Finite Group Databasebeta

← #20285848

Subgroups — #20285848

Order 6784.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811178C211yesmaximal, characteristic, complemented
2#16811171C211yesmaximal, characteristic
3#16811026C211yesmaximal, characteristic, complemented
4#16811021C211yesmaximal, characteristic, complemented
5#16810969C211yesmaximal, characteristic, complemented
6#16810946C214yesmaximal, complemented
7#16810925C214yesmaximal, complemented
8#16810674C211yesmaximal, characteristic, complemented
9C53×#486C211yesmaximal, characteristic, complemented
10C2×#11719947—21no—
11#13133217C2211yescharacteristic
12#13133217C2211yescharacteristic
13#13133208C2212yes—
14#13133204—21no—
15#13133197—21no—
16#13133193—21no—
17#13133188—21no—
18#13133188C2212yes—
19Q8×Dic53—21no—
20#13133182—21no—
21D4×Dic53—21no—
22D4×Dic53C2212yes—
23#13133175C2212yes—
24C42×D53C2211yescharacteristic
25#13133150C2214yes—
26#13133149C2214yes—
27#13133146C2212yes—
28#13133138C2214yes—
29#13133129C2212yes—
30C4×#88322C2211yescharacteristic
31C4⋊Q8×C53C2211yescharacteristic
32#127×C53C2211yescharacteristic
33#102×C53C2214yes—
34C4×C8×C53C2211yescharacteristic
35C2×Q8×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53C2312yes—
39C2×C22⋊D53—22no—
40C22×Dic53—22no—
41#11719947—44no—
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C2×C4×D53C2311yescharacteristic
45C2×Dic106—21no—
46C2×Dic106—21no—
47C2×C8×C53C2312yes—
48C4⋊C4×C53—21no—
49C4⋊C4×C53C2312yes—
50C42×C53C2311yescharacteristic
51C22⋊Dic53—22no—
52C22⋊Dic53—22no—
53#11719921—22no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53—22no—
57C4⋊Dic53C2312yes—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60C4×Dic53—22no—
61C4×Dic53D411yescharacteristic
62C4×Dic53D411yescharacteristic
63C4×Dic53C2311yescharacteristic
64C2×#88322C2312yes—
65C23×C53—22no—
66C22×D53C2×D411yescharacteristic
67Q8×C53—42no—
68D4×C53—24no—
69D4×C53—42no—
70D4×C53—42no—
71C2×C4×C53—22no—
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2411yescharacteristic
75C22⋊D53—44no—
76C2×Dic53—22no—
77C2×Dic53—22no—
78C2×Dic53—44no—
79C2×Dic53C2×D411yescharacteristic
80C2×Dic53C2×D411yescharacteristic
81C2×Dic53C2×D411yescharacteristic
82C4×D53—22no—
83C4×D53—22no—
84C4×D53—22no—
85Dic106—42no—
86Dic106—42no—
87C8×C53—22no—
88#88322—22no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53C22×D411yescharacteristic
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C4×C53—42no—
97C4×C53#13512yes—
98C4×C53AGL(1,ℤ/8)12yes—
99C4×C532−1+412yes—
100Dic53—21no—
101Dic53—21no—
102Dic53—21no—
103Dic53—42no—
104Dic53—42no—
105#3228—531nomaximal, Sylow
106C2×C53—82no—
107C2×C53#53611yescharacteristic
108C2×C53C2×#13511yescharacteristic
109C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
110D53—41no—
111#533—531no—
112#532—531no—
113#500—531no—
114#498—531no—
115#486—531no—
116#486D5311yescharacteristic, complemented
117#482—534no—
118#478—534no—
119#431—531no—
120C53#322811yescharacteristic, complemented, Sylow
121C2×#41—1061no—
122C4⋊Q8—531no—
123C4⋊Q8—531no—
124C4⋊Q8C2×D5311yescharacteristic
125#127C2×D5311yescharacteristic
126#125—532no—
127#124—1061no—
128C22⋊Q8—1061no—
129C22⋊Q8—1061no—
130#121—532no—
131#121—1061no—
132C4×Q8—1061no—
133C4×Q8—1061no—
134C4×D4—532no—
135C4×D4—1061no—
136#117—532no—
137C2×C42—531no—
138#107—534no—
139#106—534no—
140C4⋊C8—532no—
141#102—534no—
142#102C2×D5314yes—
143C22⋊C8—532no—
144C4×C8—531no—
145C4×C8C2×D5311yescharacteristic
146#41—2124no—
147C2×Q8—21no—
148C2×Q8—1061no—
149C2×Q8—1061no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—1062no—
153C2×D4C22×D5312yes—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—1062no—
158C2×C8—532no—
159C2×C8C22×D5312yes—
160C4⋊C4—21no—
161C4⋊C4—532no—
162C4⋊C4—532no—
163C4⋊C4—1062no—
164C4⋊C4—1062no—
165C4⋊C4—1061no—
166C4⋊C4—1061no—
167C4⋊C4C22×D5312yes—
168C22⋊C4—1062no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C42—531no—
172C42—531no—
173C42—531no—
174C42—1062no—
175C42C22×D5311yescharacteristic
176C23—22no—
177C23—531no—
178Q8—42no—
179Q8—2122no—
180Q8—2122no—
181D4—24no—
182D4—42no—
183D4—42no—
184D4—2124no—
185C2×C4—22no—
186C2×C4—531no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—1062no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—2124no—
195C2×C4D4×D5311yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C2×C4C23×D5311yescharacteristic
198C8—22no—
199C8—1062no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—1061no—
204C22—1061no—
205C22—1061no—
206C22C2×D4×D5311yescharacteristic
207C4—42no—
208C4—1061no—
209C4—1061no—
210C4—1061no—
211C4—2122no—
212C4—2122no—
213C4#1313323812yes—
214C4#1313324112yes—
215C4#1313326412yes—
216C2—82no—
217C2—2121no—
218C2#1681119311yescharacteristic
219C2C2×#1313323811yescharacteristic
220C2C2×#1313324111yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#486C2
socleC22×C53C22×D4
hypercenter#486D53
hypocenterC53#3228
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811026C2semidirect
4. #16811021C2semidirect
5. #16810969C2semidirect
6. #16810946C2semidirect
7. #16810925C2semidirect
8. #16810674C2semidirect
9. C53×#486C2semidirect
116. #486D53semidirect
120. C53#3228semidirect