Finite Group Databasebeta

← #20281024

Subgroups — #20281024

Order 6784.

544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133215C212yesmaximal, complemented
2C2×#13133194C212yesmaximal
3#16810559C211yesmaximal, characteristic
4#16810557C212yesmaximal, complemented
5#16810544C212yesmaximal, complemented
6#16810517C212yesmaximal
7#16810505C212yesmaximal, complemented
8#16810495C211yesmaximal, characteristic
9C53×#393C211yesmaximal, characteristic
10C22×Dic106C2212yes—
11C23×Dic53C2211yescharacteristic
12#13133215—28no—
13#13133194—28no—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×C22⋊Dic53C2212yes—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C4×Dic53C2212yes—
24#13133124C2211yescharacteristic
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28C22×Q8×C53C2211yescharacteristic
29C2×C22⋊C4×C53C2212yes—
30C2×C22⋊C4×C53C2211yescharacteristic
31#95×C53C2212yes—
32#95×C53C2211yescharacteristic
33C24×C53C2311yescharacteristic
34C2×Q8×C53—24no—
35C2×Q8×C53—41no—
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×Dic53—24no—
41C22×Dic53—41no—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C2×Dic106—24no—
47C2×Dic106—24no—
48C2×Dic106—42no—
49C22⋊C4×C53—24no—
50C22⋊C4×C53—41no—
51C22⋊Dic53—24no—
52C22⋊Dic53Q814yescomplemented
53C4⋊Dic53—24no—
54C4⋊Dic53—41no—
55#11719919—24no—
56#11719919—24no—
57#11719919—24no—
58#11719919—24no—
59#11719919D418yes—
60C4×Dic53—24no—
61C23×C53—22no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—22no—
65C23×C53C2×Q812yes—
66C23×C53C2411yescharacteristic
67Q8×C53—44no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53C2×D414yes—
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—44no—
88C2×Dic53C2×D414yes—
89C2×Dic53C2×D414yes—
90C2×Dic53C2×Q814yes—
91Dic106—48no—
92C22×C53—22no—
93C22×C53—24no—
94C22×C53—42no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53C22×D412yes—
98C22×C53C22×Q811yescharacteristic
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C532−1+411yescharacteristic
101C22×C532−1+412yes—
102C4×C53—24no—
103C4×C53—42no—
104C4×C53—41no—
105C4×C53—41no—
106Dic53—24no—
107Dic53—24no—
108Dic53—24no—
109Dic53—42no—
110#2677—531nomaximal, Sylow
111C2×C53—42no—
112C2×C53#53612yes—
113C2×C53#54411yescharacteristic
114C2×C53D4211yescharacteristic
115C2×C53D4×Q812yes—
116C2×C53#56311yescharacteristic
117C2×C4⋊Q8—532no—
118C2×C22⋊Q8—532no—
119#399—531no—
120#398—532no—
121#395—532no—
122#393D5311yescharacteristic
123#388—532no—
124#387—532no—
125#385—531no—
126C53#267711yescharacteristic, complemented, Sylow
127C22×Q8—532no—
128C22×Q8C2×D5311yescharacteristic
129C23×C4—531no—
130C4⋊Q8—1068no—
131C22⋊Q8—1068no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—531no—
139C2×C22⋊C4—531no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4C2×D5312yes—
142C2×C22⋊C4C2×D5311yescharacteristic
143C2×C42—532no—
144#95—532no—
145#95—531no—
146#95—532no—
147#95—532no—
148#95C2×D5312yes—
149#95C2×D5311yescharacteristic
150C24C22×D5311yescharacteristic
151C2×Q8—24no—
152C2×Q8—41no—
153C2×Q8—1064no—
154C2×Q8—1064no—
155C2×Q8—2122no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4—532no—
159C22×C4—532no—
160C22×C4—1064no—
161C22×C4—2121no—
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5312yes—
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5312yes—
166C4⋊C4—538no—
167C4⋊C4—1064no—
168C4⋊C4—1064no—
169C4⋊C4—1064no—
170C4⋊C4—1064no—
171C4⋊C4—1064no—
172C4⋊C4—2121no—
173C22⋊C4—24no—
174C22⋊C4—41no—
175C22⋊C4—534no—
176C22⋊C4—1064no—
177C42—1064no—
178C23—22no—
179C23—21no—
180C23—22no—
181C23—21no—
182C23Q8×D5312yes—
183C23C23×D5311yescharacteristic
184Q8—44no—
185Q8—2128no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—534no—
196C2×C4—534no—
197C2×C4—534no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—2124no—
208C2×C4D4×D5314yes—
209C22—24no—
210C22—22no—
211C22—41no—
212C22—42no—
213C22—41no—
214C22C2×D4×D5312yes—
215C22C2×Q8×D5311yescharacteristic
216C22#1313326211yescharacteristic
217C22#1313326412yes—
218C22#1313326411yescharacteristic
219C4—24no—
220C4—41no—
221C4—42no—
222C4—41no—
223C4—1064no—
224C4—1064no—
225C4—1064no—
226C4—2122no—
227C2—42no—
228C2#1681119312yes—
229C2#1681125211yescharacteristic
230C2#1681125511yescharacteristic
231C2#1681128612yes—
232C2#1681137311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2677
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
4. #16810557C2semidirect
5. #16810544C2semidirect
7. #16810505C2semidirect
52. C22⋊Dic53Q8semidirect
126. C53#2677semidirect