Finite Group Databasebeta

← #20280076

Subgroups — #20280076

Order 6784.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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#16810563C211yesmaximal, characteristic
4#16810562C211yesmaximal, characteristic
5#16810542C211yesmaximal, characteristic
6#16810527C212yesmaximal
7#16810518C212yesmaximal
8#16810510C212yesmaximal
9#16810509C212yesmaximal
10#16810487C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13Q8×Dic53—24no—
14#13133182—24no—
15C2×C4⋊Dic53C2212yes—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
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
50C2×Dic106—22no—
51C2×Dic106—22no—
52C2×Dic106Q814yescomplemented
53C4⋊C4×C53—22no—
54C4⋊C4×C53—24no—
55C4⋊Dic53—22no—
56C4⋊Dic53—24no—
57#11719919—22no—
58#11719919—24no—
59#11719919—22no—
60#11719919—24no—
61C4×Dic53—22no—
62C4×Dic53—22no—
63C4×Dic53—24no—
64C4×Dic53D414yes—
65C23×C53C2411yescharacteristic
66Q8×C53—28no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2×Q812yes—
79C2×C4×C53#4114yes—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—22no—
83C2×Dic53—21no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—24no—
89C2×Dic53—22no—
90C2×Dic53C2×D412yes—
91C2×Dic53C2×D412yes—
92C2×Dic53C2×Q812yes—
93C2×Dic53C2×Q812yes—
94C2×Dic53#4112yes—
95Dic106—28no—
96C22×C53C22×D411yescharacteristic
97C22×C53C22×Q811yescharacteristic
98C22×C53C2×#4111yescharacteristic
99C22×C53C2×#4112yes—
100C22×C532−1+412yes—
101C4×C53—22no—
102C4×C53—22no—
103C4×C53—24no—
104C4×C53—41no—
105C4×C53—42no—
106Dic53—22no—
107Dic53—22no—
108Dic53—41no—
109Dic53—42no—
110Dic53—42no—
111Dic53C22⋊Q814yes—
112#2581—531nomaximal, Sylow
113C2×C53C2×C22⋊Q811yescharacteristic
114C2×C53#52511yescharacteristic
115C2×C53#53611yescharacteristic
116C2×C53#55612yes—
117C2×C53#55712yes—
118C2×C4×Q8—531no—
119C2×C4×Q8—531no—
120#400—531no—
121#400—531no—
122#395—531no—
123#395D5311yescharacteristic
124#391—532no—
125#389—532no—
126#387—532no—
127#387—532no—
128#382—531no—
129C53#258111yescharacteristic, complemented, Sylow
130C22×Q8—531no—
131C22×Q8C2×D5311yescharacteristic
132C4×Q8—1064no—
133C4×Q8—1064no—
134C2×C4⋊C4—531no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—532no—
140C2×C4⋊C4C2×D5311yescharacteristic
141C2×C4⋊C4C2×D5312yes—
142C2×C42—531no—
143C2×C42—531no—
144C2×C42—531no—
145C2×C42—532no—
146#95—532no—
147#95—531no—
148#95—532no—
149#95—532no—
150#95—532no—
151#95—532no—
152#95—532no—
153#95C2×D5311yescharacteristic
154#95C2×D5312yes—
155C2×Q8—22no—
156C2×Q8—24no—
157C2×Q8—534no—
158C2×Q8—1062no—
159C2×Q8—1062no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—531no—
166C22×C4C22×D5312yes—
167C22×C4C22×D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169C22×C4C22×D5312yes—
170C22×C4C22×D5311yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—1062no—
174C4⋊C4—1064no—
175C4⋊C4—1064no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1064no—
179C42—534no—
180C42—1062no—
181C42—1064no—
182C42—1062no—
183C23C23×D5311yescharacteristic
184Q8—28no—
185Q8—1068no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—532no—
197C2×C4—532no—
198C2×C4—532no—
199C2×C4—532no—
200C2×C4—532no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—1062no—
204C2×C4—1061no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1064no—
211C2×C4D4×D5312yes—
212C2×C4#1171994714yes—
213C2×C4Q8×D5312yes—
214C22C2×D4×D5311yescharacteristic
215C22C2×Q8×D5311yescharacteristic
216C22#41×D5311yescharacteristic
217C22C2×#1171994712yes—
218C22#1313326512yes—
219C4—22no—
220C4—22no—
221C4—24no—
222C4—42no—
223C4—41no—
224C4—534no—
225C4—1062no—
226C4—1062no—
227C4—2122no—
228C4—2121no—
229C4—2122no—
230C2C22⋊Q8×D5311yescharacteristic
231C2#1681112911yescharacteristic
232C2#1681119811yescharacteristic
233C2#1681134412yes—
234C2#1681134812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2581
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
52. C2×Dic106Q8semidirect
129. C53#2581semidirect