Finite Group Databasebeta

← #20279197

Subgroups — #20279197

Order 6784.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133197C211yesmaximal, characteristic
2C2×C4⋊C4×D53C211yesmaximal, characteristic
3#16810565C214yesmaximal, complemented
4#16810529C211yesmaximal, characteristic, complemented
5#16810528C211yesmaximal, characteristic, complemented
6#16810488C211yesmaximal, characteristic, complemented
7#16810485C214yesmaximal
8C4×#11719921C211yesmaximal, characteristic
9C53×#382C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×C4×D53C2212yes—
12C22×C4×D53C2211yescharacteristic
13#13133197—24no—
14C2×C4⋊Dic53C2211yescharacteristic
15C2×C4⋊Dic53C2212yes—
16C2×#11719919C2212yes—
17C4⋊C4×D53—24no—
18C2×#11719921C2212yes—
19C2×#11719921C2212yes—
20C2×C4×Dic53C2211yescharacteristic
21#13133124C2214yes—
22#13133124C2214yes—
23#13133124C2212yes—
24#13133124C2214yes—
25C2×C4⋊C4×C53C2211yescharacteristic
26C2×C4⋊C4×C53C2211yescharacteristic
27C2×C42×C53C2211yescharacteristic
28#95×C53C2214yes—
29C23×D53C2311yescharacteristic
30C22×C4×C53C2312yes—
31C22×C4×C53C2312yes—
32C22×C4×C53C2312yes—
33C22×C4×C53C2311yescharacteristic
34C22×Dic53C2312yes—
35C22×Dic53C2312yes—
36C22×Dic53C2312yes—
37C22×Dic53C2311yescharacteristic
38C2×C4×D53—22no—
39C2×C4×D53—22no—
40C2×C4×D53—24no—
41C2×C4×D53—28no—
42C2×C4×D53D414yes—
43C2×Dic106—22no—
44C2×Dic106—22no—
45C2×Dic106—22no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53—22no—
48C42×C53—22no—
49#11719921—24no—
50#11719921—24no—
51C4⋊Dic53—22no—
52C4⋊Dic53—24no—
53#11719919—24no—
54C4×Dic53—22no—
55C23×C53C2411yescharacteristic
56C22×D53—21no—
57C22×D53—21no—
58C22×D53—22no—
59C22×D53C2×D412yes—
60C22×D53#4114yes—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—24no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—24no—
67C2×C4×C53—22no—
68C2×C4×C53C2×D412yes—
69C2×C4×C53#4112yes—
70C2×C4×C53#4112yes—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53—24no—
78C2×Dic53—22no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53C2×D412yes—
82C4×D53—28no—
83C4×D53—48no—
84Dic106—44no—
85C22×C53C22×D411yescharacteristic
86C22×C53C2×#4112yes—
87C22×C53C2×#4111yescharacteristic
88C22×C53C2×#4111yescharacteristic
89C22×C532−1+412yes—
90C2×D53—22no—
91C2×D53—22no—
92C2×D53—24no—
93C2×D53—24no—
94C2×D53#12414yes—
95C4×C53—42no—
96C4×C53—42no—
97C4×C53—42no—
98C4×C53#12314yes—
99Dic53—22no—
100Dic53—42no—
101Dic53—42no—
102Dic53—42no—
103#2503—531nomaximal, Sylow
104C2×C53C2×#12311yescharacteristic
105C2×C53C2×#12411yescharacteristic
106C2×C53#53611yescharacteristic
107C2×C53#55212yes—
108C2×C53#55212yes—
109D53—24no—
110C2×C22⋊Q8—531no—
111C22×C4⋊C4—531no—
112#400—534no—
113#391—531no—
114#391—531no—
115#382—531no—
116#382D5311yescharacteristic, complemented
117#381—534no—
118C4×C22⋊C4—531no—
119C53#250311yescharacteristic, complemented, Sylow
120C22×Q8—531no—
121C23×C4—531no—
122C23×C4—532no—
123C22⋊Q8—1064no—
124C2×C4⋊C4—532no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—532no—
127C2×C4⋊C4—1064no—
128C2×C4⋊C4C2×D5311yescharacteristic
129C2×C4⋊C4C2×D5311yescharacteristic
130C2×C22⋊C4—532no—
131C2×C22⋊C4—532no—
132C2×C42—531no—
133C2×C42C2×D5311yescharacteristic
134#95—532no—
135#95—534no—
136#95—534no—
137#95—534no—
138#95C2×D5314yes—
139C24—531no—
140C2×Q8—1062no—
141C2×Q8—1062no—
142C2×Q8—1062no—
143C22×C4—531no—
144C22×C4—534no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4—532no—
148C22×C4—1064no—
149C22×C4—1068no—
150C22×C4—1062no—
151C22×C4—1062no—
152C22×C4C22×D5312yes—
153C22×C4C22×D5312yes—
154C22×C4C22×D5312yes—
155C22×C4C22×D5311yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—1064no—
159C4⋊C4—1064no—
160C4⋊C4—1062no—
161C22⋊C4—1064no—
162C22⋊C4—1064no—
163C42—22no—
164C42—1062no—
165C23—532no—
166C23—534no—
167C23—1061no—
168C23—1061no—
169C23—1062no—
170C23C23×D5311yescharacteristic
171Q8—2124no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—532no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—1068no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1064no—
187C2×C4—1062no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—2128no—
192C2×C4#1171994512yes—
193C2×C4D4×D5312yes—
194C2×C4#1171994912yes—
195C22—534no—
196C22—1064no—
197C22—1064no—
198C22—1062no—
199C22—1062no—
200C22C2×D4×D5311yescharacteristic
201C22#41×D5312yes—
202C22C2×#1171994511yescharacteristic
203C22C2×#1171994911yescharacteristic
204C22#1313326412yes—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—1062no—
209C4—2122no—
210C4—2122no—
211C4—2122no—
212C4#1313320214yes—
213C2—1064no—
214C2C2×#1313320211yescharacteristic
215C2#124×D5311yescharacteristic
216C2#1681119311yescharacteristic
217C2#1681131112yes—
218C2#1681131212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#382C2
hypercenter#382D53
hypocenterC53#2503
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #16810565C2semidirect
4. #16810529C2semidirect
5. #16810528C2semidirect
6. #16810488C2semidirect
9. C53×#382C2semidirect
116. #382D53semidirect
119. C53#2503semidirect