Finite Group Databasebeta

← #20279740

Subgroups — #20279740

Order 6784.

556 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133216C212yesmaximal
2C2×#13133215C212yesmaximal
3#16810544C212yesmaximal
4#16810543C211yesmaximal, characteristic
5#16810520C212yesmaximal
6#16810518C212yesmaximal
7#16810492C211yesmaximal, characteristic
8#16810474C212yesmaximal
9C53×#395C211yesmaximal, characteristic
10C22×Dic106C2211yescharacteristic
11C22×Dic106C2212yes—
12#13133216—28no—
13#13133215—28no—
14C2×C4⋊Dic53C2212yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×C4×Dic53C2212yes—
23C2×C4×Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2211yescharacteristic
26#13133124C2212yes—
27#13133124C2212yes—
28C22×Q8×C53C2211yescharacteristic
29C2×C4⋊C4×C53C2212yes—
30C2×C4⋊C4×C53C2211yescharacteristic
31#95×C53C2211yescharacteristic
32#95×C53C2212yes—
33C2×Q8×C53—24no—
34C2×Q8×C53—41no—
35C22×C4×C53C2311yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×Dic53C2312yes—
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C22×Dic53C2312yes—
44C2×Dic106—24no—
45C2×Dic106—24no—
46C2×Dic106—24no—
47C2×Dic106—41no—
48C2×Dic106—42no—
49C4⋊C4×C53—24no—
50C4⋊C4×C53—41no—
51C4⋊Dic53—24no—
52C4⋊Dic53—41no—
53#11719919—24no—
54#11719919—24no—
55#11719919—24no—
56#11719919—24no—
57#11719919D418yes—
58#11719919Q818yescomplemented
59C4×Dic53—24no—
60C4×Dic53—24no—
61C4×Dic53Q814yescomplemented
62C23×C53C2411yescharacteristic
63Q8×C53—44no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53C2×D414yes—
75C2×C4×C53C2×Q812yes—
76C2×C4×C53C2×Q814yes—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53C2×D414yes—
85C2×Dic53C2×D414yes—
86C2×Dic53C2×Q814yes—
87C2×Dic53C2×Q814yes—
88C2×Dic53C2×Q814yes—
89Dic106—44no—
90Dic106—48no—
91C22×C53C22×D412yes—
92C22×C53C22×Q811yescharacteristic
93C22×C53C22×Q812yes—
94C22×C53H(5,𝔽2)11yescharacteristic
95C22×C532−1+411yescharacteristic
96C4×C53—22no—
97C4×C53—24no—
98C4×C53—24no—
99C4×C53—41no—
100C4×C53—41no—
101Dic53—24no—
102Dic53—24no—
103Dic53—42no—
104Dic53C4⋊Q818yescomplemented
105#2551—531nomaximal, Sylow
106C2×C53C2×C4⋊Q812yes—
107C2×C53#54411yescharacteristic
108C2×C53D4211yescharacteristic
109C2×C53D4×Q812yes—
110C2×C53Q8211yescharacteristic
111C2×C4⋊Q8—532no—
112C2×C4⋊Q8—532no—
113#395—531no—
114#395—532no—
115#395D5311yescharacteristic
116#389—532no—
117#389—532no—
118#384—531no—
119C4×C4⋊C4—532no—
120C53#255111yescharacteristic, complemented, Sylow
121C22×Q8—531no—
122C22×Q8—532no—
123C22×Q8C2×D5311yescharacteristic
124C4⋊Q8—1068no—
125C4⋊Q8—1068no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—532no—
128C2×C4⋊C4—532no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4C2×D5312yes—
135C2×C4⋊C4C2×D5311yescharacteristic
136C2×C42—531no—
137C2×C42—532no—
138C2×C42—532no—
139#95—532no—
140#95—531no—
141#95—532no—
142#95C2×D5312yes—
143#95C2×D5311yescharacteristic
144C2×Q8—24no—
145C2×Q8—41no—
146C2×Q8—1064no—
147C2×Q8—1064no—
148C2×Q8—1064no—
149C2×Q8—2122no—
150C2×Q8—2121no—
151C22×C4—532no—
152C22×C4—532no—
153C22×C4—532no—
154C22×C4—532no—
155C22×C4C22×D5311yescharacteristic
156C22×C4C22×D5311yescharacteristic
157C22×C4C22×D5312yes—
158C22×C4C22×D5311yescharacteristic
159C22×C4C22×D5312yes—
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—538no—
163C4⋊C4—538no—
164C4⋊C4—1064no—
165C4⋊C4—1064no—
166C4⋊C4—1064no—
167C4⋊C4—1064no—
168C4⋊C4—1064no—
169C4⋊C4—2121no—
170C42—534no—
171C42—1064no—
172C42—1064no—
173C23C23×D5311yescharacteristic
174Q8—44no—
175Q8—2128no—
176Q8—2124no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—534no—
188C2×C4—534no—
189C2×C4—534no—
190C2×C4—534no—
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×C4D4×D5314yes—
200C2×C4Q8×D5312yes—
201C2×C4Q8×D5314yes—
202C22C2×D4×D5312yes—
203C22C2×Q8×D5312yes—
204C22C2×Q8×D5311yescharacteristic
205C22#1313326211yescharacteristic
206C22#1313326411yescharacteristic
207C4—24no—
208C4—22no—
209C4—24no—
210C4—41no—
211C4—41no—
212C4—538no—
213C4—1064no—
214C4—1064no—
215C4—2122no—
216C2C4⋊Q8×D5312yes—
217C2#1681125211yescharacteristic
218C2#1681125511yescharacteristic
219C2#1681128612yes—
220C2#1681135311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2551
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
58. #11719919Q8semidirect
61. C4×Dic53Q8semidirect
104. Dic53C4⋊Q8semidirect
120. C53#2551semidirect