16C2×#11719919C2212yes—
17C2×C22⋊Dic53C2212yes—
18C2×C22⋊Dic53C2211yescharacteristic
19#13133124C2211yescharacteristic
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
26C2×C4⋊C4×C53C2212yes—
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C4×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30#95×C53C2211yescharacteristic
31C24×C53C2311yescharacteristic
32C22×C4×C53C2311yescharacteristic
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×C4×C53C2312yes—
36C22×Dic53—41no—
37C22×Dic53—41no—
38C22×Dic53Q814yescomplemented
39C22×Dic53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C2×Dic106—41no—
44C2×Dic106—42no—
45C4⋊C4×C53—42no—
46C4⋊C4×C53—41no—
47C22⋊C4×C53—41no—
48C22⋊C4×C53—42no—
49C22⋊Dic53—42no—
50C22⋊Dic53—41no—
51C4⋊Dic53—42no—
52#11719919—42no—
53#11719919—42no—
54#11719919—42no—
55#11719919—42no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—22no—
59C23×C53—22no—
60C23×C53C2×Q812yes—
61C23×C53C2411yescharacteristic
62C2×C4×C53—21no—
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—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×Dic53—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—44no—
84C2×Dic53C2×Q814yes—
85Dic106—44no—
86C22×C53—22no—
87C22×C53—24no—
88C22×C53—41no—
89C22×C53—42no—
90C22×C53—41no—
91C22×C53C22×Q811yescharacteristic
92C22×C53H(5,𝔽2)11yescharacteristic
93C22×C53H(5,𝔽2)12yes—
94C22×C53H(5,𝔽2)11yescharacteristic
95C22×C532−1+412yes—
96C4×C53—41no—
97C4×C53—42no—
98C4×C53—42no—
99C4×C53—41no—
100Dic53—24no—
101Dic53—42no—
102Dic53—42no—
103Dic53—42no—
104#2841—531nomaximal, Sylow
105C2×C53—42no—
106C2×C53#54311yescharacteristic
107C2×C53#54412yes—
108C2×C53#56011yescharacteristic
109C2×C53#56212yes—
110C2×C53#56311yescharacteristic
111#400—532no—
112#399D5311yescharacteristic
113#398—532no—
114#398—532no—
115#396—532no—
116#395—532no—
117#393—531no—
118#381—531no—
119#380—532no—
120C53#284111yescharacteristic, complemented, Sylow
121C22×Q8—531no—
122C23×C4—531no—
123C2×C4⋊C4—532no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4C2×D5312yes—
129C2×C4⋊C4C2×D5311yescharacteristic
130C2×C22⋊C4—532no—
131C2×C22⋊C4—531no—
132C2×C22⋊C4C2×D5312yes—
133C2×C22⋊C4C2×D5311yescharacteristic
134#95—532no—
135#95—532no—
136#95—532no—
137#95—532no—
138#95—531no—
139#95—532no—
140#95—532no—
141#95C2×D5311yescharacteristic
142C24C22×D5311yescharacteristic
143C2×Q8—2121no—
144C2×Q8—2122no—
145C22×C4—532no—
146C22×C4—534no—
147C22×C4—532no—
148C22×C4—532no—
149C22×C4—532no—
150C22×C4—2121no—
151C22×C4—2121no—
152C22×C4C22×D5311yescharacteristic
153C22×C4C22×D5311yescharacteristic
154C22×C4C22×D5312yes—
155C22×C4C22×D5312yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—2122no—
159C4⋊C4—2122no—
160C4⋊C4—2122no—
161C4⋊C4—2122no—
162C4⋊C4—2122no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—2122no—
166C22⋊C4—2121no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D5312yes—
172C23C23×D5311yescharacteristic
173Q8—2124no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
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×C4—1062no—
196C2×C4—2124no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D5311yescharacteristic
203C22#1313326212yes—
204C22#1313326211yescharacteristic
205C22#1313326311yescharacteristic
206C22#1313326412yes—
207C4—42no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—1064no—
212C4—2122no—
213C4—2122no—
214C4—2122no—
215C2—42no—
216C2#1681124811yescharacteristic
217C2#1681125212yes—
218C2#1681136011yescharacteristic
219C2#1681137012yes—
220C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2841
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
2. #16810556C2semidirect
3. #16810553C2semidirect
5. #16810543C2semidirect
38. C22×Dic53Q8semidirect
120. C53#2841semidirect