Finite Group Databasebeta

← #20280662

Subgroups — #20280662

Order 6784.

416 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133216C211yesmaximal, characteristic
2C2×#13133207C211yesmaximal, characteristic
3#16810565C212yesmaximal
4#16810564C212yesmaximal
5#16810563C211yesmaximal, characteristic
6#16810562C211yesmaximal, characteristic
7#16810561C211yesmaximal, characteristic
8#16810527C212yesmaximal
9#16810527C212yesmaximal
10#16810468C211yesmaximal, characteristic
11C53×#400C211yesmaximal, characteristic
12C22×Dic106C2212yes—
13#13133216—24no—
14#13133207—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×C4×Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×Dic106—24no—
49C2×Dic106—28no—
50C4⋊C4×C53—22no—
51C4⋊C4×C53—22no—
52C4⋊Dic53—22no—
53C4⋊Dic53—22no—
54#11719919—22no—
55#11719919—22no—
56#11719919—22no—
57#11719919—22no—
58#11719919—24no—
59C4×Dic53—24no—
60C4×Dic53D414yes—
61C23×C53C2411yescharacteristic
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—24no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53C2×D412yes—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—22no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—24no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88C2×Dic53#4114yes—
89Dic106—48no—
90C22×C53C22×D411yescharacteristic
91C22×C53C2×#4112yes—
92C22×C532−1+412yes—
93C22×C532−1+411yescharacteristic
94C22×C532−1+411yescharacteristic
95C4×C53—22no—
96C4×C53—42no—
97C4×C53—41no—
98C4×C53—41no—
99C4×C53—42no—
100Dic53—22no—
101Dic53—42no—
102Dic53—41no—
103Dic53—41no—
104Dic53—42no—
105Dic53#12414yes—
106#2642—531nomaximal, Sylow
107C2×C53C2×#12411yescharacteristic
108C2×C53#53611yescharacteristic
109C2×C53#53611yescharacteristic
110C2×C53#54112yes—
111C2×C53#54112yes—
112C2×C4⋊Q8—531no—
113C2×#125—531no—
114#400—531no—
115#400—531no—
116#400—531no—
117#400—532no—
118#400—532no—
119#400D5311yescharacteristic
120#391—532no—
121#391—532no—
122#376—531no—
123C53#264211yescharacteristic, complemented, Sylow
124C22×Q8—532no—
125C4⋊Q8—1064no—
126#125—1064no—
127C2×C4⋊C4—531no—
128C2×C4⋊C4—531no—
129C2×C4⋊C4—531no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4C2×D5311yescharacteristic
135C2×C4⋊C4C2×D5311yescharacteristic
136C2×C42—531no—
137C2×C42—532no—
138#95—532no—
139#95—531no—
140#95—532no—
141#95—532no—
142#95—532no—
143#95—532no—
144#95—532no—
145#95—532no—
146#95C2×D5311yescharacteristic
147#95C2×D5312yes—
148#95C2×D5312yes—
149C2×Q8—1064no—
150C2×Q8—1068no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—532no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—532no—
157C22×C4C22×D5312yes—
158C22×C4C22×D5311yescharacteristic
159C22×C4C22×D5311yescharacteristic
160C22×C4C22×D5312yes—
161C22×C4C22×D5311yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—1062no—
165C4⋊C4—1062no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—1064no—
169C4⋊C4—1062no—
170C4⋊C4—1062no—
171C42—534no—
172C42—1064no—
173C23C23×D5311yescharacteristic
174Q8—2128no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—532no—
187C2×C4—534no—
188C2×C4—532no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1064no—
196C2×C4—1062no—
197C2×C4—1061no—
198C2×C4—1062no—
199C2×C4—1061no—
200C2×C4—1062no—
201C2×C4D4×D5312yes—
202C22C2×D4×D5311yescharacteristic
203C22#41×D5312yes—
204C22#1313326412yes—
205C22#1313326411yescharacteristic
206C22#1313326511yescharacteristic
207C4—22no—
208C4—42no—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—534no—
213C4—1062no—
214C4—2122no—
215C4—2121no—
216C4—2121no—
217C4—2122no—
218C2#124×D5311yescharacteristic
219C2#1681119511yescharacteristic
220C2#1681119611yescharacteristic
221C2#1681123012yes—
222C2#1681123212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2642
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
123. C53#2642semidirect