Finite Group Databasebeta

← #20280596

Subgroups — #20280596

Order 6784.

416 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 131 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×#13133208C211yesmaximal, characteristic
2C2×#13133182C211yesmaximal, characteristic
3#16810564C212yesmaximal
4#16810563C211yesmaximal, characteristic
5#16810562C211yesmaximal, characteristic
6#16810554C211yesmaximal, characteristic
7#16810543C212yesmaximal
8#16810510C212yesmaximal
9#16810509C212yesmaximal
10#16810490C211yesmaximal, characteristic
11C53×#398C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13#13133208—24no—
14#13133182—24no—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2211yescharacteristic
24#13133124C2212yes—
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×Dic106—22no—
49C2×Dic106—22no—
50C2×Dic106—22no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—24no—
54C4⋊Dic53—22no—
55C4⋊Dic53—24no—
56#11719919—22no—
57#11719919—22no—
58#11719919—24no—
59#11719919—24no—
60C4×Dic53—22no—
61C4×Dic53—22no—
62C4×Dic53D414yes—
63C23×C53C2411yescharacteristic
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53#4112yes—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—24no—
88C2×Dic53—22no—
89C2×Dic53C2×D412yes—
90C2×Dic53C2×D412yes—
91C2×Dic53#4112yes—
92Dic106—28no—
93C22×C53C22×D411yescharacteristic
94C22×C53C2×#4111yescharacteristic
95C22×C53C2×#4111yescharacteristic
96C22×C532−1+412yes—
97C22×C532−1+412yes—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—42no—
101C4×C53—41no—
102C4×C53—42no—
103Dic53—22no—
104Dic53—22no—
105Dic53—22no—
106Dic53—41no—
107Dic53—42no—
108Dic53—42no—
109#2634—531nomaximal, Sylow
110C2×C53#52511yescharacteristic
111C2×C53#53611yescharacteristic
112C2×C53#53611yescharacteristic
113C2×C53#54112yes—
114C2×C53#54112yes—
115C2×#125—531no—
116C2×C4×Q8—531no—
117#400—531no—
118#400—531no—
119#400—532no—
120#398—531no—
121#398D5311yescharacteristic
122#395—532no—
123#387—532no—
124#387—532no—
125#383—531no—
126C53#263411yescharacteristic, complemented, Sylow
127C22×Q8—531no—
128#125—1064no—
129C4×Q8—1064no—
130C2×C4⋊C4—531no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—531no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4C2×D5311yescharacteristic
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C4⋊C4C2×D5312yes—
139C2×C42—531no—
140C2×C42—531no—
141C2×C42—531no—
142#95—532no—
143#95—532no—
144#95—531no—
145#95—532no—
146#95—532no—
147#95—532no—
148#95—532no—
149#95—532no—
150#95C2×D5311yescharacteristic
151#95C2×D5312yes—
152C2×Q8—1062no—
153C2×Q8—1062no—
154C2×Q8—1062no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—532no—
159C22×C4—531no—
160C22×C4—532no—
161C22×C4C22×D5312yes—
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5312yes—
164C22×C4C22×D5311yescharacteristic
165C22×C4C22×D5311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—24no—
169C4⋊C4—1062no—
170C4⋊C4—1064no—
171C4⋊C4—1064no—
172C4⋊C4—1064no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C42—534no—
176C42—1062no—
177C42—1062no—
178C23C23×D5311yescharacteristic
179Q8—1068no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—532no—
192C2×C4—532no—
193C2×C4—532no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1062no—
199C2×C4—1061no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1064no—
205C2×C4—1062no—
206C2×C4D4×D5312yes—
207C2×C4#1171994712yes—
208C22C2×D4×D5311yescharacteristic
209C22#41×D5311yescharacteristic
210C22C2×#1171994711yescharacteristic
211C22#1313326412yes—
212C22#1313326512yes—
213C4—22no—
214C4—22no—
215C4—42no—
216C4—41no—
217C4—42no—
218C4—1062no—
219C4—1062no—
220C4—1062no—
221C4—2121no—
222C4—2122no—
223C4—2122no—
224C2#1681113311yescharacteristic
225C2#1681119511yescharacteristic
226C2#1681119811yescharacteristic
227C2#1681123212yes—
228C2#1681123312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2634
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
126. C53#2634semidirect