Finite Group Databasebeta

← #20277336

Subgroups — #20277336

Order 6784.

600 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 229 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×#13133175C211yesmaximal, characteristic
2C2×#13133174C211yesmaximal, characteristic
3C2×C4⋊C4×D53C211yesmaximal, characteristic
4#16810518C214yesmaximal, complemented
5#16810494C214yesmaximal
6#16810492C211yesmaximal, characteristic, complemented
7#16810489C211yesmaximal, characteristic, complemented
8#16810488C211yesmaximal, characteristic, complemented
9C53×#384C211yesmaximal, characteristic, complemented
10C22×C4×D53C2212yes—
11C22×C4×D53C2211yescharacteristic
12#13133175—24no—
13#13133174C418yescomplemented
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2214yes—
18C2×#11719919C2212yes—
19C4⋊C4×D53—24no—
20C2×#11719921C2212yes—
21C2×#11719921C2212yes—
22C2×C4×Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2214yes—
25#13133124C2214yes—
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2214yes—
29C2×C42×C53C2211yescharacteristic
30C23×D53C2311yescharacteristic
31C22×C4×C53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2312yes—
34C22×C4×C53C2311yescharacteristic
35C22×Dic53C2312yes—
36C22×Dic53C2312yes—
37C22×Dic53C2312yes—
38C22×Dic53C2311yescharacteristic
39C2×C4×D53—24no—
40C2×C4×D53—44no—
41C2×C4×D53C2×C414yes—
42C2×C4×D53D414yes—
43C2×C4×D53Q814yes—
44C4⋊C4×C53—22no—
45C4⋊C4×C53—22no—
46C4⋊C4×C53—44no—
47C42×C53C2×C414yescomplemented
48#11719921—24no—
49#11719921C2×C418yes—
50C4⋊Dic53—22no—
51C4⋊Dic53—22no—
52#11719919—24no—
53#11719919—44no—
54#11719919C2×C418yescomplemented
55C4×Dic53—24no—
56C4×Dic53C2×C414yescomplemented
57C23×C53C2411yescharacteristic
58C22×D53—22no—
59C22×D53—22no—
60C22×D53C22×C412yes—
61C22×D53C2×D412yes—
62C22×D53C2×Q812yes—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—24no—
68C2×C4×C53—24no—
69C2×C4×C53C22×C414yes—
70C2×C4×C53C22×C412yes—
71C2×C4×C53C2×D412yes—
72C2×C4×C53C2×Q812yes—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—24no—
78C2×Dic53—24no—
79C2×Dic53C22×C414yes—
80C2×Dic53C22×C412yes—
81C2×Dic53C2×D412yes—
82C2×Dic53C2×Q812yes—
83C4×D53—48no—
84C4×D53C4⋊C4116yescomplemented
85C22×C53C23×C411yescharacteristic
86C22×C53C22×D411yescharacteristic
87C22×C53C22×Q811yescharacteristic
88C22×C53H(5,𝔽2)12yes—
89C22×C532−1+412yes—
90C2×D53—24no—
91C2×D53—22no—
92C2×D53—22no—
93C2×D53—24no—
94C2×D53C2×C4⋊C414yes—
95C4×C53—24no—
96C4×C53—42no—
97C4×C53—42no—
98C4×C53C2×C4⋊C414yes—
99Dic53—24no—
100Dic53—42no—
101Dic53—42no—
102Dic53C2×C4⋊C414yes—
103#2317—531nomaximal, Sylow
104C2×C53C22×C4⋊C411yescharacteristic
105C2×C53#52012yes—
106C2×C53#53511yescharacteristic
107C2×C53#53611yescharacteristic
108C2×C53#54412yes—
109D53—24no—
110C2×#117—531no—
111C2×#117—531no—
112C22×C4⋊C4—531no—
113#389—534no—
114#385—534no—
115#384—531no—
116#384D5311yescharacteristic, complemented
117#382—531no—
118#382—531no—
119C53#231711yescharacteristic, complemented, Sylow
120C23×C4—531no—
121C23×C4—532no—
122#117—538no—
123#117—1064no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—532no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4—531no—
128C2×C4⋊C4—534no—
129C2×C4⋊C4—1064no—
130C2×C4⋊C4C2×D5311yescharacteristic
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C4⋊C4C2×D5314yes—
133C2×C22⋊C4—532no—
134C2×C22⋊C4—532no—
135C2×C42—531no—
136C2×C42—532no—
137C2×C42C2×D5311yescharacteristic
138#95—534no—
139#95—534no—
140C24—531no—
141C22×C4—531no—
142C22×C4—534no—
143C22×C4—534no—
144C22×C4—534no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4—532no—
148C22×C4—1064no—
149C22×C4—2124no—
150C22×C4C22×D5312yes—
151C22×C4C22×D5312yes—
152C22×C4C22×D5312yes—
153C22×C4C22×D5311yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—538no—
158C4⋊C4—1064no—
159C4⋊C4—1062no—
160C4⋊C4—1062no—
161C4⋊C4—2124no—
162C22⋊C4—538no—
163C22⋊C4—1064no—
164C42—534no—
165C42—1064no—
166C42C4×D5314yescomplemented
167C23—532no—
168C23—532no—
169C23—532no—
170C23—1062no—
171C23—1062no—
172C23C23×D5311yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—532no—
180C2×C4—532no—
181C2×C4—532no—
182C2×C4—5316no—
183C2×C4—534no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1064no—
189C2×C4—1064no—
190C2×C4—2128no—
191C2×C4C2×C4×D5314yes—
192C2×C4C2×C4×D5312yes—
193C2×C4D4×D5312yes—
194C2×C4Q8×D5312yes—
195C22—534no—
196C22—1064no—
197C22—1062no—
198C22—1062no—
199C22—1064no—
200C22C22×C4×D5311yescharacteristic
201C22C2×D4×D5311yescharacteristic
202C22C2×Q8×D5311yescharacteristic
203C22#1313326212yes—
204C22#1313326512yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—534no—
209C4—1064no—
210C4—2122no—
211C4—2122no—
212C4C4⋊C4×D5314yes—
213C2—1064no—
214C2C2×C4⋊C4×D5311yescharacteristic
215C2#1681110412yes—
216C2#1681118611yescharacteristic
217C2#1681119811yescharacteristic
218C2#1681125012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#384C2
socleC23×C53C24
hypercenter#384D53
hypocenterC53#2317
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
4. #16810518C2semidirect
6. #16810492C2semidirect
7. #16810489C2semidirect
8. #16810488C2semidirect
9. C53×#384C2semidirect
13. #13133174C4semidirect
47. C42×C53C2×C4semidirect
54. #11719919C2×C4semidirect
56. C4×Dic53C2×C4semidirect
84. C4×D53C4⋊C4semidirect
116. #384D53semidirect
119. C53#2317semidirect
166. C42C4×D53semidirect