Finite Group Databasebeta

← #20278007

Subgroups — #20278007

Order 6784.

504 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 163 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×#13133182C211yesmaximal, characteristic
2C2×#13133182C211yesmaximal, characteristic
3#16810557C212yesmaximal
4#16810554C211yesmaximal, characteristic
5#16810544C211yesmaximal, characteristic
6#16810543C211yesmaximal, characteristic
7#16810543C212yesmaximal
8C4⋊C4×Dic53C212yesmaximal
9#16810474C212yesmaximal
10#16810468C211yesmaximal, characteristic
11C53×#398C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14#13133182—24no—
15#13133182—24no—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C22×C4×C53C2311yescharacteristic
38C22×C4×C53C2312yes—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×Dic53C2311yescharacteristic
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2312yes—
47C22×Dic53C2311yescharacteristic
48C2×Dic106—22no—
49C2×Dic106—22no—
50C2×Dic106—22no—
51C2×Dic106—22no—
52C2×Dic106—24no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—24no—
56C4⋊Dic53—24no—
57#11719919—22no—
58#11719919—24no—
59#11719919—22no—
60#11719919—24no—
61#11719919—24no—
62C4×Dic53—22no—
63C4×Dic53—24no—
64C4×Dic53—22no—
65C4×Dic53D414yes—
66C4×Dic53Q818yescomplemented
67C23×C53C2411yescharacteristic
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×Q814yes—
80C2×C4×C53#4112yes—
81C2×Dic53—22no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—22no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53C2×D412yes—
92C2×Dic53C2×D412yes—
93C2×Dic53C2×Q814yes—
94C2×Dic53C2×Q814yes—
95C2×Dic53#4112yes—
96C2×Dic53#4114yes—
97Dic106—28no—
98Dic106—28no—
99C22×C53C22×D411yescharacteristic
100C22×C53C22×Q812yes—
101C22×C53C2×#4112yes—
102C22×C53C2×#4111yescharacteristic
103C22×C53C2×#4111yescharacteristic
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—24no—
107C4×C53—41no—
108C4×C53—42no—
109Dic53—22no—
110Dic53—24no—
111Dic53—22no—
112Dic53—24no—
113Dic53—41no—
114Dic53C4⋊Q814yes—
115#2399—531nomaximal, Sylow
116C2×C53#52511yescharacteristic
117C2×C53#52511yescharacteristic
118C2×C53C2×C4⋊Q811yescharacteristic
119C2×C53#53312yes—
120C2×C53#53312yes—
121C2×C4×Q8—531no—
122C2×C4×Q8—531no—
123#398—531no—
124#398—532no—
125#398D5311yescharacteristic
126#395—532no—
127#395—531no—
128#395—531no—
129C4×C4⋊C4—532no—
130C4×C4⋊C4—532no—
131#376—531no—
132C53#239911yescharacteristic, complemented, Sylow
133C22×Q8—531no—
134C22×Q8—531no—
135C4×Q8—1064no—
136C4×Q8—1064no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—532no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4C2×D5311yescharacteristic
144C2×C4⋊C4C2×D5311yescharacteristic
145C2×C4⋊C4C2×D5312yes—
146C2×C42—531no—
147C2×C42—532no—
148C2×C42—532no—
149C2×C42—531no—
150C2×C42—531no—
151#95—531no—
152#95—532no—
153#95—532no—
154#95—532no—
155#95—532no—
156#95C2×D5311yescharacteristic
157#95C2×D5312yes—
158C2×Q8—1062no—
159C2×Q8—1064no—
160C2×Q8—1062no—
161C2×Q8—1062no—
162C2×Q8—1062no—
163C22×C4—531no—
164C22×C4—532no—
165C22×C4—531no—
166C22×C4—532no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4C22×D5312yes—
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5311yescharacteristic
172C22×C4C22×D5311yescharacteristic
173C22×C4C22×D5312yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—1064no—
178C4⋊C4—1062no—
179C4⋊C4—1064no—
180C4⋊C4—1064no—
181C4⋊C4—1064no—
182C4⋊C4—1062no—
183C42—534no—
184C42—538no—
185C42—1062no—
186C42—1062no—
187C42—1064no—
188C23C23×D5311yescharacteristic
189Q8—1068no—
190Q8—1068no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—532no—
202C2×C4—534no—
203C2×C4—532no—
204C2×C4—534no—
205C2×C4—534no—
206C2×C4—532no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—1062no—
217C2×C4D4×D5312yes—
218C2×C4#1171994712yes—
219C2×C4Q8×D5314yes—
220C22C2×D4×D5311yescharacteristic
221C22C2×Q8×D5312yes—
222C22#41×D5312yes—
223C22#41×D5311yescharacteristic
224C22C2×#1171994711yescharacteristic
225C4—22no—
226C4—22no—
227C4—24no—
228C4—41no—
229C4—42no—
230C4—534no—
231C4—1064no—
232C4—1062no—
233C4—1062no—
234C4—1064no—
235C4—2121no—
236C2#1681113111yescharacteristic
237C2#1681113311yescharacteristic
238C2C4⋊Q8×D5311yescharacteristic
239C2#1681117512yes—
240C2#1681117912yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2399
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
66. C4×Dic53Q8semidirect
132. C53#2399semidirect