Finite Group Databasebeta

← #20277473

Subgroups — #20277473

Order 6784.

484 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 217 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#16810516C212yesmaximal
3#16810516C212yesmaximal
4#16810509C212yesmaximal, complemented
5#16810509C212yesmaximal, complemented
6#16810486C211yesmaximal, characteristic
7C4⋊C4×Dic53C212yesmaximal, complemented
8C4×C22⋊Dic53C211yesmaximal, characteristic
9C2×C22⋊Q8×C53C211yesmaximal, characteristic
10C2×C4⋊Dic53C2212yes—
11C2×C4⋊Dic53C2212yes—
12C2×C22⋊Dic53C2212yes—
13C2×C22⋊Dic53C2211yescharacteristic
14C2×C22⋊Dic53C2211yescharacteristic
15C2×C4×Dic53C2212yes—
16C2×C4×Dic53C2211yescharacteristic
17C2×C4×Dic53C2212yes—
18C2×C4×Dic53C2211yescharacteristic
19#13133124C2212yes—
20#13133124C2212yes—
21#13133124C2212yes—
22#13133124C2212yes—
23#13133124C2212yes—
24#13133124C2212yes—
25#13133124C2212yes—
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—41no—
34C2×Q8×C53—41no—
35C2×Q8×C53C2×C414yes—
36C22×C4×C53—41no—
37C22×C4×C53—41no—
38C22×C4×C53C2×C414yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2312yes—
47C4⋊C4×C53C2×C414yes—
48C4⋊C4×C53C2×C418yes—
49C22⋊C4×C53C2×C418yes—
50C22⋊Dic53—24no—
51C22⋊Dic53—41no—
52C22⋊Dic53—41no—
53C4⋊Dic53—24no—
54C4⋊Dic53—42no—
55C4×Dic53—24no—
56C4×Dic53—41no—
57C4×Dic53—42no—
58C4×Dic53—41no—
59C23×C53—22no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—22no—
63C23×C53C22×C412yes—
64C23×C53C2411yescharacteristic
65Q8×C53—44no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—44no—
75C2×C4×C53C22×C412yes—
76C2×C4×C53C22×C412yes—
77C2×C4×C53C22×C414yes—
78C2×C4×C53C22×C414yes—
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—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53#4114yes—
91C22×C53—22no—
92C22×C53—22no—
93C22×C53—22no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53C23×C411yescharacteristic
97C22×C53C2×#4112yes—
98C22×C53H(5,𝔽2)11yescharacteristic
99C22×C53H(5,𝔽2)11yescharacteristic
100C22×C532−1+412yes—
101C4×C53—24no—
102C4×C53—22no—
103C4×C53—22no—
104C4×C53—24no—
105Dic53—24no—
106Dic53—42no—
107Dic53—42no—
108Dic53—42no—
109#2332—531nomaximal, Sylow
110C2×C53—42no—
111C2×C53C4×#4111yescharacteristic
112C2×C53#51811yescharacteristic
113C2×C53#51911yescharacteristic
114C2×C53#53912yes—
115C2×C53#54212yes—
116C2×C22⋊Q8D5311yescharacteristic
117#391—532no—
118#388—532no—
119#388—532no—
120#387—532no—
121#387—532no—
122#381—531no—
123C4×C4⋊C4—532no—
124C4×C22⋊C4—531no—
125C53#233211yescharacteristic, complemented, Sylow
126C22×Q8C2×D5311yescharacteristic
127C23×C4C2×D5311yescharacteristic
128C22⋊Q8Dic5318yescomplemented
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4C2×D5312yes—
132C2×C4⋊C4C2×D5311yescharacteristic
133C2×C22⋊C4—532no—
134C2×C22⋊C4—531no—
135C2×C22⋊C4—531no—
136C2×C22⋊C4C2×D5312yes—
137C2×C42—532no—
138C2×C42—531no—
139C2×C42—531no—
140C2×C42—532no—
141#95—532no—
142#95—532no—
143#95—532no—
144#95—532no—
145#95—532no—
146#95—532no—
147#95—532no—
148C24C22×D5311yescharacteristic
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8C2×Dic5314yes—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—532no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—532no—
158C22×C4C2×Dic5314yes—
159C22×C4C22×D5311yescharacteristic
160C22×C4C22×D5311yescharacteristic
161C22×C4C22×D5312yes—
162C22×C4C22×D5312yes—
163C4⋊C4—1064no—
164C4⋊C4—2122no—
165C4⋊C4C2×Dic5314yes—
166C4⋊C4C2×Dic5318yes—
167C22⋊C4—1064no—
168C22⋊C4—2121no—
169C22⋊C4—2121no—
170C22⋊C4C2×Dic5318yes—
171C42—1064no—
172C42—2121no—
173C42—2121no—
174C42—2122no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C22×Dic5312yes—
180C23C23×D5311yescharacteristic
181Q8—44no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—534no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4C22×Dic5314yes—
204C2×C4C22×Dic5314yes—
205C2×C4C22×Dic5312yes—
206C2×C4C22×Dic5312yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C23×Dic5311yescharacteristic
213C22#41×D5312yes—
214C22#1313326211yescharacteristic
215C22#1313326211yescharacteristic
216C22#1313326512yes—
217C4—24no—
218C4—22no—
219C4—22no—
220C4—24no—
221C4—1064no—
222C4—2122no—
223C4—2122no—
224C4—2122no—
225C2—42no—
226C2#41×Dic5311yescharacteristic
227C2#1681109811yescharacteristic
228C2#1681110111yescharacteristic
229C2#1681121912yes—
230C2#1681124312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC2×C22⋊Q8×C53C2
socleC23×C53C24
hypercenterC2×C22⋊Q8D53
hypocenterC53#2332
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
4. #16810509C2semidirect
5. #16810509C2semidirect
7. C4⋊C4×Dic53C2semidirect
28. C22⋊Q8×C53C4semidirect
125. C53#2332semidirect
128. C22⋊Q8Dic53semidirect