Finite Group Databasebeta

← #20280590

Subgroups — #20280590

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×#13133207C211yesmaximal, characteristic
2C2×Q8×Dic53C211yesmaximal, characteristic
3#16810565C212yesmaximal
4#16810563C211yesmaximal, characteristic
5#16810562C211yesmaximal, characteristic
6#16810556C212yesmaximal
7#16810542C211yesmaximal, characteristic
8#16810509C212yesmaximal
9#16810505C212yesmaximal
10#16810490C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic
12#13133207—24no—
13Q8×Dic53—24no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×C4×Dic53C2212yes—
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2211yescharacteristic
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C2×Q8×C53—22no—
38C2×Q8×C53—24no—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C4⋊C4×C53—22no—
51C4⋊C4×C53—24no—
52C4⋊Dic53—22no—
53C4⋊Dic53—24no—
54C4⋊Dic53—22no—
55#11719919—22no—
56#11719919—24no—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60C4×Dic53—24no—
61C4×Dic53D414yes—
62C23×C53C2411yescharacteristic
63Q8×C53—28no—
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—21no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53#4114yes—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—21no—
86C2×Dic53—24no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53C2×D412yes—
91C2×Dic53C2×D412yes—
92C22×C53C22×D411yescharacteristic
93C22×C53C2×#4112yes—
94C22×C532−1+411yescharacteristic
95C22×C532−1+411yescharacteristic
96C22×C532−1+412yes—
97C4×C53—22no—
98C4×C53—24no—
99C4×C53—41no—
100C4×C53—41no—
101C4×C53—42no—
102Dic53—22no—
103Dic53—22no—
104Dic53—41no—
105Dic53—41no—
106Dic53—42no—
107Dic53—42no—
108#2634—531nomaximal, Sylow
109C2×C53#52511yescharacteristic
110C2×C53#53611yescharacteristic
111C2×C53#53611yescharacteristic
112C2×C53#54112yes—
113C2×C53#54112yes—
114C2×#125—531no—
115C2×C4×Q8—531no—
116#400—531no—
117#400—531no—
118#400—532no—
119#398—532no—
120#395—531no—
121#395D5311yescharacteristic
122#387—532no—
123#387—532no—
124#383—531no—
125C53#263411yescharacteristic, complemented, Sylow
126C22×Q8C2×D5311yescharacteristic
127#125—1064no—
128C4×Q8—1064no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—531no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4—531no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C4⋊C4C2×D5312yes—
139C2×C42—531no—
140C2×C42—532no—
141#95—532no—
142#95—531no—
143#95—532no—
144#95—532no—
145#95—532no—
146#95—532no—
147#95—532no—
148#95—532no—
149#95C2×D5311yescharacteristic
150#95C2×D5312yes—
151C2×Q8—22no—
152C2×Q8—24no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—532no—
156C22×C4—532no—
157C22×C4—531no—
158C22×C4—531no—
159C22×C4C22×D5312yes—
160C22×C4C22×D5311yescharacteristic
161C22×C4C22×D5311yescharacteristic
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5312yes—
164C4⋊C4—22no—
165C4⋊C4—24no—
166C4⋊C4—1064no—
167C4⋊C4—1062no—
168C4⋊C4—1062no—
169C4⋊C4—1062no—
170C4⋊C4—1062no—
171C4⋊C4—1062no—
172C4⋊C4—1062no—
173C4⋊C4—1064no—
174C42—534no—
175C42—1064no—
176C23C23×D5311yescharacteristic
177Q8—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—532no—
190C2×C4—532no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1062no—
194C2×C4—1062no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1062no—
198C2×C4—1062no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1064no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4D4×D5312yes—
205C2×C4#1171994714yes—
206C22C2×D4×D5311yescharacteristic
207C22C2×#1171994712yes—
208C22#1313326411yescharacteristic
209C22#1313326511yescharacteristic
210C22#1313326512yes—
211C4—22no—
212C4—24no—
213C4—41no—
214C4—41no—
215C4—42no—
216C4—1062no—
217C4—1062no—
218C4—2121no—
219C4—2121no—
220C4—2122no—
221C4—2122no—
222C2#1681112911yescharacteristic
223C2#1681119611yescharacteristic
224C2#1681119811yescharacteristic
225C2#1681123312yes—
226C2#1681123412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2634
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
125. C53#2634semidirect