Finite Group Databasebeta

← #20279167

Subgroups — #20279167

Order 6784.

484 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133193C211yesmaximal, characteristic
2C2×#13133175C211yesmaximal, characteristic
3#16810552C214yesmaximal
4#16810530C211yesmaximal, characteristic, complemented
5#16810505C214yesmaximal, complemented
6#16810492C211yesmaximal, characteristic, complemented
7#16810489C211yesmaximal, characteristic, complemented
8C4×#11719921C211yesmaximal, characteristic
9C53×#391C211yesmaximal, characteristic, complemented
10C22×C4×D53C2211yescharacteristic
11#13133193—24no—
12#13133175—24no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×C4⋊Dic53C2211yescharacteristic
15C2×#11719919C2212yes—
16C2×#11719919C2214yes—
17C2×#11719921C2212yes—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×C4×Dic53C2212yes—
21C2×C4×Dic53C2211yescharacteristic
22#13133124C2212yes—
23#13133124C2214yes—
24#13133124C2214yes—
25C22×Q8×C53C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C42×C53C2211yescharacteristic
28#95×C53C2214yes—
29C23×D53C2311yescharacteristic
30C2×Q8×C53—22no—
31C2×Q8×C53—22no—
32C2×Q8×C53—22no—
33C22×C4×C53C2312yes—
34C22×C4×C53C2312yes—
35C22×C4×C53C2312yes—
36C22×C4×C53C2311yescharacteristic
37C22×Dic53C2312yes—
38C22×Dic53C2312yes—
39C22×Dic53C2311yescharacteristic
40C22×Dic53C2312yes—
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C2×C4×D53D414yes—
44C4⋊C4×C53—22no—
45C42×C53—22no—
46#11719921—24no—
47#11719921—24no—
48#11719921—24no—
49C4⋊Dic53—22no—
50C4⋊Dic53—22no—
51#11719919—24no—
52#11719919—28no—
53C4×Dic53—22no—
54C4×Dic53—24no—
55C23×C53C2411yescharacteristic
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—22no—
59C22×D53—22no—
60C22×D53C2×D412yes—
61Q8×C53—44no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—24no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53C2×D412yes—
71C2×C4×C53#4112yes—
72C2×C4×C53#4112yes—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—24no—
80C2×Dic53—24no—
81C2×Dic53C2×D412yes—
82C2×Dic53#4114yes—
83C4×D53—28no—
84C22×C53C22×D411yescharacteristic
85C22×C53C2×#4112yes—
86C22×C53C2×#4111yescharacteristic
87C22×C53C2×#4111yescharacteristic
88C22×C532−1+412yes—
89C2×D53—22no—
90C2×D53—22no—
91C2×D53—22no—
92C2×D53—42no—
93C2×D53—42no—
94C4×C53—42no—
95C4×C53—42no—
96C4×C53—42no—
97C4×C53#12314yes—
98Dic53—22no—
99Dic53—24no—
100Dic53—42no—
101Dic53—42no—
102#2499—531nomaximal, Sylow
103C2×C53C2×#12311yescharacteristic
104C2×C53#53211yescharacteristic
105C2×C53#53611yescharacteristic
106C2×C53#55212yes—
107C2×C53#55612yes—
108D53—42no—
109C2×C22⋊Q8—531no—
110C2×#117—531no—
111#397—534no—
112#391—531no—
113#391D5311yescharacteristic, complemented
114#387—534no—
115#384—531no—
116#382—531no—
117C4×C22⋊C4—531no—
118C53#249911yescharacteristic, complemented, Sylow
119C22×Q8C2×D5311yescharacteristic
120C23×C4—531no—
121C22⋊Q8—1064no—
122#117—1064no—
123C2×C4⋊C4—531no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—534no—
127C2×C4⋊C4C2×D5311yescharacteristic
128C2×C22⋊C4—532no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4—532no—
131C2×C42—532no—
132C2×C42—531no—
133C2×C42C2×D5311yescharacteristic
134#95—532no—
135#95—534no—
136#95—534no—
137#95C2×D5314yes—
138C24—531no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4—532no—
143C22×C4—531no—
144C22×C4—534no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4—1062no—
148C22×C4—1062no—
149C22×C4C22×D5312yes—
150C22×C4C22×D5312yes—
151C22×C4C22×D5312yes—
152C22×C4C22×D5311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—1064no—
155C4⋊C4—1068no—
156C4⋊C4—1062no—
157C4⋊C4—1062no—
158C22⋊C4—1064no—
159C22⋊C4—1064no—
160C22⋊C4—1064no—
161C42—22no—
162C42—1064no—
163C42—1062no—
164C23—532no—
165C23—1062no—
166C23—1061no—
167C23—1061no—
168C23—1062no—
169C23C23×D5311yescharacteristic
170Q8—44no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—532no—
180C2×C4—534no—
181C2×C4—1061no—
182C2×C4—1062no—
183C2×C4—1061no—
184C2×C4—1062no—
185C2×C4—1068no—
186C2×C4—1062no—
187C2×C4—1062no—
188C2×C4—1064no—
189C2×C4—1064no—
190C2×C4#1171994512yes—
191C2×C4D4×D5312yes—
192C2×C4#1171994712yes—
193C22—1062no—
194C22—1062no—
195C22—1062no—
196C22—2122no—
197C22—2122no—
198C22C2×D4×D5311yescharacteristic
199C22#41×D5312yes—
200C22C2×#1171994711yescharacteristic
201C22C2×#1171994511yescharacteristic
202C22#1313326512yes—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—1064no—
207C4—1062no—
208C4—2122no—
209C4—2122no—
210C4#1313319914yes—
211C2—2122no—
212C2C2×#1313319911yescharacteristic
213C2#1681117211yescharacteristic
214C2#1681119811yescharacteristic
215C2#1681131412yes—
216C2#1681134412yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2499
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #16810530C2semidirect
5. #16810505C2semidirect
6. #16810492C2semidirect
7. #16810489C2semidirect
9. C53×#391C2semidirect
113. #391D53semidirect
118. C53#2499semidirect