Finite Group Databasebeta

← #20278005

Subgroups — #20278005

Order 6784.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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×Q8×Dic53C211yesmaximal, characteristic
2C2×#13133182C211yesmaximal, characteristic
3#16810556C212yesmaximal
4#16810555C211yesmaximal, characteristic
5#16810553C211yesmaximal, characteristic
6#16810544C212yesmaximal
7#16810542C211yesmaximal, characteristic
8C4⋊C4×Dic53C212yesmaximal
9#16810474C212yesmaximal
10#16810468C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13Q8×Dic53—24no—
14#13133182—24no—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2212yes—
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2211yescharacteristic
24C2×C4×Dic53C2212yes—
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2212yes—
27#13133124C2212yes—
28#13133124C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C2×Q8×C53—22no—
38C2×Q8×C53—24no—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×Dic53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×Dic106—22no—
51C2×Dic106—22no—
52C2×Dic106—22no—
53C4⋊C4×C53—22no—
54C4⋊C4×C53—24no—
55C4⋊Dic53—22no—
56C4⋊Dic53—24no—
57C4⋊Dic53—24no—
58#11719919—22no—
59#11719919—24no—
60#11719919—24no—
61#11719919—22no—
62C4×Dic53—22no—
63C4×Dic53—22no—
64C4×Dic53—24no—
65C4×Dic53D414yes—
66C4×Dic53Q818yescomplemented
67C23×C53C2411yescharacteristic
68Q8×C53—28no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53C2×D412yes—
79C2×C4×C53C2×Q814yes—
80C2×C4×C53#4114yes—
81C2×C4×C53#4112yes—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—21no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53C2×D412yes—
94C2×Dic53C2×D412yes—
95C2×Dic53C2×Q814yes—
96C2×Dic53C2×Q814yes—
97C2×Dic53#4112yes—
98Dic106—28no—
99C22×C53C22×D411yescharacteristic
100C22×C53C22×Q812yes—
101C22×C53C2×#4111yescharacteristic
102C22×C53C2×#4112yes—
103C22×C53C2×#4111yescharacteristic
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—24no—
107C4×C53—24no—
108C4×C53—41no—
109Dic53—22no—
110Dic53—22no—
111Dic53—24no—
112Dic53—41no—
113Dic53—42no—
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#398—531no—
126#395—531no—
127#395—532no—
128#395D5311yescharacteristic
129C4×C4⋊C4—532no—
130C4×C4⋊C4—532no—
131#376—531no—
132C53#239911yescharacteristic, complemented, Sylow
133C22×Q8—531no—
134C22×Q8C2×D5311yescharacteristic
135C4×Q8—1064no—
136C4×Q8—1064no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4C2×D5311yescharacteristic
145C2×C4⋊C4C2×D5312yes—
146C2×C42—531no—
147C2×C42—532no—
148C2×C42—531no—
149C2×C42—532no—
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—22no—
159C2×Q8—24no—
160C2×Q8—1062no—
161C2×Q8—1062no—
162C2×Q8—1062no—
163C22×C4—531no—
164C22×C4—532no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—532no—
168C22×C4—531no—
169C22×C4C22×D5311yescharacteristic
170C22×C4C22×D5311yescharacteristic
171C22×C4C22×D5312yes—
172C22×C4C22×D5312yes—
173C22×C4C22×D5311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—1062no—
177C4⋊C4—1064no—
178C4⋊C4—1064no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—1064no—
182C4⋊C4—1064no—
183C42—534no—
184C42—538no—
185C42—1062no—
186C42—1064no—
187C42—1062no—
188C23C23×D5311yescharacteristic
189Q8—28no—
190Q8—1068no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—532no—
201C2×C4—534no—
202C2×C4—532no—
203C2×C4—534no—
204C2×C4—532no—
205C2×C4—1061no—
206C2×C4—1061no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4D4×D5312yes—
217C2×C4#1171994714yes—
218C2×C4Q8×D5314yes—
219C2×C4#1171994912yes—
220C22C2×D4×D5311yescharacteristic
221C22C2×Q8×D5312yes—
222C22#41×D5311yescharacteristic
223C22C2×#1171994712yes—
224C22C2×#1171994911yescharacteristic
225C4—22no—
226C4—24no—
227C4—22no—
228C4—24no—
229C4—41no—
230C4—534no—
231C4—1062no—
232C4—1062no—
233C4—1064no—
234C4—2121no—
235C4—2122no—
236C2#1681112911yescharacteristic
237C2#1681113611yescharacteristic
238C2C4⋊Q8×D5311yescharacteristic
239C2#1681117512yes—
240C2#1681117812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2399
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

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