Finite Group Databasebeta

← #20285636

Subgroups — #20285636

Order 6784.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#16811171C211yesmaximal, characteristic
2C4⋊Q8×D53C211yesmaximal, characteristic, complemented
3#16810995C211yesmaximal, characteristic, complemented
4#16810974C211yesmaximal, characteristic, complemented
5#16810973C211yesmaximal, characteristic, complemented
6#16810826C214yesmaximal, complemented
7#16810760C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#492C211yesmaximal, characteristic, complemented
10C2×#11719947—21no—
11C2×Q8×D53C2212yes—
12C2×#11719923C2214yes—
13#13133217C2211yescharacteristic
14#13133217C2211yescharacteristic
15#13133216—21no—
16#13133216—21no—
17#13133215C2212yes—
18#13133204—21no—
19#13133188—21no—
20#13133188C2212yes—
21D4×Dic53—21no—
22D4×Dic53C2212yes—
23C4⋊C4×D53—21no—
24C4⋊C4×D53—21no—
25C42×D53C2211yescharacteristic
26#13133146C2212yes—
27#13133142C2214yes—
28#13133141C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C2×SD16×C53C2214yes—
32C4⋊Q8×C53C2211yescharacteristic
33#127×C53C2211yescharacteristic
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53C2312yes—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53C2312yes—
39C2×C22⋊D53—22no—
40C22×Dic53—22no—
41Q8×D53—44no—
42#11719947—44no—
43C2×C4×D53—22no—
44C2×C4×D53D411yescharacteristic
45C2×C4×D53D411yescharacteristic
46C2×C4×D53C2311yescharacteristic
47C2×Dic106—21no—
48C2×Dic106—22no—
49C2×Dic106C2312yes—
50SD16×C53—44no—
51C2×C8×C53C2312yes—
52C4⋊C4×C53—21no—
53C4⋊C4×C53—21no—
54C42×C53C2311yescharacteristic
55C22⋊Dic53—22no—
56C22⋊Dic53—22no—
57#11719923—44no—
58C4⋊Dic53—21no—
59C4⋊Dic53—21no—
60C4⋊Dic53—21no—
61C4⋊Dic53C2312yes—
62#11719919—22no—
63#11719919—22no—
64C4×Dic53D411yescharacteristic
65C4×Dic53D411yescharacteristic
66C4×Dic53C2311yescharacteristic
67C2×#88322C2312yes—
68C23×C53—22no—
69C22×D53C2×D411yescharacteristic
70Q8×C53—24no—
71D4×C53—24no—
72D4×C53—42no—
73D4×C53—42no—
74C2×C4×C53—22no—
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2411yescharacteristic
78C22⋊D53—44no—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—44no—
82C2×Dic53C2×D411yescharacteristic
83C2×Dic53C2×D411yescharacteristic
84C2×Dic53C2×D411yescharacteristic
85C4×D53—22no—
86C4×D53—22no—
87C4×D53—44no—
88C4×D53SD1614yescomplemented
89Dic106—24no—
90Dic106—44no—
91Dic106—42no—
92C8×C53—22no—
93#88322—22no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—42no—
97C22×C53C22×D411yescharacteristic
98C2×D53—21no—
99C2×D53—21no—
100C2×D53C2×SD1612yes—
101C4×C53—42no—
102C4×C53C2×SD1612yes—
103C4×C53#13712yes—
104C4×C53H(5,𝔽2)12yes—
105Dic53—21no—
106Dic53—21no—
107Dic53—42no—
108Dic53—42no—
109Dic53C2×SD1612yes—
110#3206—531nomaximal, Sylow
111C2×C53—82no—
112C2×C53#53511yescharacteristic
113C2×C53C22×SD1611yescharacteristic
114C2×C53C2×#13711yescharacteristic
115D53—22no—
116#532—531no—
117C2×C4⋊Q8—531no—
118#492—531no—
119#492D5311yescharacteristic, complemented
120#487—531no—
121#487—531no—
122#461—534no—
123#448—534no—
124#431—531no—
125C53#320611yescharacteristic, complemented, Sylow
126C2×#41—1061no—
127C22×Q8—532no—
128C2×SD16—534no—
129C2×SD16C2×D5314yes—
130C4⋊Q8—532no—
131C4⋊Q8—531no—
132C4⋊Q8—531no—
133C4⋊Q8—1061no—
134C4⋊Q8—1061no—
135C4⋊Q8C2×D5311yescharacteristic
136#127C2×D5311yescharacteristic
137#124—1061no—
138#121—532no—
139#121—1061no—
140C4×D4—532no—
141C4×D4—1061no—
142C2×C4⋊C4—1061no—
143C2×C4⋊C4—1061no—
144C2×C42—531no—
145C4⋊C8—532no—
146#103—534no—
147#103—534no—
148C22⋊C8—532no—
149C4×C8—531no—
150C4×C8C2×D5311yescharacteristic
151#41—2124no—
152C2×Q8—532no—
153C2×Q8—1062no—
154C2×Q8—1061no—
155C2×Q8—2124no—
156C2×Q8C22×D5312yes—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—1062no—
160C2×D4C22×D5312yes—
161C22×C4—531no—
162C22×C4—531no—
163C22×C4—531no—
164C22×C4—1062no—
165C22×C4—1062no—
166SD16—44no—
167SD16—2124no—
168C2×C8—532no—
169C2×C8C22×D5312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—532no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4—1061no—
176C4⋊C4—1061no—
177C4⋊C4—1061no—
178C22⋊C4—1062no—
179C22⋊C4—1062no—
180C42—531no—
181C42—531no—
182C42—531no—
183C42C22×D5311yescharacteristic
184C23—22no—
185C23—531no—
186Q8—24no—
187Q8—1064no—
188Q8—2124no—
189Q8—2122no—
190D4—24no—
191D4—42no—
192D4—42no—
193D4—2124no—
194C2×C4—22no—
195C2×C4—531no—
196C2×C4—534no—
197C2×C4—531no—
198C2×C4—531no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—2124no—
204C2×C4—2124no—
205C2×C4D4×D5311yescharacteristic
206C2×C4D4×D5311yescharacteristic
207C2×C4C23×D5311yescharacteristic
208C8—22no—
209C8—1062no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—532no—
214C22—1061no—
215C22—1061no—
216C22C2×D4×D5311yescharacteristic
217C4—42no—
218C4—532no—
219C4—1061no—
220C4—1061no—
221C4—2122no—
222C4—2122no—
223C4SD16×D5312yes—
224C4#1313324612yes—
225C4#1313326212yes—
226C2—82no—
227C2—1062no—
228C2#1681118611yescharacteristic
229C2C2×SD16×D5311yescharacteristic
230C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#492C2
socleC22×C53C22×D4
hypercenter#492D53
hypocenterC53#3206
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D53C2semidirect
3. #16810995C2semidirect
4. #16810974C2semidirect
5. #16810973C2semidirect
6. #16810826C2semidirect
7. #16810760C2semidirect
8. #16810674C2semidirect
9. C53×#492C2semidirect
88. C4×D53SD16semidirect
119. #492D53semidirect
125. C53#3206semidirect